GENOTYPE × AGE INTERACTION, AND THE INSULIN-LIKE GROWTH

FACTOR I AXIS IN THE SAN ANTONIO FAMILY HEART STUDY: A STUDY IN HUMAN SENESCENCE

BY

VINCENT PAUL DIEGO

BA, University of Guam, 1995

MA, Binghamton University (SUNY), 2001

DISSERTATION

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Anthropology

in the Graduate School Binghamton University

State University of New York 2005

ii

© 2005 by Vincent P. Diego. All rights reserved

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Accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Anthropology

in the Graduate School of Binghamton University

State University of New York 2005

April 15, 2005

Ralph M. Garruto, Department of Anthropology, Binghamton University

Jean W. MacCluer, Department of Genetics, Southwest Foundation for Biomedical Research

Michael A. Little, Department of Anthropology, Binghamton University

John Blangero, Department of Genetics,

Southwest Foundation for Biomedical Research

John Relethford, Department of Anthropology, Binghamton University (Adjunct), Department of Anthropology, SUNY at Oneonta

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ABSTRACT

This dissertation approaches human senescence from a statistical genetic

perspective and works with data provided by the San Antonio Family Heart Study

(SAFHS). It is shown how statistical genetics provides a logical foundation for

traditional approaches to studying human senescence. For analytic tractability, the

insulin-like growth factor I (IGF-I) axis is adopted as the main physiological system of

interest. In theory, however, the statistical genetic approach used in this research can be

applied to any physiological system. Working from within the statistical genetic

framework, the basic model therein is improved upon and extended to include genotype ×

age interaction. Genotype × age interaction was found to be important in the overall

behavior of the IGF-I axis in the SAFHS. The statistical genetic, biomedical and

evolutionary implications of this finding are explored. The theory of genotype × age

interaction is then extended to include mitochondrial effects, which are known to play

important roles in human senescence. Lastly, the findings of this dissertation research are

summarized.

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Dedicated to my parents, Frank Paulino Diego and Terrisita Leon Guerrero Taitague Diego, my siblings, Eileen Diego Meno, Michael Diego, Patrick Diego, Bernadette Diego

Lujan, and Frank P. Diego, Jr., and their families.

In His infinite wisdom and mercy, God knew I was weak, so He gave me a loving family.

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ACKNOWLEDGEMENTS

My doctoral journey starts with Professor Gary M. Heathcote at the University of

Guam. He introduced me to biological anthropology, nurtured my growth in said field,

and then treated me early on like a full-fledged colleague of his. If it were not for his

early, positive influence, I would not have pursued graduate studies in biological

anthropology. And it was he who introduced me to Professor Ralph M. Garruto at

Binghamton University (State University of New York), about whom I have more to say

below. I thank Gary also for being a close friend of mine over the years, a sounding

board whenever I needed one, an academic ally and promoter when I had none, a man in

my academic corner whenever I felt dejected and down-trodden, and someone competent

to share my academic ideas and dreams with, no matter how outlandish they might have

been. Drs. Jane Underwood at the University of Arizona at Tucson and Alexander Kerr

now at the University of Guam and my good friend Frank Camacho have similarly been

there for me along the way and I am thankful to them for their warm friendship.

So I came to Binghamton University (SUNY) to study with Dr. Ralph M. Garruto.

What can I say about the man? To begin with, he’s a great human being. That’s what

Dr. Jane Underwood said about him when I had inquired with her about his personality

when I was a prospective student. Fortunately, I came to the same view on my own.

Ralph always made sure that I pursued what I was interested in, not what he was

interested in. Early in my first year in 1999 in the fall, I told him that I was interested in

doing my dissertation research on the statistical genetics of the complex diseases

associated with aging. It just so happened that the annual meeting for the physical

anthropology and human biology societies were being held in San Antonio in the spring

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of 1999. Ralph took the time to meet with the folks at the Department of Genetics,

Southwest Foundation for Biomedical Research (SFBR) at San Antonio and inquire with

them about the possibility of training a student of his (me of course). I remember vividly

our meeting in his office upon his return from San Antonio. He told me that he had

spoken with some people at the SFBR and that they had expressed interest in taking on

one of his students. He mentioned two names in particular, Drs. Jean W. MacCluer and

John Blangero. I will talk more about the SFBR and these latter two individuals below,

but for now I continue on with Ralph and my time at Binghamton. So I progressed

through the master’s program in due time and graduated in 2001. Now, I am still

embarrassed to report a certain height of absent-mindedness of mine, but it is a necessary

part of my later story. I had completely forgotten to apply for financial support from the

Department of Anthropology in the spring of 2000 for the following academic year.

When this became known, Ralph was extremely upset and angry and I was feeling very

down-trodden. While the first half of that summer immediately following was extremely

trying, the second half held out hope. It just so happened that Jean was at this time

looking for a pre-doctoral-level research assistant and that this person would be trained in

statistical genetics, and have their pick of studies being carried out by the Department of

Genetics. So when Jean had approached Ralph and Professor Michael A. Little about

taking on a student of theirs, it came to pass that I took up the position and made my way

to San Antonio. They say you reap what you sow, and I guess we were reaping what

Ralph had sown in the spring of 1999. Before moving onto my San Antonio phase I

should say a little more about Professor Little. I greatly appreciate Mike’s sincere

interest in my academic development and his help in this regard. Mike served on my

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master’s committee and now also on my dissertation committee. I should also thank

Professor John Relethford, who also served on my dissertation committee. Dr.

Relethford is the only person I know who can lecture on Hardy-Weinberg Equilibrium

and have the class cracking up most of the way through. It is such an honor for me to be

able to say that these three distinguished biological anthropologists were on my

dissertation committee. But it gets better!

So I moved to San Antonio to learn statistical genetics under Drs. Jean W.

MacCluer and John Blangero at the SFBR. I’ve been there ever since and am just now

finished with my dissertation research, the spring of 2005. Jean is the nicest, sweetest

scientist I know. She has always made me feel at home in the Department of Genetics

and, perhaps more importantly, that my work was valuable and my thoughts were worth

discussing. It’s important to realize that Jean is a highly-respected human geneticist and

is the principal investigator (PI) and co-PI of several multi-million dollar research grants.

Yet, she is always humble and unassuming in her conversations and always willing to

listen to what you have to say. John Blangero taught me what I know in statistical

genetics, which is still a little to be sure but much more than what I had coming in. Dr.

Ravindranath Duggirala, who is a Scientist in the department, and myself call John

“Maha Guru”, which is Indian for “Great Teacher”. The dude is straight-up brilliant and

his knack for real-world problem-solving in statistical genetics never ceases to amaze me.

Also, if there be any complaints on the mathematical nature of my dissertation research,

the proper person to complain to is Dr. Blangero.

Now there are the friends and family to thank. I’ll take my friends first. I am

happy to thank my friends in Binghamton: David Hopwood, Nasser Malit, Bretton Giles,

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Marie Marley, Jen Bauder, Patrick Clarkin, Stephanie Rutledge, Bridgette Zavala, Felix

Acuto, Alex Novgloski, Julia McCausland Gaines, Ralph Quam (Dr. P!), Tom Beasom,

Tom Pearson, Laura Soloway, and Helene van Berge-Landry. You all made Binghamton

a bearable and warm place for a Pacific Islander, even during the harsh winter. A special

thanks goes out to David Hopwood for going the extra mile a number of times for me . . .

David is from Canada (not Canadia!). So for Dave: Go Canada!! I’m also happy to

thank my friends at San Antonio. At the SFBR, I would like to thank Nico Guoin,

Prakash Nair, and the Population Genetics Office people, Linda Freeman-Shade, Amuche

Ezeilo, Kent Polk, Debbie Newman (lifetime member) and Cheryl Reindl (honorary

member). There are many others but we have a big department. I would also like to

thank my two pool shooting friends who helped me to keep sane, Jonathan Camacho and

Art Williams. At my church, Freedom Baptist Church, I would like to thank Preacher

Lamb, and brothers V, Nacho, Thomas, Rob, Randy, Roy, Ben, Joseph, Sam, Rudy,

Eakin, Henry, Miguel, Reggie, Nate and others for their fellowship in the Lord.

I have to acknowledge the love and support that my family has given me

throughout the years. I would like to thank my mom and dad for being wonderful, loving

parents. I cannot thank them enough. They taught me the value of hard work and of

humility and it is these traits in particular that have brought me to this point. I would also

like to thank my brothers and sisters but especially my oldest sister Eileen, who has bore

the brunt of my vacation visits. It was really important for me to see family once in a

while as I was working on my dissertation. Lastly, I thank God, Jesus Christ, and the

Holy Spirit. I was lost in darkness, and Jesus brought me back to live in the light of His

righteousness. I offer my life’s work as my humble service and in honor of the Lord.

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Table of Contents

Title Page…………………………………………………………………………………..i Copyright Notice………………………………………………………………………….ii Signature Page……………………………………………………………………………iii Abstract…………………………………………………………………………………...iv Dedication………………………………………………………………………………....v Acknowledgements……………………………………………………………………….vi Table of Contents………………………………………………………………………….x List of Tables…………………………………………………………………………….xii List of Figures…………………………………………………………………………...xiii Chapter 1. Introduction…...………………………………………………………………1 Chapter 2. Background: Mathematical Biology of Senescence…..………………………6 Chapter 3. Background: Endocrinology of the IGF-I Axis in Relation to Senescence…32 Chapter 4. Background: The Study Population and Epidemiological Patterns…………44 Chapter 5. Methods I: Sampling Design, Pedigrees, and Phenotypes…………………..61 Chapter 6. Methods II: The Multivariate Mixed Effects Linear and Polygenic

Models……………………………………………………………………………72 Methods II: Theory and Model of Genotype × Environment Interaction……......77

Chapter 7. Methods III: Likelihood Theory and Maximum Likelihood

Estimation……...………………………………………………………………...93 Methods III: Hypotheses and Statistical Inference……………………………..101 Methods III: Power and Alternative Test Statistics………………………….…112 Chapter 8. Results: Statistical Behavior of the Phenotypes..…………………………..116 Results: Model Results………………………………………………………....116 Results: Power Analyses of the Genotype × Age Interaction Model…………..132 Chapter 9. Discussion: Statistical Genetic Finding……...……………………………..144 Discussion: Biomedical Ramifications…………………………………………153 Relation to Metabolism in Adulthood and the Metabolic

Syndrome……………………………………………………….154 Ontogeny, Aging, and Neuroendocrine Cascades……………...159

Discussion: Evolutionary Ramifications……………………………………….166 Chapter 10. Conclusions: Caveats….……...…………………………………………..172 Conclusions: Prospectus………………………………………………………..172 Conclusions: Conclusions………………………………………………………175

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Appendix A: A Geometric Proof of the G × E Interaction Theorem…………………...178 Appendix B: Relation to the Gaussian and Ornstein-Uhlenbeck Stochastic

Processes………………………………………………………………………..188 Appendix C: Derivation of the Elements in the Expected Fisher Information

Matrix……….…………………………………………………………………..196 Appendix D: Geometry of the Likelihood Function……………………………………237 References………………………………………………………………………………240

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List of Tables

•Table 1. Numbers of Relative Pairs in the SAFHS…………………………………….63 •Table 2. Descriptive Statistics of Raw Data……………………………………………67 •Table 3. Descriptive Statistics of Log-Transformed Data……………………………...67 •Table 4. Genome-wide expectations for alleles identical by descent (IBD)…………...75 •Table 5. Trait Heritabilities of the IGF-I Axis Components in the SAFHS…………..125 •Table 6. Models: Polygenic versus Genotype × Age Interaction……………………..125 •Table 7. Model Fitting for Log IGF-I under the Genotype × Age Interaction

Model…………………………………………………………………………...126 •Table 8. Model Fitting for Log IGFBP-3 under the Genotype × Age Interaction

Model…………………………………………………………………………...126 •Table 9. Model Fitting for Log Ratio3 under the Genotype × Age Interaction

Model…………………………………………………………………………...127 •Table 10. Power Analyses: Parameter Sets…………………………………………...135

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List of Figures

•Figure 1. Linear decline with age in physiological variables…………………………....8 •Figure 2. Brown and Forbes model of the mortality process with increasing age……..11 •Figure 3. Reliability Structures for (a) technical devices and (b) complex organisms...14 •Figure 4. First-order Taylor approximations…………………………………………...18 •Figure 5. Relation between fitness, measured by r, and the level of investement in somatic maintenance, s……..………………………………………………….30 •Figure 6. The main endocrine axes in aging and senescence…………………………..33 •Figure 7. IGF-I secretion pattern early in the human life span………………………...35 •Figure 8. Schematic description of the IGF-I axis and the two major sites of IGF-I secretion…………...……………………………..…………………………….36 •Figure 9. Endocrine, paracrine, and autocrine modes of IGF-I action…………………38 •Figure 10. The somatomedin hypotheses………………………………………………39 •Figure 11. Schematic of a gene expression network…………………………………...43 •Figure 12. Map of Bexar County in Texas……………………………………………..45 •Figure 13. Map of San Antonio in Bexar County……………………………………...46 •Figure 14. Rise of the Hispanic population in San Antonio, 1900-2000………………47 •Figure 15. Relative increase of the Hispanic population in Bexar County, Texas, 1990-2030…………….……………………………………………………..…48 •Figure 16. Relative standardized mortality ratios (RSMR) for total mentions of

diabetes mellitus: Spanish surname and non-Hispanic whites age 30 and over by sex………...…….………………………………………………...50

•Figure 17. Change in T2D incidence in San Antonio, Texas………………………………..51 •Figure 18. T2D and CVD mortality in San Antonio…………………………………………..53 •Figure 19. Heart disease and T2D mortality in Bexar County, 2002…………………..55 •Figure 20. Schematic diagram of the epidemiologic transition………………………...56 •Figure 21. SAFHS recruitment area……………………………………………………62 •Figure 22. Schematic pedigree structure for the typical extended family unit in the

SAFHS…………………………………………………………………………63 •Figure 23. Histograms of raw IGF-I and IGFBP-1 data………………………………..68 •Figure 24. Histograms of raw IGFBP-3 and Ratio3 data………………………………69 •Figure 25. Histograms of log transformed IGF-I and IGFBP-1 data…………………..70 •Figure 26. Histograms of log transformed IGFBP-3 and Ratio3 data………………….71 •Figure 27. G × E interaction variance under heteroscedasticity and homoscedasticity..84 •Figure 28. Graphical representation of exponential functions………………………..108 •Figure 29. One- and two-tailed tests on the assumption that maximum likelihood

estimates (MLEs) are normally distributed ( )1,0N 2 =σ=μ …………………111 •Figure 30. Age-specific means and variances in IGF-I levels (ng/ml)………………..117 •Figure 31. IGF-I versus age and BMI………………………………………………...118 •Figure 32. Age-specific means and variances in IGFBP-1 levels (ng/ml)……………119 •Figure 33. IGFBP-1 versus age and BMI……………………………………………..120 •Figure 34. Age-specific means and variances in IGFBP-3 levels (ng/ml)……………121 •Figure 35. IGFBP-3 versus age and BMI……………………………………………..122 •Figure 36. Age-specific means and variances in Ratio3……………………………...123 •Figure 37. Ratio3 versus age and BMI………………………………………………..124

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•Figure 38. Genotype × age interaction for log IGF-I in SAFHS Mexican Americans (variance functions)………………………………………………128

•Figure 39. Genotype × age interaction for log IGF-I in SAFHS Mexican Americans: The same genetic correlation function is displayed on 2 scales…129

•Figure 40. Questionable genotype × age interaction for log IGFBP-3 in SAFHS Mexican Americans: Phenotypic, additive genetic and environmental variance functions…………………………………………………………….130

•Figure 41. Apparent genotype × age interaction due to significant heteroscedasticity in the environmental variance function for log Ratio3 in the environmental variance in SAFHS Mexican Americans……………………………………..131

•Figure 42. Age-distribution used for power analyses…………………………………134 •Figure 43. Power analyses: additive genetic variance……………………………...…136 •Figure 44. Power analyses: genetic correlation……………………………………….137 •Figure 45. Power analyses: environmental variance………………………………….138 •Figure 46. Additive genetic variance in fetal ultrasound morphometrics in the

baboon, Papio hamadryas (spp.)……………………………………………..146 •Figure 47. Additive genetic variances in phenotypes associated with

atherosclerosis………………………………………………………………...147 •Figure 48. Genetic parameters for atherosclerosis risk factors in the Framingham

Heart Study…………………………………………………………………...148 •Figure 49. Power to detect G × E interaction by ANOVA……………………………150 •Figure 50. Power analysis of G × E interaction in samples of twin pairs…………….152 •Figure 51. Schematic diagram of changes in rank and scale along n segments of a

continuous environment. I.…………………………………………………..157 •Figure 52. Schematic diagram of changes in rank and scale along n segments of a

continuous environment. II…………………………………………………..163 •Figure 53. Additive genetic variance in ln mortality in Drosophila melanogaster…...170 •Figure A1. Schematic Representation of Vector Space in 2ℜ ………………………..179 •Figure A2. Geometry of Heteroscedasticity I: The Degenerate Triangle in Vector

Space………………………………………………………………………….184 •Figure A3. Geometry of Heteroscedasticity II: The Law of Pythagoras in Vector

Space………………………………………………………………………….184 •Figure A4. Geometry of Heteroscedasticity III: The Degenerate Triangle in Vector

Space………………………………………………………………………….185 •Figure A5. Geometry of Homoscedasticity I: The Zero Vector in Vector Space…….186 •Figure A6. Geometry of Homoscedasticity II: The Law of Pythagoras in Vector

Space………………………………………………………………………….187 •Figure A7. Geometry of Homoscedasticity III: The Degenerate Triangle in Vector

Space………………………………………………………………………….187 •Figure D1. Geometry of the Ln-Likelihood Function………………………………...238

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Chapter 1

Introduction

Historically, biological anthropology has maintained a deep and abiding interest

in the genetics of complex traits (MacCluer, 1993; Weiss, 1993, 1998a&b, 2000;

Williams-Blangero and Blangero, 1993; Blangero, 1993; Rogers et al., 1999). Yet only

as two decades ago have the analytical methods needed to study the genetics of complex

traits in anthropological settings come to fruition. This is not a criticism of the field, but,

rather, a reflection of the difficulties inherent in studying complex traits. Effecting a

breach of the seemingly insurmountable difficulties has required nothing short of

scientific revolutions in molecular and statistical genetics, and in mathematical and

computational statistics. Now that these experimental and analytical methods have been

developed, biological anthropology can examine anew its subordinate interests with

respect to the larger category of complex traits. In essence, the objective of the present

dissertation is to examine a traditional topic of interest, under the larger category of

complex traits, from the perspective of modern statistical genetics. To fully understand

the goals of this research, the developments just discussed need to be taken

contrapuntally with other, intimately-related developments within the field of biological

anthropology itself, which are specifically increasing interests in research on aging

(Crews, 1993, 1997; Crews and Garruto, 1994) and in biomedical problems (Garruto et

al., 1989, 1999; Little and Haas, 1989; Little and Garruto, 2000). For the purposes of this

dissertation, one can combine all of these developments into one theme, namely the

statistical genetics of human senescence.

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The approach of Blangero (1993), which is framed in terms of the statistical

genetic theory of genotype × environment (G × E) interaction, is used to address human

senescence. Because of the interest in senescence, this dissertation focuses on a specific

class of G × E interaction, namely genotype × age interaction, where the age continuum

has commonly been conceptualized as a special class of continuous environments

(Hegele, 1992; Zerba and Sing, 1992; Zerba et al., 1996, 2000; Jaquish et al., 1997;

Duggirala et al., 2000). For analytic tractability, the growth hormone/insulin-like growth

factor I (GH/IGF-I) axis is used as a microcosm of the complex physiology of

senescence. Thus, this dissertation is specifically on genotype × age interaction in the

GH/IGF-I axis in relation to the biology of senescence. It is common to focus on the

components of the GH/IGF-I axis involving just IGF-I and its binding proteins because it

is difficult to get a useful measure of GH without requiring overnight stays on the part of

study individuals (Neely and Rosenfeld, 1994). For this reason, the GH/IGF-I axis is

hereon referred to as the IGF-I axis. This dissertation is also a small part of a

comprehensive research project on the statistical genetics of cardiovascular disease

(CVD), namely the San Antonio Family Heart Study (SAFHS). CVD is considered to be

one of the major diseases of the metabolic syndrome (Reaven, 1988, 1993, 1995, 1999).

Given that the overall metabolic dysfunction encompassed by the metabolic syndrome is

known to be strongly age-related (Liese et al., 1998), it is perhaps safely assumed that the

metabolic syndrome is one of the more complex manifestations of senescence. In other

words, the metabolic syndrome is studied here from the perspective of the biology of

senescence.

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Statement of the Problem

This dissertation research will characterize the statistical genetics of the IGF-I

axis in relation to age in the development of the metabolic syndrome in Mexican

Americans of San Antonio participating in the SAFHS.

Specific Aims and Hypotheses

The statistical genetic characteristics of the IGF-I axis along the age continuum

are analyzed under a sequence of models. Firstly, the simplest statistical genetic model,

known as the polygenic model, is used to establish whether or not genetic factors are

important in the phenotypic determination of the components of the IGF-I axis. If the

heritability—which is taken as an indicator of genetic influence—of a given component

is found to be significant, then that component will be analyzed further using the

genotype × age interaction model. The null hypothesis under the theory of genotype ×

age interaction is that the gene expression network (GEN) underlying the IGF-I axis is

insensitive to changes in age. That is, changes in age have no effect on the GEN of the

IGF-I axis. Supposing that significant genotype × age interaction effects are found, it

will be interesting to establish whether these arise from either of two sources of genotype

× age interaction, which are variance heterogeneity and a genetic correlation coefficient

significantly different from 1, or from the two sources acting jointly. The chapters to

follow will develop the background knowledge and statistical genetic theory needed to

better understand these aims.

The aims, hypotheses, and statistical analyses used in this study are as follows:

Specific Aim 1: It became apparent early on to the author, in carrying out this dissertation research, that the relation between statistical genetics on the one hand and more established approaches towards studying senescence on the other was not at all clear. Therefore, the first aim of this dissertation is to show how the statistical genetic approach

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is related to other approaches to studying senescence. This aim can be addressed by showing how the major theories of senescence can be unified and then by showing how statistical genetics provides both a foundation and extension of this unified structure. Specific Aim 2: To determine if the components of the IGF-I axis are significantly influenced by genetic factors. •Hypothesis 1: The heritabilities of the components of the IGF-I axis are significant. Specific Aim 3: To determine if the components of the IGF-I axis are influenced by age effects. •Hypothesis 2: The components of the IGF-I axis are more consistent with the genotype × age interaction model than with the simple, polygenic model. Specific Aim 4: To describe in terms of statistical genetic parameters how the behavior of the IGF-I axis is sensitive to the age continuum. This requires having found significant heritability and then significant genotype × age interaction effects. •Hypothesis 3: The additive genetic variance significantly changes with age.

•Hypothesis 4: The genetic correlation coefficient is significantly different from 1.

Outline of the Dissertation

It is perhaps worthwhile to discuss the organizational structure of this dissertation.

Important background concepts are introduced and developed in the first three ensuing

chapters. The first background chapter covers the mathematical biology of senescence,

which includes proximate and ultimate mathematical models of senescence. It will be

pointed out in this chapter how major theories of senescence can be unified and how

statistical genetics provides a foundation for the unified structure. The second

background chapter delves into the physiology of senescence with a focus on the role

played by the IGF-I axis. The third background chapter discusses the basic population

biology for this study, including the study population and important epidemiological

concepts. The ensuing chapters generally follow the traditional organization of methods,

results, discussion, and conclusions, with a prospectus section being included in the

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conclusions chapter. The methods are discussed in a sequence of three chapters. The

first of the methods chapters treats the more empirical aspects of the present work, which

includes the sampling design, pedigree and relationship structures, demographics, and the

phenotypes. The next two methods chapters follow the logical structure of statistical

inference in that the first of these develops the statistical genetic models employed in this

research and the next discusses the elegant machinery of likelihood-based statistical

inference, which includes maximum likelihood estimation, hypothesis testing by recourse

to the likelihood ratio test statistic, and statistical power calculations. As regards the

chapter on statistical genetic models, it will be shown in that chapter how the basic model

can be improved by allowing for G × E interaction in general and genotype × age

interaction in particular. The next two chapters focus on the results and discussion. In

the prospectus section of the conclusions chapter, an extension of the genotype × age

interaction model in relation to mitochondrial theories of senescence is developed. The

conclusions of this dissertation research are then summarized in the section just

following. It should be pointed out that there are four appendices that follow the main

body of the text. These appendices at once enable a more coherent and flowing structure

in the main body of the text and a forum for the discussion of concepts and the derivation

of equations that are not immediately necessary for understanding the dissertation

research. The appendices are introduced in the development of the main body of the text.

6

Chapter 2

Background: Mathematical Biology of Senescence

Senescence can be approached from diverse perspectives. Indeed, according to

Medvedev (1990), there are more than 300 theories of aging and senescence. It will be

argued in this chapter that these diverse perspectives can be unified and that statistical

genetics offers a strong foundation for this unified structure. There are two categories of

theories of senescence, which may be called proximate and ultimate explanations of

senescence, after Mayr’s (1961) dichotomy of the explanation of biological phenomena

(on a similar approach to senescence, cf. Finch and Rose, 1995; Masoro, 1999: ch. 5).

One way to unify these two categories is to show explicitly how the proximate-level

models can be used to build up, as it were, to the ultimate-level models. To unify the

proximate and ultimate categories along these lines, two ideas are needed, which are

Cannon’s (1929, 1939a) concept of physiological homeostasis and Simms’s (1942a)

observation that the linear decline in homeostasis with increasing age can be related to

the mortality risk observed in animal populations. Although the concept of homeostasis

is original with Cannon (1929), a preferable definition is given by Shock (1977) as the

systemic regulation of physiological functions such that organism-level integration is

achieved in the face of a dynamic environment. Senescence is defined as the

physiological deterioration associated with aging (Finch, 1990), which is brought about

by the age-related decline in homeostasis. Canon (1939a&b, 1942) also originally

proposed the view that senescence is characterized by an age-associated deterioration in

the ability to maintain homeostasis against continual perturbations, extrinsic or intrinsic

to the organism. Cannon’s view became a principle that was widely invoked in the fields

7

of physiology (Simms, 1940, 1942a&b, 1946; Shock, 1952, 1961, 1969, 1974, 1977;

Comfort, 1956, 1968; Kenney, 1982), and clinical science (Selye, 1946, 1950, 1951,

1955, 1956, 1970a&b, 1976; Kohn, 1963, 1978, 1982, 1985; Selye and Prioreschi, 1972).

Indeed, Dilman (1981) proclaimed the above principle to be a biological law, “the law of

deviation of homeostasis”. This principle is of importance because it suggests that aging

individuals are increasingly predisposed to succumbing to perturbations in homeostasis

(Strehler, 1977; Kohn, 1978). In fact, the interaction of stress and homeostasis in relation

to disease and aging formed a central component of Selye’s theory of the “general

adaptation syndrome” (Selye, 1946, 1950, 1951, 1955, 1956, 1970a&b, 1976; Selye and

Prioreschi, 1972; for recent reformulations, see Frolkis, 1993; McEwen and Stellar, 1993;

McEwen, 1998). Simms (1942a) made another advance when he suggested that the

observation of a gradual or linear decline in homeostasis (Canon, 1939a; Simms, 1940,

1942a, 1946; Shock, 1955, 1961, 1977, 1985; Kohn, 1963, 1978, 1985; Fig. 1) can be

logically related to the exponential mortality rate in animal populations. Simms’s

(1942a) observation encouraged a number of theories of senescence relating the

physiological characteristics of populations to their mortality rate, which is taken to be a

proxy of the senescence rate (for historical reviews, see Strehler, 1959, 1977; Mildvan

and Strehler, 1960; Kohn, 1978; Economos, 1982). As will be seen, linear decline in

homeostasis or, more usually, an inversely proportional linear increase in physiological

damage, thought to accrue under declining homeostasis, is often the critical assumption in

proximate-level models that predict a fairly universal mortality pattern.

This universal mortality pattern is known as the Gompertz and Gompertz-

Makeham mortality functions (Gompertz, 1825; Olshansky and Carnes, 1997), which are

8

Figure 1. Linear decline with age in physiological variables. All values were standardized against the value at 30 years of age and so percent remaining means deviation from that value. Source: Strehler (1959).

respectively given as:

( ) xAexm α= , Eq. 1

and

( ) EAexm x += α , Eq. 2

where A and α are constants to be determined by data, and E is a correction term that

accounts for mortality due to sources extrinsic to the organism such as accidents and

infectious diseases. There are several models that have derived the Gompertz on the

basis of general assumptions and that can be applied to physiological systems in general.

These are the models provided by Sacher and Trucco (Sacher, 1956, 1966, 1978; Sacher

9

and Trucco, 1962; Trucco, 1963a&b), Brown and Forbes (1974a&b, 1975, 1976),

Koltover (1982, 1983, 1992, 1996, 1997, 2004), Gavrilov and Gavrilova (2001,

2002a&b), van Leeuween et al. (2002), and Mangel and Bonsall (2004). Because the

models of Sacher and Trucco and of Brown and Forbes are mathematically equivalent, a

review of one of them will suffice. The Gavrilov and Gavrilova model is radically

different and so this too will be reviewed. The models by Koltover, van Leeuween and

colleagues, and Mangel and Bonsall are conceptualized in relation to oxidative stress.

After making some introductory remarks on the roles of oxidative stress and

mitochondrial dysfunction in senescence, these models will be discussed together. It is

perhaps encouraging that very different perspectives lead to the same outcome.

Brown and Forbes (1974a) developed a model that is mathematically equivalent

to the model of Sacher and Trucco (see also extensions of the model in Brown and

Forbes, 1974b, 1975, 1976). The assumptions of the Brown-Forbes model are: 1) The

state of physiological injury that may lead to death, if severe enough, is inversely and

linearly related to the decline in homeostasis. 2) The observed state of physiological

injury xy at corresponding age, x , (satisfying assumption 1) may be taken as an

observation from a Gaussian random variable, Y , where ( )2,N~Y σμ , where by

convention upper case will be taken to denote the random variable, Y , and lower case,

xy , denotes an observation on Y , μ is the mean of the normal distribution and 2σ is the

variance. As such, observations on Y may be modeled as a linear regression on age:

( ) 0E ; xyx =ε⋅β+α=ε+μ= , Eq. 3

where ε is a random error term with expectation zero, and α and β are respectively the

intercept and slope of the regression line of xy on x . 3) There is a theoretical cut-off

10

level giving the absolute level of physiological injury that may be sustained before dying.

This cut-off level is represented by a horizontal line that gives some constant high value

of y for all x and that lies above and is approached from below by the regression line.

Therefore, the cut-off level, denoted by cy , satisfies the constant function:

ℜ∈∀= x ; xyc . Eq. 4

Conceive now of a series of normal distribution curves, where each one is centered on a

point on the regression line given by Equation 3. Since the regression line is approaching

the horizontal line cy from below, the normal curves will come to have increasing area

falling above cy . By the assumptions of the model, the area for any given xy falling

above cy will give the probability of mortality, ( )xm , at the corresponding age, x (Fig.

2). Therefore, integration under the appropriate interval, from cy to ∞+ (in the y-axis),

will give ( )xm . Now, Equation 3 allows ( )xm to be written as:

( ) [ ]( ) x

y

2 x2

yd xy2

1exp

2

1xm

c∫

∞

⎭⎬⎫

⎩⎨⎧

⋅β+α−σ

−πσ

= . Eq. 5

Generally, random variable Y may be expressed in terms of the standard normal

distribution, ( )1,0N 2 =σ=μ , by way of the Z-transformation:

σμ−= YZ . Eq. 6

To prepare for use of the Z-transformation in the present context, define the distances:

( )xycx ⋅β+α−=ε , Eq. 7

and

α−=ε c0 y . Eq. 8

11

xy ⋅β+α=

xy c = ( )xm

y

x

Figure 2. Brown and Forbes model of the mortality process with increasing age. Redrawn from Brown and Forbes (1974a).

To evaluate the integral in the interval [ )+∞,yc , we may now take the Z-transformed

version of cy , i.e., σ

εx , and evaluate it under the standard normal, noting that here 0=μ

and 12 =σ :

( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎥⎦

⎤⎢⎣

⎡

σ

ε−

πε

σ≈

⎭⎬⎫

⎩⎨⎧−

π= ∫

∞

σ

ε

2x

xx

2x

2

1exp

2ydy

2

1exp

2

1xm

x. Eq. 9

where the solution is an approximation of the integral (from Feller, 1968: 166). On

expressing Equation 7 in terms of Equation 8, the quadratic term in Equation 9 may be

written as:

( ) ( ) ( )2

2 0

20

2

2 0

2

2x xx2x

σ

⋅β−+⋅β−ε+ε=

σ

⋅β−ε=

σ

ε. Eq. 10

At large values of xy , the inequality:

12

σ

ε<<

σ

⋅β− 0x, Eq. 11

holds true such that ( )

2

2 x

σ

⋅β− can be assumed in this case to make a negligible

contribution in the expansion of the quadratic term in Equation 10. Consequently, we

arrive at:

( )2

020

2

2x x2

σ

⋅β−ε+ε≈

σ

ε, Eq. 12

and

000000x x ;

x2x2ε<<⋅β∀

σ

ε≈

σ

⋅β⋅ε−ε≈

σ

⋅β⋅ε−

σ

ε≈

σ

ε. Eq. 13

These approximations can be substituted into Equation 9 as follows:

( ) ( )

. x

exp2

1exp

2

x2

2

1exp

2xm

20

2 0

0

20

20

0

⎟⎟⎠

⎞⎜⎜⎝

⎛

σ

⋅β⋅ε⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎥⎦

⎤⎢⎣

⎡

σ

ε−

πε

σ≈

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

σ

⋅β−ε+ε−

πε

σ≈

Eq. 14

At age 0x = , define ( ) 0m0m = and ( ) 0y0y = . In analogy to Equation 9, an expression

for the mortality function at age 0x = is:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎥⎦

⎤⎢⎣

⎡

σ

ε−

πε

σ≈

⎭⎬⎫

⎩⎨⎧−

π= ∫

∞

σ

ε

20

00

200

2

1exp

2ydy

2

1exp

2

1m

0, Eq. 15

where we see that the first two terms in the right hand side of Equation 14 are in fact

identical to 0m . On putting 2

0*

σ

ε⋅β=α , Am0 = , and appropriately substituting in

Equation 14, we now have:

13

( ) x*Aexm α≈ , Eq. 16

which is the Gompertz mortality function.

We now turn to the model developed by Gavrilov and Gavrilova (2001,

2002a&b), which is based on reliability theory. The application of reliability theory to

the problem of aging in organisms was independently pioneered by several groups in the

late seventies (Rosenberg et al., 1973; Skurnick and Kemeny, 1978, 1979; Abernethy,

1979; see also the work of Gavrilov and Gavrilova (cited in their 1991 book) in Russian

publications in the late seventies). Reliability models sensu stricto of aging in organisms

were developed by Abernethy (1979, 1998), Ďoubal (1982), Koltover (1982, 1983, 1992,

1996, 1997, 2004), Witten (1983, 1984a&b, 1985), Miller (1987, 1989), Ďoubal and

Klemera (1989, 1990), Gavrilov and Gavrilova (1991, 2001, 2002a&b), and Izsák and

Gavrilov (1995) (again, see the earlier work in Russian by Gavrilov and Gavrilova). The

model developed by Gavrilov and Gavrilova (hereon G&G) is fairly general and may be

taken as representative of the scope of reliability models (however, the Koltover model

will also be reviewed shortly). Exposition of the G&G model requires some terminology

from reliability theory regarding how systems are constructed (see Fig. 3). A serial or

serially-constructed system is one that requires for its correct and continued operation

that every single one of its components is correctly operating or functioning. Failure in

one component results in system failure. This brings to mind the old saying that “A chain

is only as strong as its weakest link”. In this case, component redundancy is irrelevant to

system operation. A parallel or parallel-constructed system is one that requires that at

least 1 out of n components are properly operating or functioning for its correct and

continued operation. Thus, in this case, the probability that a system remains operational

14

Figure 3. Reliability Structures for (a) technical devices and (b) complex organisms. (a) Technical devices are serially connected between and within sub-systems. The large blocks (j = 5) represent sub-systems and the small blocks represent elements therein. (b) Organisms exhibit serial connections between subsystems (represented by the larger, m = 5 vertical rectangular blocks) that themselves exhibit parallel construction (represented by the smaller, k = 10 horizontal rectangular blocks). (a) and (b) also differ in the quality of elements. Organisms can sustain an initially high degree of defects (cross-marks) whereas technical devices start out with an initially low level of defects by design. Source: Gavrilov and Gavrilova (2001).

or alive is a function of component redundancy. Under the G&G model, multicellular

organisms exhibit both types of construction in that organisms are serially constructed out

of sub-systems (each one necessary for survival of the organism) but each sub-system can

be described as being parallel constructed. However, as senescence takes its toll, the

redundancy at the sub-system level becomes completely exhausted and the organism

degenerates into a serially-constructed system at which point any new instance of damage

is sufficient to cause system failure or death. Further, at this point, the mortality rate

becomes constant; that is, a mortality plateau is produced. It will be useful at this point to

briefly state some of the fundamental concepts common to demography and reliability

15

engineering, as these concepts will form a common underlying theoretical basis (see Cox,

1962; Gross and Clark, 1975; Elandt-Johnson and Johnson, 1980; Crowder et al., 1991).

Let X be a random variable representing the lifetime of individuals. Then the lifetime

cumulative distribution function is defined as:

( ) ( )xXPrxF ≤= , Eq. 17

and the survivorship function, now denoted by ( )xs , is defined relative to ( )xF as:

( ) ( ) ( )xXPrxF1xs >=−= . Eq. 18

Now, the lifetime probability density function, ( )xf , as for all probability density

functions, is found by taking the first derivative of the corresponding cumulative

distribution function:

( ) ( ) ( )[ ] ( )dx

xdsdx

xs1ddx

xdFxf

−=

−== . Eq. 19

The mortality function, ( )xm , is defined as:

( ) ( )( )

( )( )

( )( )

( )dx

xslnddx1

xsxds

xs1

dxxds

xsxf

xm−

=⎥⎦⎤

⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−== . Eq. 20

If ( )xF is known or taken as given, then ( )xs , ( )xf , and ( )xm are determined by

Equations 18-20.

The G&G model starts by deriving the mortality rate for blocks that are parallel

constructed out of k mutually substitutable elements, each described by a constant

failure rate φ . The cumulative distribution function for block failure, ( )x,,kFb φ , is

assumed to be:

( ) ( )kx b e1x,,kF φ−−=φ . Eq. 21

From the relations mentioned just above, we have for the survivorship function:

16

( ) ( )kx b e11x,,ks φ−−−=φ , Eq. 22

for the probability density function:

( ) [ ]( ) ( ) 1kx x k x

b e1ekdt

e11dx,,kf −φ−φ−φ−

−⋅φ=−−−=φ , Eq. 23

and for the mortality function:

( ) ( )( )

( )( )kx

1kx x

b

bb

e11e1ek

x,,ksx,,kfx,,km

φ−

−φ−φ−

−−−⋅φ=

φφ=φ . Eq. 24

For a serially constructed system comprised of j blocks made up of k elements, the

mortality function of the system is found by simply summing the block mortality rates:

( ) ( ) ( ) ( )( )kx

1kx x

b

j

1hbs

e11e1ejkx,,kmjhmx,,kmφ−

−φ−φ−

= −−−⋅φ=φ⋅==φ ∑ . Eq. 25

Now consider the more realistic case for organisms wherein which blocks are

comprised of mutually substitutable elements, each of which may be defective or

functional. For the distribution of the number of functional elements, denoted by i , out

of k total elements, G&G postulate a truncated Poisson distribution:

⎪⎩

⎪⎨

⎧

=∀λ

+++=∀= λ− ,k,...,3,2,1i ;

!ice

,...,3k,2k,1k,0i ; 0P i

i Eq. 26

where

∑ ∞

+=λ−λ− λ−−

=

1ki i !iee1

1c , Eq. 27

where λ is the parameter of the Poisson distribution and c is a normalizing factor

ensuring that the probabilities of all possible outcomes sum to unity:

17

∑=

=k

1ii 1P . Eq. 28

The Poisson distribution is truncated at the left as stipulated to acknowledge the fact that

organisms cannot survive with zero functional elements and is truncated at the right as

stipulated because the number of functional elements cannot exceed the total number of

elements. Note that the normalizing constant accounts for the cases when 0i = and

∞+++= ,...,3k,2k,1ki . In this case, the mortality rate for such a system is given by:

( ) ( ) ( ) ( )∑∑∑=

λ−

==

λ===φk

1i

bik

1ibi

j

1hbs !i

imjceimjPh,imx,,km , Eq. 29

where the mortality rate of blocks with i initially functional elements, denoted by ( )imb ,

is given by an expression analogous to Equation 24:

( ) ( )( )ix

1ix x

be11

e1eiimφ−

−φ−φ−

−−−⋅φ= . Eq. 30

In view of Equation 30, Equation 29 may be rewritten as:

( ) ( )( ) ( )[ ]∑

= φ−

−φ−−φ−λ−

−−−

−λφλ=φ

k

1i i x

1i x 1ix

s

e11! 1i

e1ejcex,,km . Eq. 31

The situation seems rather messy at this point. However, some approximations and

simplifications are possible. For ( ) xe1xg φ−−= , where ( )⋅g will be hereon referred to as

the function for which the Taylor approximation is to be applied, we have:

( ) [ ] xxx

edx

de

dx

e1dxg φ−

φ−φ−

φ=−

=−

=′ . Eq. 32

The first-order Taylor approximation of ( ) xe1xg φ−−= about the point ( )ax − at 0a = is

given as:

18

( ) ( ) ( )( ) xx0ax0g0gxg φ=φ+=−′+≈ . Eq. 33

It is important to note that since φ gives the failure rate, the approximation assumes that

the failure rate is linear in x (see Fig. 4: Left Panel). Using Equation 33 in the numerator

in Equation 31, we can write the following expression, which employs another first-order

Taylor approximation justified below:

( ) ( )( )( )

( )( )∑∑

=

−λ−

=φ−

−φ−λ−

−

λφφλ=

−

λφφλ≈φ

k

1i

1i k

1ix

1i x

s! 1i

xjce

! 1ie

xejcex,,km . Eq. 34

On comparing Equations 31 and 34, it would seem that the critical part of this last

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

20 30 40 50 60 70 80

Age (years)

f(x)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

20 30 40 50 60 70 80

Age (years)

f(x)

Figure 4. First-order Taylor approximations. Left Panel: the function ( ) xxg φ= (solid line) is a good approximation of the function ( ) xe1xg φ−−= (diamonds). Right Panel: for

1x <<φ , the function ( ) ( ) ⎥⎦⎤⎢⎣⎡ −−= φ− i x e11xg (solid line) is well approximated by the

function ( ) xexg φ−= (diamonds).

19

approximation appearing in the denominator is:

( ) ( )xexpe11i x φ−≈⎥⎦⎤

⎢⎣⎡ −− φ− . Eq. 35

Equation 35 is justified as follows. The first derivative of the left hand side of Equation

35 is:

( )( ) ( ) ( ) ( )

( ) ( ) . e1iedx

dee1i

dx

e1de1i

dx

e1d

dx

e11dxg

1i x x x

1i x

x 1i x

i x i x

−φ−φ−φ−

−φ−

φ−−φ−

φ−φ−

−φ−=−

⎥⎦⎤

⎢⎣⎡ −−=

−⎥⎦⎤

⎢⎣⎡ −−=

⎥⎦⎤

⎢⎣⎡ −−

=⎥⎦⎤

⎢⎣⎡ −−

=′

Eq. 36

For 0a = , the first-order Taylor approximation about ( )ax − of the left hand side of

Equation 35 is:

( ) ( ) ( )( ) 101ax0g0gxg =+=−′+≈ . Eq. 37

The first derivative of the right hand side of Equation 35 is:

( ) xx

edx

dexg φ−

φ−

φ−==′ . Eq. 38

For 0a = , the first-order Taylor approximation about ( )ax − of the right hand side of

Equation 35 is:

( ) ( ) ( )( ) x1ax0g0gxg φ−=−′+≈ . Eq. 39

Therefore, for 1x <<φ , the approximation is legitimate (Fig. 4: Right Panel). The

summation in the right most term in Equation 34 can be expressed as the difference of

infinite series:

( )( )

( )( )

( )( )∑∑∑

∞

+=

−∞

=

−

=

−

−

λφ−

−

λφ=

−

λφ

1ki

1i

1i

1ik

1i

1i

! 1i

x

! 1i

x

! 1i

x, Eq. 40

20

where the first term on the right hand side of Equation 40 is the power series definition of

the exponential function, which in general is given as:

( )∑∑

∞

=

−∞

= −==++++=

1i

1i

0i

i32x

! 1i

x

! i

x . . .

! 3

x

! 2

xx1e . Eq. 41

Thus, Equation 34 may be rewritten as follows:

( ) ( )( )

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

−

α−=

−

λφφλ≈φ ∑∑

∞

+=

−α

=

−λ−

1ki

1ix

k

1i

1i

s! 1i

xeA

! 1i

xjcex,,km , Eq. 42

where λ−φλ= jceA , and φλ=α . On noting the limit:

( )( )

0! 1i

xlim

1ki

1i

0xk

=−

α∑∞

+=

−

→∞→

, Eq. 43

we find that:

( ) xs Aex,,km α≈φ , Eq. 44

which is the Gompertz mortality function again. However, this holds to the extent that

the approximation holds, which in turn is valid early in the lifespan (Fig. 4).

Later in the life span, the system degenerates to a serially-constructed system, in

which case the number of functional elements given by k approaches 1. Note also that

the postulation of a Poisson distribution for the functional elements is no longer

necessary. Therefore, for 1k ≈ , we have from Equation 25:

( ) ( )( )

( )φ=

+−

−⋅φ≈

−−

−⋅φ=φ

φ−

φ−φ−

φ−

−φ−φ−

je11

e1ej

e11

e1ejkx,,km

x

0x x

kx

1kx x

s . Eq. 45

Thus, the phenomenon of the mortality plateau is observed.

Oxidative stress and mitochondrial dysfunction are increasingly thought to play

major roles in senescence (Wallace, 1992a&b, 1995, 1999; Ames et al., 1993; Shigenaga

21

et al., 1994; Sohal and Weindruch, 1996; Beckman and Ames, 1998; Lenaz, 1998; Wei,

1998; Wei et al., 1998; Ashok and Ali, 1999; Cortopassi and Wong, 1999; Finkel and

Holbrook, 2000; Grune and Davies, 2001; Van Remmen and Richardson, 2001; Lenaz et

al., 2002; Reis, 2003; Sastre et al., 2003; Singh et al., 2003; Barja, 2004). The two

processes of oxidative stress and mitochondrial dysfunction are logically connected

because mitochondria are by far the predominant source of reactive oxygen species

(ROS), which cause oxidative stress (Cadenas and Davies, 2000; Grune and Davies,

2001; Sastre et al., 2003; Singh et al., 2003; Turrens, 2003), although there are other

sources of ROS. The modern view ultimately derives from Harman’s (1956) original

“free radical theory” of senescence, which Harman (1972, 1983) himself first extended to

also incorporate mitochondrial effects (see reviews in Harman, 1981, 1991, 1992, 2001).

There are at least two differential equation models (van Leeuween et al., 2002;

Mangel and Bonsall, 2004) and a reliability model (Koltover, 1982, 1983, 1992, 1996,

1997, 2004) that can recover the Gompertz mortality function on the basis of general

assumptions at the proximate level and in terms of oxidative stress. Of the two

differential equation models, only the model developed by van Leeuwen et al. (2002)

admits a straightforward analytic solution whereas the model by Mangel and Bonsall

(2004) is a little more complicated and must be solved numerically. Even more

encouraging, both the van Leeuwen et al. (2002) and Koltover (1982, 1983, 1992, 1996,

1997, 2004) models assume linearity in the oxidative damage accruing with age. It will

be instructive to review the van Leeuwen et al. (2002) and Koltover (1982, 1983, 1992,

1996, 1997, 2004) models.

For mortality risk, van Leeuween et al. (2002) propose the following model:

22

( ) ( )( )tV

tDtm β= , Eq. 46

where ( )tD is the amount of oxidative damage, ( )tV is the structural volume, and β is

the damage-specific killing rate. As will be seen, ( )tV need not interest us here.

However, it should be noted that the model of van Leeuwen et al. (2002) was developed

within the framework of what has been called a “Dynamic Energy Budget” (DEB)

approach (for reviews, see Kooijman, 2001; Lika and Kooijman, 2003). It is through

( )tV that the model of van Leeuween et al. (2002) is coupled to the DEB approach.

Having stated this, the present focus is to derive an expression for ( )tD . The model

makes four assumptions. The first assumption is that the ROS-generation rate is

proportional to the catabolic rate:

( ) ( ) ( )tCttJ α=+ , Eq. 47

where ( )tJ+ is the ROS-production rate, ( )tα is the amount of ROS produced per utilized

reserve unit, and ( )tC is the catabolic rate, which itself satisfies:

( ) ( ) .maxetctC = , Eq. 48

where ( )tc is the scaled catabolic rate taken as a product with the maximum energy

reserve density, .maxe . The second assumption is that ROS reactivity is effectively

instantaneous following ROS production such that the ROS-generation rate translates

immediately into the ROS-reaction rate. ROS reactivity, however, is reduced or

eliminated by antioxidant defenses at rate ( )tJ− . Therefore, the total ROS-reaction rate,

denoted by ( )tJr , is given by:

( ) ( ) ( ) ( )tJtJtJtJr +−+ γ=−= , Eq. 49

23

where ( )tJ+γ gives the fraction of ROS actually reacting. The third assumption is that

the rate of oxidative damage is a linear combination of the fraction of ROS-reactions

actually inducing damage, given by ( )tzJr , the amplification to the oxidative damage rate

due to cellular and intracellular damage, occurring at rate ( )tx , and the repair rate, ( )ty .

Therefore, on suppressing the function notation, a preliminary differential equation is

given as:

yDxDzJdt

dDr −+= . Eq. 50

The fourth assumption is that α is a linear function in D. Here once again is the crucial

assumption of linearity in physiological damage or its inverse. As regards oxidative

stress, this assumption appears to be supported at least in humans (Jones et al., 2002;

Junqueira et al., 2004) and rats (Driver et al., 2000). The assumption is formulated as:

[ ]CDJ 10 α+α=+ . Eq. 51

Using Equations 49 and 51, Equation 50 may be written as:

( )[ ]DyxCzCzdt

dD10 −+γα+γα= . Eq. 52

Define new, compound parameters ( )yx −=ψ , .max1 ezγα=φ , and .max0 ezγα=ϕ . Using

these definitions and Equation 48, Equation 52 becomes:

[ ] cDcdt

dDϕ+φ+ψ= . Eq. 53

On supposing that *cc = , a constant, and 0=ϕ , Equation 53 becomes:

[ ]Dcdt

dD*φ+ψ= , Eq. 54

which is a separable differential equation and is solved as follows:

24

[ ]

[ ]

[ ]( ) 1eD ; tcexpDD

tcDln

dtcD

D

0*0

1*

*

κ=∀φ+ψ=

⇒κ+φ+ψ=

⇒φ+ψ=′∫∫

Eq. 55

where 1κ is a constant of integration. On the assumption that ( )tV is constant and given

by *V , use of Equation 55 in Equation 46 gives:

( ) [ ]( ) t*

*

0 *AetcexpV

Dtm α=φ+ψβ= , Eq. 56

which is the Gompertz mortality function, and where *

0

V

DA β= , and [ ]** cφ+ψ=α .

The reliability approach discussed earlier has also been conceptualized in terms of

oxidative stress (Koltover, 1982, 1983, 1992, 1996, 1997, 2004). Koltover (1992) noted

the necessity of relating the linear increase in oxidative damage to the Gompertz

mortality function (on the linear increase in oxidative damage, see also Driver et al.,

2000; Jones et al., 2002; Junqueira et al., 2004). Although oxidative damage occurs

linearly, the distribution of damaged structures that are critical to survival is what matters

most under Koltover’s approach (this argument goes back to Simms, 1942a; the argument

is reiterated explicitly in Sacher and Trucco, 1962; Brown and Forbes, 1974a). To derive

the Gompertz mortality function in terms of oxidative damage, Koltover developed the

following model. Koltover postulates the existence of Q critical structures, each one

essential for life. Therefore, the organism is conceptualized as being serially constructed

out of Q critical systems. Koltover motivates the model by considering the jth critical

system, where Q,,2,1j K= . Define jm as the number of defective elements due to

25

oxidative damage in the jth critical system and cm as a critical threshold in the number

of defects due to oxidative damage that the jth system can sustain. By these definitions,

the difference ( )jc mm − can be seen to be a safety margin defined on the interval:

cj mm0 ≤≤ . For simplicity, assume that cm is the same for all Qj∈ . Now imagine a

process in which jm accumulates in time so that in all that follows ( ) jj mtm ≡ (that is,

jm is now a function of time). For the jth system, the time of failure-free functioning,

denoted by jτ , is assumed to be proportional to the safety margin, and is given as:

( )jcj mmb −=τ , Eq. 57

where b is a constant of proportionality. In general, the time of failure-free functioning is

given as:

( ) Qj ; mmbt c ∈∀−= , Eq. 58

which of course implies that:

b

tmm c −= . Eq. 59

On supposing jm to be a random variable, the Palm-Khintchine Theorem (Khintchine,

1969: ch. 5; Koltover, 1982) suggests that the exponential distribution will suffice as the

probability law governing jm . The idea of a critical threshold given by cm , however,

requires a truncated exponential distribution; that is, an exponential distribution that is

truncated at cm . From these considerations, Koltover (1997; and implicitly in his related

works) suggested the following density distribution function:

( ) ( )( ) c

c

mm0 ; amexp1

amexpamf <<∀

−−

−= , Eq. 60

26

where a is a parameter of the exponential distribution. From the relation between the

density and cumulative distribution functions (see Eq. 19), we have for the cumulative

distribution function:

( )( )

( )( )

( ) ( )( ) 1amexp

amexpamexp

mt

mt

amexp1

atexpdt

amexp1

atexpa

c

cc

c

m

m c

c

−−

−−−=

=

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−

−−=

−−

−∫ . Eq. 61

Denote the above cumulative distribution function by ( )tG j . Writing the second term in

the numerator in ( )tG j in terms of cm (Eq. 59), we have:

( ) ( ) [ ]( )( )

( ) ( ) ( )( )

( ) ( )[ ]( )

( )( ) ( )[ ]

( )( )

. 1amexp

1batexp

1amexpamexp

batexp1

1amexp

batexp1amexp

1amexp

batexpamexpamexp

1amexp

btmaexpamexptG

cccc

c

c

cc

c

ccj

−

−=

−−

−=

−−

−−=

−−

−−−=

−−

−−−−=

Eq. 62

From the relation between the failure cumulative distribution and survivorship functions

(see Eq. 18), and the assumption that cm is the same for all Qj∈ , the following

survivorship function for individuals is derived as:

( ) ( )[ ] ( )[ ] Q Q

1jj tG1tG1ts −=−= ∏

=

. Eq. 63

Koltover suggested the following approximation for the survivorship function:

( )[ ] ( )[ ]tQGexptG1 Q −≈− . Eq. 64

Since ( )[ ] ( )[ ]( )Q tGexptQGexp −=− , it is sufficient show that:

( )[ ] ( )[ ]tGexptG1 −≈− . Eq. 65

Starting with the left hand side of Equation 65, the first derivative is:

27

( ) ( )[ ] ( )( ) ( )

( )[ ]

( )( )[ ]

. 1amexpb

batexpa

dt

1batexpd

1amexp

1

1amexp

1batexp

dt

d

dt

tG1dtg

c

cc

−−=

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−−=

−=′

Eq. 66

At 0x = , the first-order Taylor approximation about the point ( )xt − is:

( ) ( ) ( )( )( )[ ]1amexpb

at1xt0g0gtg

c −−=−′+≈ . Eq. 67

For the right hand side of Equation 65, the first derivative is:

( ) ( )[ ] ( )( )

( )( )[ ]

( )( )

. 1amexp

1batexpexp

1amexpb

batexpa

1amexp

1batexpexp

dt

d

dt

tGexpdtg

cc

c

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−−

−−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−−=

−=′

Eq. 68

Therefore, the first-order Taylor approximation about the point ( )xt − at 0x = is:

( ) ( ) ( )( )( )[ ]1amexpb

at1xt0g0gtg

c −−=−′+≈ . Eq. 69

We have just seen that the first-order Taylor approximations are identical. Hence,

Koltover’s approximation can be said to hold true up to first order. On writing b

a=α

and ( ) 1amexp

Q

c −=β , the survivorship function becomes:

( ) ( )[ ] ( )[ ] 1texpexptQGexpts −αβ−=−≈ . Eq. 70

Finally, from the relation between the mortality and survivorship functions, we find:

( ) ( ) ( )[ ] tAedt

texpd

dt

tslndtm α=

β+αβ−−≈

−= , Eq. 71

28

which is the Gompertz mortality function yet again for αβ=A . Koltover (1982, 1983,

1992, 1996, 2004; see also Koltover et al., 1993) empirically tested this model against

data on oxidative damage available in the literature and found an excellent fit between

data and predictions under the model.

Given a Gompertzian mortality function, ultimate-level models can easily explain

the evolution of senescence. The following discussion is a selective account of

evolutionary approaches to senescence (see Rose, 1991 for a comprehensive account).

The deterioration in homeostasis can be understood in ecological evolutionary terms

using the disposable soma (DS) theory of the evolution of senescence. The DS theory

was developed by T. B. L. Kirkwood and colleagues (Kirkwood, 1977, 1981, 1987, 1990,

1996, 1997, 2002; Kirkwood and Holiday, 1979, 1986; Kirkwood and Cremer, 1982;

Kirkwood and Rose, 1991), and is predicated on the life history tradeoff in the allocation

of resources to reproduction and to growth and maintenance (Perrin and Sibly, 1993; Zera

and Harshman, 2001). It is important to note that Kowald and Kirkwood (1994, 1996,

2000; see also Kirkwood and Kowald, 1997) have begun to show how the DS model can

be connected with the cellular-level processes of oxidative stress and mitochondrial

dysfunction. Kirkwood and Rose (1991) developed an elegant mathematical model of

the DS theory (cf. similar models in Kirkwood and Holliday, 1986; Kirkwood, 1990).

The DS model starts with the Euler-Lotka Equation:

( ) ( ) 1dxes,xMs,xL rx

0 =⋅⋅ −∞

∫ , Eq. 72

where ( )s,xL and ( )s,xM are respectively survivorship and fecundity functions of age,

denoted by x, and of the level of investment in somatic maintenance, denoted by s, and r

is the intrinsic rate of increase (note that the notation here follows Kirkwood and Rose for

29

investment in somatic maintenance). Once ( )s,xL and ( )s,xM are specified, r can be

solved for by standard methods (Charlesworth, 1994a). Note that survivorship and

mortality have the following relation:

( ) ( ) ⎥⎦⎤

⎢⎣⎡−= ∫ dx xmexpxL . Eq. 73

Using the Gompertz-Makeham in Equation 47, and integrating across the interval from

the age at which reproduction begins, denoted by a, to x, we have:

( ) ( )

( ) ( ) ( ) . axEeeA

exp tE eA

exp

Edt dtAeexpdt EAeexpxL

axxt

at

xt

at

t

x

a

x

a

t x

a

t

⎥⎦

⎤⎢⎣

⎡ −−−α

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⋅−

α−=

⎥⎦⎤

⎢⎣⎡ −−=⎥⎦

⎤⎢⎣⎡ +−=

αα=

=

=

=

α

αα ∫∫∫ Eq. 74

Assuming that the total juvenile mortality is given by V, the DS model specifies the adult

survivorship function as:

( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣

⎡ −−−α

−−= αα axEeeA

expV1s,xL ax . Eq. 75

To specify the fecundity function, the DS model assumes that fecundity declines like a

survivorship function following the Gompertz mortality function:

( ) ( )⎥⎦⎤

⎢⎣

⎡ −α

−= αα ax eeA

exphs,xM , Eq. 76

where h is the reproduction rate. The DS model requires that the parameters α and h are

given by increasing functions in s, and that the parameter a is given by a decreasing

function in s. Candidate forms for the α , h, and a functions are given respectively as:

⎪⎩

⎪⎨

⎧

′≥∀

′<∀⎟⎟⎠

⎞⎜⎜⎝

⎛

−

′α

=αss ; 0

ss ; 1s

s0 , Eq. 77

30

( )s1hh max −= , Eq. 78

and

( )s1aa min

−= , Eq. 79

where the region from s′ to 1 defines a non-senescence region. The DS model shows

that the optimal level of investment in somatic maintenance is lower than the amount of

investment required to be in the non-senescence region (Fig. 5). Taking r as a measure of

fitness, the DS model also shows that senescence is a consequence of the optimal life

history strategy (cf. more sophisticated models by Abrams and Ludwig, 1995; Cichoń,

1997; Cichoń and Kozłowski, 2000; Shanley and Kirkwood, 2000; Mangel, 2001;

Novoseltev et al., 2002). Since the amount of investment in somatic maintenance is less

0 1s* s’

intri

nsi c

rate

of i

ncre

ase,

r

investment in somatic maintenance, s

0dsdr =

Figure 5. Relation between fitness, measured by r, and the level of investement in somatic maintenance, s. The optimal amount of investment corresponding to the maximum fitness (maximum intrinsic rate of increase, r) is denoted by s*. Redrawn from Kirkwood and Rose (1991).

31

than what is required for the non-senescence phenotype (i.e., nearly-perfect to perfect

fidelity in somatic maintenance), it follows that the soma would accumulate defects with

increasing age and that homeostasis would progressively deteriorate.

To recapitulate, senescence can be understood from unified proximate and

ultimate perspectives. Recall that the DS model assumes the Gompertz mortality

function and that the derivations of the Gompertz mortality function assume linearity in

either homeostatic decline or an inversely proportional increase in the damage or injury

accruing thereto. It has been remarked that linearity appears in so many biological

processes because the linear terms of their respective Taylor approximations tend to

dominate the overall behavior (Starmer and Starmer, 2002). Economos (1982) argued

that the linear decline is a frame of mind. Similarly, Finch (1990) holds such a pattern to

be “untrue” (see Finch, 1990: 155). However, in a review of 469 studies, Sehl and Yates

(2001: B200) noted “We did encounter some cases of curvilinear loss. However, the

linear term in most polynomial fits carried most of the weight.” Thus, the general

argument of Starmer and Starmer (2002) appears to empirically validated, at least for the

case of senescence. This observation of linearity has significant ramifications for this

dissertation because it implies that the linear model derived from Fisher (1918), on which

all of contemporary statistical genetics is predicated, is a sufficient basis for the statistical

genetic investigation of processes that are fundamental to senescence. It must be pointed

out, however, that the statistical genetics approach, more than providing a foundation,

also makes a valuable extension to traditional approaches to studying senescence by

accounting for genetic variation among individuals of a given population.

32

Chapter 3

Background: Endocrinology of the IGF-I Axis in Relation to Senescence

This chapter is an extension of the discussion of senescence in the previous

chapter, but with a focus on the physiological approach to senescence and on the role

played therein by the IGF-I axis. The physiological basis of the statistical genetic

hypothesis to be tested, which was briefly mentioned in the introduction, is discussed in

this chapter as well.

From the corpus of work on the physiology and clinical biology of senescence, we

know that the deterioration in homeostasis is causally related to the development of age-

related pathology and disease (Strehler, 1977; Dilman, 1981, 1992 1994; Kohn, 1978,

1982; Kenney, 1982). Since the pathophysiology associated with senescence is

exceedingly complex, one can take the reductionism route. A major undertaking in this

direction is provided by the neuroendocrine theory of senescence, which has been

elaborated by Finch (1975, 1976, 1977, 1979, 1987, 1988, 1990, 1993; Finch and

Landfield, 1985) among others (see also Frolkis, 1966, 1968, 1972, 1976, 1981; Dilman,

1971, 1976, 1979, 1981, 1984, 1986, 1992 1994; Everitt, 1973, 1976a&b; 1980a&b;

Dilman and Anisimov, 1979; Dilman and Berstein, 1979; Dilman et al., 1979a&b, 1986).

According to Finch’s theory, senescence involves neuroendocrine cascades that are

dysfunctional, late-life occurrences of the same physiological control systems responsible

for maintaining homeostasis in earlier ontogeny. Under this view, the neuroendocrine

cascades may be seen as inducers of pathology or as inefficacious mechanisms for

restoring homeostasis. The neuroendocrine cascades refer to the cascading interactions

of the two main effector arms of the central nervous system (CNS) that are responsible

33

for the maintenance of homeostasis, which are the autonomic nervous system (ANS) and

endocrine arms, hence the name “neuroendocrine”. The neuroendocrine cascades theory

of senescence will be taken as the general physiological foundation for the current

approach. One can focus further still on one of the three main endocrine axes involved in

aging processes, which are the IGF-I, sex hormone, and the hypothalamic-pituitary-

adrenal (HPA) axes (Fig. 6). With a view towards understanding senescence, the IGF-I

axis across the age continuum will be taken as the system of study. Indeed, Finch and

colleagues suggest that a focus on the IGF-I axis in relation to senescence may well be

profitable (Finch and Ruvkun, 2001; Longo and Finch, 2002, 2003).

Figure 6. The main endocrine axes in aging and senescence. Left: The IGF-I axis. Middle: The Sex hormone axis. Right: The HPA axis. See text. Source: Lamberts et al. (1997).

34

The IGF-I axis—a complex network of hormones, binding proteins, proteases and

receptors (Sara and Hall, 1990; Werner et al., 1994; Jones and Clemmons, 1995; Collett-

Solberg and Cohen, 1996)—is an important regulator of prenatal development

(Gluckman, 1986; Gluckman and Pinal, 2003), postnatal growth (Daughaday, 2000; Lupu

et al., 2001), aging processes (Barbieri et al., 2003; Tatar et al., 2003) and metabolism

(Liu and Barrett, 2002; Murphy, 2003). Moreover, the IGF-I axis plays critical roles in

osteoporosis (Geusens and Boonen, 2002; Žofková, 2003), sarcopenia and muscle

atrophy (Borst and Lowenthal, 1997; Grounds, 2002), a number of cancers (LeRoith and

Roberts, 2003; Fürstenberger and Senn, 2003), a number of neurodegenerative disorders

(Gasparini and Xu, 2003; Trejo et al., 2004) and the four components of the metabolic

syndrome, namely T2D, CVD, hypertension and obesity (Raines and Ross, 1995, 1996;

Sowers and Epstein, 1995; Froesch, 1997; Bayes-Genis et al., 2000; Maccario et al.,

2000; Hausman et al., 2001; Frystyk et al., 2002; Holt et al., 2003). Thus, a study of the

IGF-I axis leads naturally to the more general concern of senescence.

A fund of studies on a wide range of human populations have established that the

pattern of IGF-I secretion follows a rise from low levels during early postnatal growth to

maximal levels at puberty, declines shortly thereafter and culminates at relatively lower

levels at older ages (Hall et al., 1980, 1981; Bala et al., 1981; Luna et al., 1983;

Rosenfield et al., 1983; Hall and Sara, 1984; Furlanetto and Carra, 1986; Cara et al.,

1987; Savage et al., 1992; Argente et al., 1993; Hesse et al., 1994; Juul et al., 1994, 1995;

Olivié et al., 1995; Yamada et al., 1998; Kawai et al., 1999; Barrios et al., 2000; Löqvist

et al., 2001; Low et al., 2001; reviewed in Juul, 2003). Data provided by Diagnostic

Systems Laboratory (DSL) for 1700 boys and 1700 girls from 3 to 17 years of age are

35

plotted in Figure 7 (the data may be obtained from their web page at the following URL:

http://www.dslabs.com). Figure 7 shows the archetypical secretion pattern up until

shortly after puberty. Data from the SAFHS will demonstrate the continued decline at

older ages (reported below). Generally, females achieve their peak IGF-I secretion height

before males at puberty, which is consistent with general patterns of pubertal growth

(Tanner, 1978; Bogin, 1999). The general features of this IGF-I secretion pattern over

the life span has been documented in baboons (Copeland et al., 1981, 1982; Crawford

and Handelsman, 1996; Crawford et al., 1997), chimpanzees (Copeland et al., 1985),

rhesus macaques (Liu et al., 1991; Styne, 1991) and gibbons (Suzuki et al., 2003).

0

100

200

300

400

500

600

700

3 5 7 9 11 13 15 17

Age (years)

Mea

n IG

F-I (

ng/m

l)

Boys Girls Sex-Averaged

Figure 7. IGF-I secretion pattern early in the human life span. Note that girls typically achieve their peak secretion height earlier than boys. Data are from Diagnostic Systems Laboratories for 1700 boys and 1700 girls from 3 to 17 years of age.

36

The liver is by far the main source of systemic IGF-I and accounts for around

80% of the total IGF-I pool in circulation (Sara and Hall, 1990; Jones and Clemmons,

1995). In the circulatory system, IGF-I may form several complexes with its binding

proteins (IGFBPs), of which six are known, designated as IGFBP-1 to IGFBP-6, and an

acid labile sub-unit (ALS) (Fig. 8; Rechler, 1993; Clemmons, 1999; Baxter, 2000). Of

the IGFBPs, IGFBP-1 and IGFBP-3 are considered to be the most important in

determining the availability of free IGF-I to tissues (Clemmons, 1999; Baxter, 2000).

The system is more complicated than depicted in Figure 8 because there are also

proteases and phosphorylating proteins that modulate IGFBP activity (Coverly and

Baxter, 1997; Bunn and Fowlkes, 2003). All of these proteins constitute a complex

Figure 8. Schematic description of the IGF-I axis and the two major sites of IGF-I secretion. GHRH – GH release hormone; SS – Somatostatin; GH-R – GH receptor; IGF-1 = IGF-I; IGF-1R – IGF-I receptor (see text). Source: Carter et al. (2002a).

37

system operating under dynamic biochemical equilibria that modulate the tissue-level

availability of free IGF-I. Liver secretion of IGF-I is stimulated mainly by growth

hormone (GH), which is secreted by the somatotrophs of the anterior pituitary (Corpas et

al., 1993; Giustina and Veldhuis, 1998; Müller et al., 1999). It should be noted that

insulin and nutritional factors are also important in stimulating liver secretion of IGF-I

(Clemmons and Underwood, 1991; Thissen et al., 1994; Jones and Clemmons, 1995;

Ketelslegers et al., 1995). Shortly after sexual maturation, the decline in circulating IGF-

I is mediated foremost by negative feedback regulation of GH secretion by two well-

known pathways: 1) the short-loop pathway, which refers to the action of IGF-I directly

at the somatotrophs and 2) the long-loop pathway, which refers to the actions of IGF-I at

the hypothalamus, namely down-regulation of GH release hormone and up-regulation of

somatostatin, which are respectively positive and negative regulators of somatotroph

secretion of GH (Corpas et al., 1993; Giustina and Veldhuis, 1998; Müller et al., 1999).

IGF-I is also expressed and regulated in virtually all other tissue types, as first

demonstrated by the work of D’Ercole and colleagues on the tissue distribution of IGF-I

synthesis in the human fetus and the rat (D’Ercole et al., 1980a&b, 1984; D’Ercole and

Underwood, 1981, 1986; Van Wyk et al., 1981; Underwood et al., 1984, 1986; D’Ercole,

1996). Similarly, work by Isaksson and colleagues on the endocrinology of bone growth

suggested that GH promotes the local expression and regulation of IGF-I in bone tissue

(Isaksson et al., 1982, 1985, 1987, 2000; Ohlsson et al., 1998, 1999). These and similar

such findings established the concept that the IGF-I axis has an autocrine/paracrine mode

of action in addition to its classical endocrine mode (i.e., via liver-secreted IGF-I) (Fig. 9;

Underwood et al., 1986; Holly and Waas, 1989; Chatelain et al., 1991). The original

38

GH

Pituitary

Liver

Circulation

IGF-IIGFBP-3

IGF-I

Nutrition &

ALS

IGF-I

Other Factors

IGF-I

AutocrineParacrineEndocrine

Figure 9. Endocrine, paracrine, and autocrine modes of IGF-I action. Courtesy of Dr. A. J. D’Ercole.

somatomedin hypothesis (Fig. 10a; Salmon and Daughaday, 1957; Daughaday and

Garland, 1972; Daughaday et al., 1972), which claimed that GH exerts its effects solely

through the mediating actions of liver-produced IGF-I (originally named sulfation factor

and then somatomedin by Daughaday and colleagues), was accordingly revised to

acknowledge the ubiquitous autocrine/paracrine mode of action. Under the revised

somatomedin hypothesis, it was still maintained that the predominant effects of GH arise

through the endocrine mode (Fig. 10b; Daughaday and Rotwein, 1989; Daughaday, 1989,

1997, 2000; Spagnoli and Rosenfeld, 1996; Salmon and Burkhalter, 1997).

The revised somatomedin hypothesis has come under scrutiny because it appears

from work on transgenic mice that the endocrine mode is not at all essential for normal

39

Figure 10. The somatomedin hypotheses. (a) The original somatomedin hypothesis. (b) The revised somatomedin hypothesis. (c) The current somatomedin hypothesis. Igf1-/- – Transgenic IGF-I double-negative mutant mice that are unable to synthesize liver IGF-I following fetal development. Such mice can be used to study the effects of postnatal ablation in liver secretion of IGF-I. All other abbreviations are as mentioned previously. See text for discussion. Source: LeRoith et al. (2001a).

growth whereas the autocrine/paracrine mode is both sufficient and necessary to this end

(Fig. 10c; Liu and LeRoith, 1999; Sjögren et al., 1999, 2002a-c; Yakar et al., 1999, 2000;

Ohlsson et al., 2000a&b; Liu et al., 2000; Butler and LeRoith, 2001a&b; Isaksson et al.,

2001a&b; LeRoith et al, 2001a&b; Butler et al., 2002). Still, the data in favor of the

concept that the endocrine mode is important in somatic growth is compelling, such as

the clinical observations that reduced growth is incurred under systemic IGF-I deficiency

and/or resistance (Spagnoli and Rosenfeld, 1996; Hintz, 1999; Laron, 1999, 2002; Zapf

and Froesch, 1999; Daughaday, 2000; López-Bermejo et al., 2000; Camacho-Hübner and

40

Savage, 2001; Reiter and Rosenfeld, 2003; Rosenfeld, 2003) and in disparate human

pygmy populations (Merimee et al., 1981, 1982; Jain et al., 1998; Clavano-Harding et al.,

1999; Dávila et al., 2002). Further, the transgenic mouse studies are subject to several

ambiguities of interpretation, with the consequence that they cannot clearly reject an

important role for endocrine IGF-I in somatic growth (D’Ercole and Calikoglu, 2001;

Robson et al., 2002; van der Eerden et al., 2003). Further still, a recent transgenic mouse

study along the lines of somatic growth regulation has found that there appears to be a

critical threshold-level for circulating IGF-I below which longitudinal bone growth and

bone density are severely affected (Yakar et al., 2002a&b; Yakar and Rosen, 2003).

In an earlier review of the above debate by Gluckman et al. (1991) it was thought

that the main role of endocrine IGF-I was in the regulation of whole-body protein

metabolism. In this regard, it is noteworthy that studies using isotopic tracer infusions of

the essential amino acid leucine as a marker of whole-body protein metabolic activity

have found that IGF-I promotes the protein anabolism typical of pubertal growth

(Arslanian and Kalhan, 1996; Mauras et al., 1996; Mauras, 1999). These results are

consistent, moreover, with the well-known anabolic effects of GH and IGF-I in skeletal

muscle metabolism (Fryburg, 1994; Florini et al., 1995, 1996; Fryburg and Barrett, 1995;

Liu and Barrett, 2002; Rennie et al., 2004). Further, other transgenic mouse studies have

also demonstrated that liver-derived IGF-I plays an important role in the metabolic

regulation of carbohydrate and lipids (Fernández et al., 2001; Sjögren et al., 2001;

Wallenius et al., 2001; Yakar et al., 2001, 2002b; 2004; Haluzik et al., 2003; Clemmons,

2004). This said, it should be recalled that IGF-I had long been thought to be a regulator

of at least glucose homeostasis due largely to the seminal work of Froesch and colleagues

41

(Froesch et al., 1963, 1966, 1967). These earlier studies are relevant to the current debate

because they were carried out on what would later be identified as IGF-I extracts from

human serum (Froesch et al., 1985, 1996a). Because liver-derived IGF-I constitutes the

vast majority of the IGF-I pool in circulation (Sara and Hall, 1990; Jones and Clemmons,

1995), the early studies are consistent with an important endocrine mode of action.

One balanced conceptual model that has developed out of this lively debate is that

both modes of action of the IGF-I axis are important in somatic growth regulation and

their relative importance will vary according to developmental stage and tissue-type

(D’Ercole and Calikoglu, 2001; see also D’Ercole and Underwood, 1986; D’Ercole, 1996

for an earlier version of this model). D’Ercole and Calikoglu (2001) postulated that the

autocrine/paracrine mode predominates during early fetal development and the endocrine

mode becomes increasingly important over the course of postnatal growth. In statistical

genetic studies on the mouse, it has been demonstrated that there are at least two different

gene systems controlling the overall dynamics of growth, one operative early in ontogeny

and the other later (Cheverud et al., 1996; Atchley and Zhu, 1997; Vaughn et al., 1999).

Cheverud et al. (1996) hypothesized that the genetic system controlling late growth was

related to IGF-I and suggested that their results are consistent with the model proposed by

D’Ercole and Underwood (1986). More recently, using GH-deficient lit/lit mutant mice

and IGF-I knockout mice, Mohan et al. (2003) demonstrated that GH-independent

mechanisms controlled prepubertal bone growth whereas GH-dependent IGF-I was

largely responsible for pubertal bone growth, which is consistent with the 2-phase model.

If one may paraphrase the conclusions of D’Ercole and Calikoglu (2001) in which

they proposed a subtle extension of the above 2-phase model and integrate these with

42

general tenets of physiological ecology, we come now to a development-oriented 3-phase

model for the behavior of the IGF-I axis according to which: 1) the autocrine/paracrine

mode predominates during late fetal development; 2) the endocrine mode becomes

increasingly important for somatic growth and is maximally important for the pubertal

growth spurt; and 3) the endocrine mode undergoes a transition from being a regulator of

somatic growth to being a regulator of metabolism and somatic maintenance over the

course of adulthood. Phase 3 at once has the potential to resolve the debate regarding the

“true” role of endocrine IGF-I and highlights the likely role that the IGF-I axis might play

in the physiological mechanisms underlying the well-known life history tradeoffs

obtaining amongst growth, reproduction and somatic maintenance (on the theory of life

history tradeoffs in relation to senescence, see Kirkwood and Rose, 1991; Partridge and

Barton, 1993; Abrams and Ludwig, 1995; Cichoń, 1997; Cichoń and Kozłowski, 2000).

The observation that the IGF-I axis is important throughout all the major stages of

the life span for a diverse array of physiological phenomena would seem to indicate the

existence of a dynamic gene expression network (GEN) (sensu Wyrick and Young, 2002;

see Fig. 11) reflecting the behavior of the IGF-I axis. This is to be expected from the

tenet of endocrinology that hormones initiate signal transduction networks that, in turn,

modulate the behavior of a GEN. If the 3-phase model of the behavior of the IGF-I axis

has any credence, then the predicted shifts should be reflected by the behavior of the IGF-

I axis GEN translated along the age continuum. Here, with data from the SAFHS for the

relevant age range, the statistical genetics of the third phase will be addressed. A

simplified genetic model of the IGF-I axis may be postulated. Under this model, the

IGF-I axis interacts with an underlying GEN, and the genes of the GEN exhibit

43

pleiotropy and variation in age-specific effects (cf. Cheverud et al., 1996). Minimally,

this theoretical model consists of two testable hypotheses: 1) The IGF-I axis GEN is

pleiotropic. 2) The components of the IGF-I axis GEN exhibit age-specific effects. Both

of these hypotheses can be rigorously addressed using current statistical genetic models.

This dissertation will focus on the second hypothesis.

Figure 11. Schematic of a gene expression network. The IGF-I axis can be thought of as occupying the arrow connecting an environmental stimulus to a signal transduction network, with control over a number of transcriptional activators. The transcriptional activators in turn control the expression of a number of genes, either in one-to-one fashion as in the left set of examples or in multiple-to-one fashion as in the middle and right set of examples. Source: Wyrick and Young (2002).

44

Chapter 4

Background: The Study Population and Epidemiological Patterns

The study population is derived from the San Antonio Family Heart Study

(SAFHS). The demographic and epidemiological foundations of the SAFHS will be

reviewed. The SAFHS is comprised of Mexican Americans recruited from low-income

barrios in San Antonio, Texas. San Antonio is the largest city of Bexar County, Texas,

and Bexar County is itself situated in southcentral Texas (Figs. 12 and 13). San Antonio

is currently the second largest city in Texas (after Houston and Dallas) and the ninth

largest in the United States (U.S.) (cf. Fehrenbach, 2002). Arreola (2002: 131) points out

that “among large cities, San Antonio is the urban area with the highest proportion of

Mexican Americans in the country; the Hispanic subgroup was 59 percent of the city in

2000.” According to statistics from the San Antonio Metropolitan Health District

(SAMHD), the San Antonio population grew from 1,185,394 residents in 1990 to

1,392,931 residents in 2000 at a rate of about 17.5% (SAMHD, 2000). An important

point for the current study is that most of the San Antonio population growth is

attributable to the 29% growth in the Hispanic population, which, in turn, was due to both

a relatively high Hispanic birth rate and a high Mexican immigration rate (SAMHD,

2000). The rise of those of Mexican ancestry in San Antonio from 1900 to 2000 is

depicted in Figure 14. This agrees roughly with the projected change in the ethnic

composition in Bexar County from 1990 to 2030 (Fig. 15). At both the city and county

level, Hispanics were the majority population by the year 2000. According to Figure 15,

this pattern in the dominance of Hispanics in the contribution to the total population looks

to be increasing up until at least the year 2030.

45

Figure 12. Map of Bexar County in Texas. Blow-up at bottom left shows Bexar County surrounded by seven counties.

46

Figure 13. Map of San Antonio in Bexar County. Yellow indicates the inner city—encircled by Loop 410—and green indicates the city limits.

47

San Antonio Population

0200400600800

100012001400

1900 1920 1940 1960 1980 2000

Decades

Popu

latio

n x

1000

Hispanic Total

San Antonio Hispanic Population

010203040506070

1900 1920 1940 1960 1980 2000

Decades

% H

ispa

nic

Figure 14. Rise of the Hispanic population in San Antonio, 1900-2000. Top Panel: Population growth for the total and Hispanic population in San Antonio. Botom Panel: Percentage of total population growth due to Hispanic population growth. Data are from Arreola (2002: Table 7.3, p. 145).

48

Bexar County Population

0200400600800

100012001400

1990 2000 2010 2020 2030

Decades

Popu

latio

n x

1000

Hispanic NH White Black Other

Bexar County Hispanic Population

45

50

55

60

65

1990 2000 2010 2020 2030

Decades

% H

ispa

nic

Figure 15. Relative increase of the Hispanic population in Bexar County, Texas, 1990-2030. Data and projections are from the U.S. Census and Texas State Data Center as reported in SAMHD (2000).

49

The main goal of the SAFHS is to discover the genetic determinants of

atherosclerosis Mexican Americans, focusing on the Mexican American population of

San Antonio. Atherosclerosis is perhaps the most important cause of mortality under the

more general category of cardiovascular disease (CVD). CVD in turn is one of the major

diseases of the metabolic syndrome. A brief discussion of the historical and current

epidemiology of the metabolic syndrome in San Antonio will show the SAFHS to be a

logical step in addressing these problems.

All of the metabolic syndrome components, namely CVD, type 2 diabetes (T2D),

obesity and hypertension, are classic diseases of modernization. Their etiologies, known

to be physiologically related (Reaven, 1988, 1993, 1995, 1999), involve environmental

effects associated with modernization and a poorly understood genetic predisposition

(Zimmet and Thomas, 2003; for representative work particularly on the Mexican

American population of San Antonio, see Diehl and Stern, 1989; Stern and Haffner,

1990; Stern et al., 1991, 1992; Mitchell et al., 1996a&b, 1999; MacCluer et al., 1999;

Hixson and Blangero, 2000). Early features of the metabolic syndrome involve deranged

carbohydrate and lipid metabolism, which promote progression to T2D and obesity

(Haffner et al., 1992; Liese et al., 1997, 1998; Reaven, 1999). Moreover, T2D is one of

the more important predictors of CVD and type 2 diabetics are at higher risks for CVD

morbidity and mortality relative to nondiabetics (Laakso and Lehto, 1997; Howard and

Magee, 2000; Laakso, 2001; Resnick and Howard, 2002; Laakso and Kuusisto, 2003;

Nesto, 2003, 2004). The fact that T2D has an earlier age of onset than CVD will be

important below.

50

Judging from the historical epidemiology of T2D, the current problems associated

with the metabolic syndrome in Mexican Americans of San Antonio appear to have

started shortly after 1940 (Fig. 16; Ellis, 1962; Carey et al., 1992; Bradshaw et al., 1995).

For simplicity, it is here assumed that the historical works referred to herein were

speaking of what we today recognize as T2D, which is justifiable because the criterion of

diabetes mellitus in adults was often met or implied in these reports. In one of the earliest

studies focusing on Mexican American mortality in Bexar County, Ellis (1962) reported

that Spanish-surname men and women had T2D mortality rates of 16.97 and 22.27

0

1

2

3

4

5

6

7

8

1 2 3 4 5

Decades

RSM

R

Spanish surname female Spanish surname maleNon-Hispanic white female Non-Hispanic white male

Figure 16. Relative standardized mortality ratios (RSMR) for total mentions of diabetes mellitus: Spanish surname and non-Hispanic whites age 30 and over by sex. Data are from multiple-cause-of-death records for Bexar County, Texas, 1935-1944 to 1975-84. Decades 1-5 correspond to 1935-1944, 1945-1954, 1955-1964, 1965-1974, and 1975-1984, respectively. Modified from Bradshaw et al. (1995).

51

whereas their other-white counterparts had rates of 10.75 and 9.02, respectively. The

differentials in T2D mortality in Mexican Americans and non-Hispanic whites appear to

have started shortly after 1940 for females and shortly after 1950 for males. These

differentials would increase with time (Fig. 16; Carey et al., 1992; Bradshaw et al., 1995).

The negative trend continued unabated, as evidenced by reports of increased T2D

incidence from the late 1970s to the late 1980s (Haffner et al., 1991, 1992; Fig. 17).

These results are from earlier phases of the San Antonio Heart Study (SAHS) (not to be

confused with the SAFHS). In a more recent phase of the SAHS, Burke et al. (1999)

found that the incidence of T2D for 7- to 8-year follow-up examinations carried out from

1987 to 1996 for cohorts enrolled from 1979 to 1988 increased in both Mexican

Americans and non-Hispanic whites (Fig. 17).

T2D in San Antonio, 1979-1996

5.7

15.7

2.6

9.4

02468

101214161820

1979-1988 1987-1996Cohorts

Inci

denc

e, %

MA CA

Figure 17. Change in T2D incidence in San Antonio, Texas. Data are T2D incidence (%) by ethnicity, Mexican American (MA) and Caucasian American (CA). Data were reported in Burke et al. (1999).

52

One related and potentially contentious issue needs to be considered in some

detail. It was once widely held that Mexican Americans have lower CVD mortality than

non-Hispanic whites, which is somewhat paradoxical given that Mexican Americans

have higher T2D mortality and/or morbidity indices and lower socioeconomic status

(SES) indices relative to non-Hispanic whites (Ellis, 1962; Kautz, 1982; Bradshaw et al.,

1985; Castro et al., 1985; Markides and Coreil, 1986; Diehl and Stern, 1989; Rosenwaike

and Bradshaw, 1989; Stern and Haffner, 1990; Bradshaw and Liese, 1991; Bradshaw and

Frisbie, 1992; Carey et al., 1992; Mitchell et al., 1992; Stern, 1993). This pattern has

been called the Hispanic Paradox (Hunt et al., 2002, 2003). The Hispanic Paradox is

really more complex than can be adequately described here (for more comprehensive

treatments, see Franzini et al., 2001; Palloni and Morenoff, 2001; Morales et al., 2002).

As can be seen in Figure 18, Mexican Americans of San Antonio had advantages

over or were comparable to Caucasian Americans of San Antonio in CVD mortality

overall while the situation is reversed with respect to T2D. Stern and Wei (1999) argued

that the pattern is spurious because it is based on vital statistics and these data tend to

underestimate deaths in minority segments of the population (see also Wei et al., 1996).

Their analyses of risk factor distributions derived from the SAHS indicate that Mexican

Americans have higher CVD mortality than non-Hispanic whites. More recent work

from the SAHS by Hunt et al. (2002, 2003) hasconfirmed these findings and the

argument that vital statistics data underestimate mortality in minorities was reiterated.

However, Espino et al. (1994), from analyses of Bexar County death certificates, found

that elderly Mexican Americans had higher mortality risks of T2D and CVD than their

non-Hispanic white counterparts. Thus, while the pattern uncovered by Espino et al.

T2D

00.5

11.5

22.5

33.5

44.5

1940 1950 1960 1970 1980Decades

SMR

CAM CAF MAM MAF MIM MIF

Acute Myocardial Infarction

0

0.2

0.4

0.6

0.8

1

1.2

1940 1950 1960 1970 1980Decades

SMR

CAM CAF MAM MAF MIM MIF

Chronic Ischemia

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1940 1950 1960 1970 1980Decades

SMR

CAM CAF MAM MAF MIM MIF

Other Circulatory

0

0.5

1

1.5

2

1940 1950 1960 1970 1980Decades

SMR

CAM CAF MAM MAF MIM MIF

Figure 18. T2D and CVD mortality in San Antonio. The last letters in the legend stand for male (M) or female (F). CA and MA stand for Caucasian American and Mexican American, respectively. MI stands for Mexican immigrant. Data are for standardized mortality ratios (SMR) for total mentions of the cause of mortality, where the values for CAM and CAF over the decade beginning at the year 1960 are taken as the standards. Data are from Carey et al. (1992).

54

(1994) is consistent with the current reports from the SAHS, the argument that the bias in

vital statistics data gives rise to the Hispanic Paradox is not entirely accurate. Indeed, the

most recent community health review by the SAMHD (2002) reported, on the basis of

vital statistics data, that Hispanics have higher T2D and CVD mortality rates than non-

Hispanic whites, although the differential in heart disease mortality may not be

statistically significant until the oldest age group (Fig. 19). Similar findings were

reported from the Corpus Christi Heart Project in which higher CVD mortality rates in

Mexican Americans relative to non-Hispanic whites were also observed (Goff et al.,

1994; Pandey et al., 2001). Thus, the Hispanic Paradox would seem to be falsified for the

last decade in San Antonio (and perhaps Corpus Christi). It is perhaps significant to note

that all of the studies rejecting the Hispanic Paradox in San Antonio appeared relatively

recently whereas the studies suggesting the existence of a Hispanic Paradox all appeared

earlier. This observation suggests an alternative explanation that is explored below.

The problems with vital statistics data notwithstanding, it is possible that the

Hispanic Paradox was a real phenomenon in the past that was brought about by

heterogeneity in the position along the epidemiologic transition (see below) occupied by

the Mexican immigrant population, and the Mexican and Caucasian American

populations. Before proceeding with this argument, some tenets of the epidemiology of

modernization need to be discussed.

There are generally two versions of biomedical studies termed natural

experimental models (sensu Garruto et al., 1989, 1999) that seek to identify the health

effects of modernization and the respective etiologies of the health problems associated

with modernization. A caveat to be taken with the gross generalization to follow is that

55

Heart Disease Mortality

0

2000

4000

6000

8000

10000

12000

0-19 20-29 30-39 40-49 50-59 60-69 70-79

Age Intervals

Mor

talit

y R

ate

(per

1,0

00,0

00)

MA CA

T2D Mortality

0500

1000150020002500300035004000

0-19 20-29 30-39 40-49 50-59 60-69 70-79

Age Intervals

Mor

talit

y R

ate

(per

1,0

00,0

00)

MA CA

Figure 19. Heart disease and T2D mortality in Bexar County, 2002. Top Panel: Mortality due to heart disease. Bottome Panel: Mortality due to T2D. Data were compiled and reported by SAMHD (2002).

56

the two versions are not to be treated typologically, but rather, as ends of a populational

continuum. Under what might be called the in situ modernization model, a particular

community undergoes modernization and as the result of such change there occurs an

epidemiologic transition (Omran, 1971) from an epidemiological profile dominated by

infectious disease to one dominated by non-infectious disease (Fig. 20). The

epidemiologic transition was originally conceived with respect to developed Western

nations. Therefore, the rate at which such nations progressed through the three stages of

the epidemiologic transition (see Fig. 20) is taken as the standard case strictly for

Stage I Stage II Stage III

Noninfectiousmortality

Infectiousmortality

Modernization

Figure 20. Schematic diagram of the epidemiologic transition. As modernization increases, infectious mortality decreases while noninfectious mortality increases. At Stage I, the “age of pestilence”, infectious mortality accounts for most of the population mortality. At Stage II, the “transition stage”, infectious mortality decreases while noninfectious mortality increases. At Stage III, the “age of chronic degenerative diseases”, noninfectious mortality now accounts for most of the population mortality.

57

comparative purposes. Thus, it is commonplace to speak of a relatively rapid or delayed

epidemiologic transition for other nations or populations. Classic examples of the in situ

modernization model are the Republic of Nauru (Zimmet, 1978, 1979) and the United

States Territory of American Samoa (Baker et al., 1986), both of which are considered to

be instances of rapid modernization and, hence, rapid progression through the three

stages of the epidemiologic transition. Under what might be called the migration model,

there arises a migration link from a lesser-developed nation or community to a more-

developed nation or community. The pattern may be international, rural-to-urban or

some mix of both and the prevailing commonality is that large numbers of people

radically transform their environment by moving from one place to another (for

illustrative examples from the Pacific, see Garruto, 1990). Migration from various

sources in Mexico to various communities in the United States is a well-known

phenomenon. Not surprisingly, migration effects have figured prominently in studies of

T2D and CVD mortality and morbidity in the Hispanic population (to include both those

born in the U.S. and in Mexico) of San Antonio (Rosenwaike and Bradshaw, 1989;

Bradshaw and Frisbie, 1992; Carey et al., 1992; Wei et al., 1996; Stern and Wei, 1999;

Hunt et al., 2002).

The situation of the Mexican Americans of San Antonio would seem to fall

somewhere in between the in situ modernization and migration models. In this

population, two forces are inextricably entangled in their effects; migration from Mexico

and rural areas of south Texas and in situ modernization exacerbated by the

socioeconomic stratification and inequality “endemic” to large metropolitan areas in the

U.S. (Sen, 1993). That in situ modernization (with all its unintended problems) plays a

58

role in the causal structure of the metabolic syndrome in San Antonio is suggested by

observations that T2D prevalence in Mexican American participants in the SAHS is

inversely related to socioeconomic status (Hazuda et al., 1988; Mitchell and Stern, 1992).

The mechanism leading to the previous relation presumably involves the increase in

relative deprivation and the genesis of socioeconomic gradients in health (Marmot, 1994;

Williams and Collins, 1995; Daniels et al., 1999; Nguyen and Peschard, 2003) that are

known to occur under the epidemiologic transition (Wilkinson, 1994). Similarly, that

migration plays a role is suggested by comparisons of T2D prevalence in Mexican

American participants in SAHS and foreign-born Mexican Americans: T2D prevalence in

Mexican American participants in the SAHS is inversely related to acculturation status

independent of socioeconomic status (Hazuda et al., 1988; Stern and Haffner, 1990),

whereas foreign-born Mexican Americans tend to have relatively lower prevalences of

T2D and CVD than U.S.-born Mexican Americans in San Antonio (Bradshaw and

Frisbie, 1992; Wei et al., 1996; Hunt et al., 2002). The latter observation is consistent

with reports in the literature that foreign-born Mexican Americans have lower CVD

mortality in Texas (Rosenwaike and Bradshaw, 1989). Studies by Sundquist and

Winkelby (1999, 2000) on data from the National Health and Nutrition Examination

Survey III (NHANES III) came to similar conclusions at the national level. Sundquist

and Winkelby (1999, 2000) divided the Mexican American group from NHANES III into

three sub-groups roughly reflective of migration and acculturation status: 1) Mexico-

born, 2) U.S.-born English-speaking, and 3) U.S.-born Spanish-speaking. These sub-

groups were compared against each other and against non-Hispanic whites for a number

of CVD risk factors (e.g. BMI and T2D). Sundquist and Winkelby (1999, 2000) found

59

that overall, Mexican Americans are at higher risk for CVD than non-Hispanic whites. In

comparisons among the Mexican American sub-groups, they also found that the U.S.-

born Spanish-speaking individuals were at significantly greatest risk for CVD. Taken

together, these observations suggest that as Mexican immigrants become assimilated into

American society, the concomitant changes in environment exact increases in the risk of

T2D and CVD.

Now, migration is a complex sociocultural phenomenon (for Mexican

immigration to the U.S., see Massey, 1986; Massey and España, 1987; Durand and

Massey, 1992). Based on studies of Mexican immigration into the U.S., it appears that

the demographic structure of the migrant flow to the U.S. changes in accordance with a

three phase model of migration (Massey, 1986). In the first phase, the migrant flow is

comprised predominantly of young male adults. Inevitably, these young male adults

become well-adapted to their foreign setting, thus setting up the next two phases. From

the transition to the settlement phases, women and children, who are the families of the

young male adults, become part of the migrant flow. The important point here is that the

demographic structure of the receiving Hispanic populations in the U.S. would be

accordingly affected. Given that Bradshaw and Frisbie (1992) have demonstrated that

Mexicans were in stage II of the epidemiologic transition relative to Caucasian

Americans, it follows that the Hispanic population of San Antonio would have

characteristics that are intermediate between stages II and III of the epidemiologic

transition for most of the last century. Bradshaw and Frisbie (1992) did in fact find that

Mexican Americans of San Antonio are intermediate between Caucasian Americans and

Mexicans. An immediate corollary of this line of thinking is that only after the Mexican

60

American population becomes demographically aged—due to increasing maturation of

the demographic structure of the migrant stream as well as to in situ demographic

aging—do we begin to observe ethnic group differentials in CVD mortality that are

consistent with expectations. Under this scenario, the differential in CVD mortality

arises simply because proportionately more and more Mexican Americans are now living

to the age of onset for CVD. This scenario is consistent with the facts that Hispanics

have always had higher T2D mortality than non-Hispanic Whites and that T2D has an

earlier age of onset than CVD.

San Antonio has been the venue of a number of informative epidemiological

studies on the etiology of the metabolic syndrome (e.g., Hazuda et al., 1988; Stern et al.,

1991, 1992; Wei et al., 1996). This epidemiological work has established the

unquestionable importance of environmental factors. Besides socioeconomic status and

acculturation status, dietary behaviors related to fat and sugar intake patterns have also

been implicated as contributing risk factors to obesity and T2D (Stern and Haffner,

1990). In stark contradiction to our knowledge of the role of environmental factors, very

little is known about the genetic factors that may either predispose individuals to or be in

some way protective against the metabolic syndrome. The SAFHS seeks to redress the

dearth of knowledge on the role of genetic factors in the metabolic syndrome. This

dissertation is only one small part of this large-scale research enterprise.

61

Chapter 5

Methods I: Sampling Design, Pedigrees, and Phenotypes

The SAFHS is “the first comprehensive genetic epidemiologic study of

atherosclerosis and its correlates in Mexican Americans (Dr. J. W. MacCluer, personal

communication)”. The SAFHS is a research enterprise jointly carried out by the

Department of Genetics at the Southwest Foundation for Biomedical Research (SFBR),

San Antonio, Texas, and the School of Medicine at the University of Texas at San

Antonio Health Science Center. The findings of this research are reviewed in Mitchell et

al. (1996a&b, 1999) and MacCluer et al. (1999). Two phases have been completed so

far, designated as SAFHS1 and SAFHS2, and a third phase began in 2002. The current

study focuses on SAFHS1, but the SAFHS abbreviation will be used in the ensuing.

In general, genetic epidemiology studies require data on: 1) pedigree structure, 2)

phenotypes, 3) covariates and 4) genotypes. A description of the data is given just below.

The analytical methods are described in detail in the next two chapters. Detailed

descriptions of the study design and protocols are reported in Mitchell et al., (1996a) and

MacCluer et al. (1999). The current study focused on carrying out quantitative genetic

analyses (Lange, 1997; Thomas, 2004) as opposed to linkage analyses. Consequently,

genotype data were not required, as quantitative genetic analyses minimally require

pedigree structure and trait data.

Participants in the SAFHS were recruited from low-income barrios of greater than

90% Mexican American residency. These barrios were identified by reference to

published socioeconomic and demographic profiles of the neighborhoods of San Antonio.

The distribution of the sample population in San Antonio is shown in Figure 21.

62

SAFHSpopulation

Loop 410 IH 35

IH 10

IH 90

Figure 21. SAFHS recruitment area. The inner city of San Antonio is roughly encircled by Loop 410 (in yellow). The SAFHS population is located in the gridded area. The right boundary is formed by Interstate Highways (IH) 10 and 35.

Sampling Design, Pedigree Structure and Basic Demographics

Probands for the SAFHS were chosen from among individuals of 40 to 60 years

of age who reside in identified low-income barrios. Extended families were identified

through probands chosen because they have at least six living, first-degree relatives (i.e.,

siblings and/or age-eligible offspring) (Fig. 22). The same set of relatives of the

proband’s spouse were also recruited. The subset of the SAFHS for the current study

consists of 1,047 participants from 48 families. The mean pedigree size is 29 individuals

per family and the pedigree size ranges from 3 to 76 individuals per family. The numbers

of relationship types are reported in Table 1. The mean age is 39.5 years and the range is

from 15.5 to 94.2 years of age. There are 404 males and 643 females.

63

San Anto nio Family He art S tudy

Figure 22. Schematic pedigree structure for the typical extended family unit in the SAFHS. The arrow indicates the proband. First-, second- and third-degree relatives are in turqoise, yellow and red, respectively. Courtesy of Dr. J. W. MacCluer.

Table 1. Numbers of Relative Pairs in the SAFHS Relationship Type Number Parent-offspring 1788 Sibs 1337 Half-sibs 186 Grandparent-grandchild 1598 Great grandparent-grandchild 784 Avuncular 2686 Grand avuncular 978 Half avuncular 431 First cousins 2738 First cousins once removed 2633 Second cousins 672 Other 1172 Total Relative Pairs 17003

64

Phenotypes

All hormonal phenotypes were measured in the physiology laboratory of Dr. John

Blangero at the SFBR Department of Genetics. Circulating levels of IGF-I (ng/ml) were

measured using an IGF-I immunoradiometric assay (IRA) kit (Nichols Institute

Diagnostics, San Juan Capistrano CA). IGFBP-1 and IGFBP-3 levels (ng/ml) were

measured using IRA kits specific to the binding protein (Diagnostic Systems

Laboratories, Inc.). It is commonplace in the literature to also analyze the molar ratio of

IGF-I to IGFBP-3 (because IGFBP-3 is the main binding protein in circulation; Juul et

al., 1994, 1995). To compute the molar ratio, the molar masses of 7,649 daltons for IGF-

I and 28.5 kilodaltons for IGFBP-3 were used (Jones and Clemmons, 1995). The

resultant trait is referred to as Ratio3 for brevity. Body mass index (BMI) is commonly

used as a covariate (see below) for traits related to growth and metabolism. BMI was

computed as the ratio of weight (Kg) to height squared (m2), where weight and height

measurements were taken during the participant’s clinic visit.

Covariates

The covariates data were obtained from participant responses to the

questionnaires and interviews. These data are on age, sex, medical history, reproductive

history, smoking habits, dietary habits (based on a food frequency questionnaire), alcohol

consumption, physical activity levels (based on a modified Stanford 7-Day Physical

Activity Recall Instrument), and acculturation and socioeconomic status. Also, any of

the phenotypes may serve as covariates. The covariates in all the models were screened

for significance. The significant covariates in all the models were some combination of

age, sex, age2, sex × age, and BMI.

65

Descriptive Statistics, Transformations, and Treatment of Outliers

Generally, the raw data were significantly kurtotic and skewed and thus in

violation of the assumptions of multivariate normality and additivity (Table 2; Figs. 23-

24; see the following chapter on these assumptions). Beaty et al. (1985) demonstrated

that significant kurtosis has an adverse influence on downstream statistical inferences

derived from variance components models with more than two variance components,

which is the case for the genotype × age interaction model. Inducing univariate

normality is a reasonable first step towards satisfying multivariate normality (Looney,

1995). To this end, two remedies were sequentially employed. For all analyses, the data

were first subjected to a logarithmic transformation, which, as Wright (1968: ch.10-11)

has shown, is often sufficient to achieve normality and, as Freeman (1985) noted, is also

sufficient to induce additivity (see also just below). Following logarithmic

transformation, outliers were removed at ± 4 standard deviations from their respective

means (cf. the recommendations by Freeman, 1985). Few outliers were removed for all

traits (< 0.5 % of their respective total sample sizes). These remedies rendered the

derived traits sufficiently normally distributed as confirmed each time by inspection of

the resultant distributional properties (Table 3; Figs. 25-26).

It should be noted that using a transformation to conform to the assumptions of

normality and additivity, as opposed to seeking a variance-stabilizing transformation that

would induce constant variance, is in no way inconsistent with modeling (co)variance

heterogeneity (see the next chapter). The assumptions of normality, additivity, and

constant variance involve separate but related issues, as has been clearly delineated in

seminal works on the use of transformations in data analysis (Bartlett and Kendall, 1946;

66

Bartlett, 1947; Tukey, 1957; Box and Cox, 1964), although transformations may often

simultaneously achieve a close approximation to all three assumptions (Bartlett, 1947;

Tukey, 1957; but see Sampford, 1964). As Box and Cox (1964) pointed out in the reply

section of their article, the assumption of an underlying distribution is the logical starting

point for any parametric analysis. Subsequent to this observation, and in the context of

interaction analyses, whether or not one seeks transformations specifically to conform to

the assumptions of additivity and constant variance will largely depend on one’s

definition of statistical interaction. Thus, when Cox (1984) defined interaction as

“inconstancy of variance”, he suggested that as a first-check a variance-stabilizing

transformation should be employed, if such exists, and, similarly, when Freeman (1985)

defined interaction as nonadditivity, he suggested, again as a first-check, that a

transformation that induces additivity, if such exists, should be employed. The

assumptions of normality and additivity are maintained in the present analyses, but, for

the reasons detailed in the following chapter, (co)variance heterogeneity will be modeled.

This procedure of modeling G × E interaction is consistent with standard practice in

statistical modeling in that it represents a slightly more complex model that is firmly

predicated on a simpler model. The idea here is to incrementally increase agreement with

reality. Further, given that the logarithmic transformation may often induce close

agreement with the above three assumptions, the present search for interactions is rather

conservative.

67

Table 2. Descriptive Statistics of Raw Data

Log Trait

Mean Variance Kurtosis Skewness N

IGF-I

147.605209 14022.30 8.92850 2.29894 1001

IGFBP-1

34.36576 978.57114 13.82813 2.56707 955

IGFBP-3

3289.36581 3042352 8.33261 2.17906 1005

Ratio3

0.000501 0.000163 931.3702 30.49015 935

Table 3. Descriptive Statistics of Log-Transformed Data

Log Trait

Mean Variance Kurtosis Skewness N

IGF-I

2.04932 0.11208 0.81848 -0.38833 1001

IGFBP-1

1.37076 0.16360 0.23321 -0.43841 955

IGFBP-3

3.46626 0.04382 0.42641 -0.00125 1005

Ratio3

-5.35965 1.25415 0.78329 -0.14718 935

68

Raw IGF-I Histogram

0

50

100

150

200

250

5.68 401.1087097 796.5374194

Measurement

Freq

uenc

y

Raw IGFBP-1 Histogram

0

50

100

150

200

250

0.612 128.2853333 255.9586667

Measurement

Freq

uenc

y

Figure 23. Histograms of raw IGF-I and IGFBP-1 data.

69

Raw IGFBP-3 Histogram

020406080

100120140160180200

662.0945301 6036.487771 11410.88101

Measurement

Freq

uenc

y

Raw Ratio3 Histogram

0100200300400500600700800900

1000

4.02708E-10 0.143242348 0.286484696

Measurement

Freq

uenc

y

Figure 24. Histograms of raw IGFBP-3 and Ratio3 data.

70

Log IGF-I Histogram

0

20

40

60

80

100

120

0.75435 1.568669355 2.38298871Measurement

Freq

uenc

y

Log IGFBP-1 Histogram

0102030405060708090

100

-0.21325 0.797224667 1.807699333

Measurement

Freq

uenc

y

Figure 25. Histograms of log transformed IGF-I and IGFBP-1 data.

71

Log IGFBP-3 Histogram

0

20

40

60

80

100

120

2.82092 3.309873548 3.798827097

Measurement

Freq

uenc

y

Log Ratio3 Histogram

0

20

40

60

80

100

120

140

-9.39501 -6.099846333 -2.804682667

Measurement

Freq

uenc

y

Figure 26. Histograms of log transformed IGFBP-3 and Ratio3 data.

72

Chapter 6

Methods II: The Multivariate Mixed Effects Linear and Polygenic Models

As discussed in chapter 2, a linear model of physiological function or of

physiological damage along the life span is important in a number of the more general

theories of senescence at the proximate level. The statistical models to be employed in

this dissertation research are a special class of the multivariate linear model known as

variance components models (Searle et al., 1992; Hopper, 1993; Schork, 1993), which

ultimately derive from Fisher (1918). This chapter is divided into two sections. The first

section discusses the basic statistical genetic model whereas the second section discusses

extensions of the basic model to incorporate G × E interaction.

The variance components approach assumes that a phenotype vector of

individuals in a given pedigree, denoted by y , follows a multivariate normal (MVN)

distribution. In matrix notation, the MVN is given as:

( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ −′−−π= −−− μyΣμyΣy 1212N

21exp2f . Eq. 80

Stylized for the present context, the parameters μ and Σ are respectively the vector of

phenotype means and the variance-covariance matrix (usually just referred to as the

covariance matrix) of the phenotype variances and covariances for a single pedigree, π is

the mathematical constant (= 3.14…), N is the number of individuals in the pedigree, Σ

is the determinant of Σ , the prime indicates matrix or vector transpose, and 1−Σ is the

inverse matrix of Σ . For the MVN in general, we have: [ ] μy =E . A better model of

[ ]yE that takes into account the effects of covariates is formulated as follows:

[ ] Xβy =E , Eq. 81

73

where X is an incidence matrix augmented by a column of 1’s, and β is a vector of the

grand trait mean, μ (so that μ=0β ), and covariate effects, n1 β,...,β (Searle et al., 1992).

On conceiving of the phenotype as being additively determined by random

genetic and environmental effects (i.e., the additivity assumption), the phenotype vector

y may be expressed in terms of a multivariate mixed effects linear model:

egXβy ++= , Eq. 82

where g is a vector of random genetic effects, and e is a vector of random

environmental effects. Moreover, g and e are distributed as mutually independent

MVNs, with [ ] [ ] 0eg == EE , where 0 is the null vector (Searle et al., 1992). By these

assumptions, Equation 82 implies that the variability in y , given by the covariance

matrix Σ , is related to the variances in the random effects. Hence, our interest should

now lie in modeling the components of Σ , under the model of Equation 82.

Equation 82 is called a mixed effects model because it is comprised of fixed

effects in the component modeled by “ Xβ ” and of random effects in the component

modeled by “ eg + ”. Searle et al. (1992) discuss the history and meanings of fixed,

random, and mixed effects models (see also Rao, 1997). Equation 82 has a

complementary interpretation in that it is understood to be a probabilistic model of the

population-level behavior of a measurement of interest, which in the context of statistical

genetics is usually a phenotype. As a probabilistic model, it is comprised of deterministic

and stochastic components (see the discussion by Wackerly et al., 1996: 476-479). In the

present case, the deterministic component of the model is given by Equation 81, which

appears in the right hand side of Equation 82. Equation 81 is deterministic in that it

determines the values to be assigned in a precise, specified manner and, by itself, does

74

not allow for random error or stochasticity. To this extent, the function specifies law-like

behavior. However, it has long been known that population phenomena cannot be fully

explained by a deterministic function. That is, there is always stochastic behavior about

the law-like process implied by a candidate deterministic model. Accounting for this

stochastic behavior constitutes the second component of all probabilistic models. It

would be left to Fisher (1918) to develop a measure of the stochasticity about the

deterministic function. This measure he called the variance:

It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance of the normal population to which it refers, and we may now ascribe to the constituent causes fractions or percentages of the total variance which they together produce (emphasis mine). Fisher (1918: 399)

Moreover, Fisher (1918) is credited with the development of the linear model (see Searle

et al., 1992; Rao, 1997) and, as can be perceived from the quote, his main interest at least

in that paper lay in modeling the variance components.

On assuming that dominance and epistatic genetic effects are negligible, pair-wise

comparisons of individual phenotype, denoted by y (the scalar values of y ), define the

elements of the covariance matrix as (Lange et al., 1976; Lange, 1997):

( )⎪⎩

⎪⎨⎧

=δ≠∀σ

=δ==∀σ+σ=σ=δσ+σ=

,0 ,zx ;φ2

,1 ,1φ2 ,zx ;φ2y,yCov

xz2gxz

xzxz2e

2g

2y

xz2e

2gxzzx Eq. 83

where x and z index individuals, xzφ2 gives the expected coefficient of relationship

(where xzφ is defined below), ∑=

σ=σn

1j

2gj

2g is the additive genetic variance summed across

n loci, 2eσ is the environmental variance, and xzδ is defined as 1 when individuals x and z

are the same and 0 otherwise. For pedigrees, the matrix formulation of Equation 83 is:

75

2e

2g2 σ+σ= IΦΣ , Eq. 84

where Φ is the kinship matrix of the pedigree and I is the identity matrix. From Feller

(1957: 215-216, 221-222), random variables are mutually independent if and only if:

( ) 0e,gCov = , Eq. 85

whence the use of xzδ and I in the scalar and matrix formulations, respectively. The

model of Equations 83 and 84 will be referred to as the polygenic model, under the scalar

and matrix formulations, respectively.

For pair-wise comparisons of any two relatives in a pedigree, the expected kinship

coefficient over the genome (Malécot, 1969), denoted by xzφ , is defined as:

( )[ ]j2j1xz 2E2

1φ κ+κ= , Eq. 86

where the ijκ are coefficients giving the jth locus-specific probability that a given pair of

relatives share i alleles identical by descent (IBD) (Cotterman, 1940). Examples of

genome-wide expected probabilities for sharing 0, 1, and 2 alleles IBD, denoted by 0κ ,

1κ , and 2κ , respectively, for typical pair-wise relationships in an extended pedigree are

presented in Table 4 (note that the subscript j has been dropped). For all loci, the iκ

must satisfy the following restriction:

∑=

=κ2

0ii 1 , Eq. 87

which states that the allele sharing probabilities sum to 1. Discussions of the theory

underlying the computation of the elements in a kinship matrix, Φ , for a given pedigree

can be found in Thompson (1986, 2000), Lange (1997), and Thomas (2004).

76

Table 4. Genome-wide expectations for alleles identical by descent (IBD)Pair-Wise Relationship 0κ 1κ 2κ xzφ MZ twins 0 0 1 0.5 Parent-offspring 0 1 0 0.25 Full-sib 0.25 0.5 0.25 0.25 Half-sib-+-first-cousin 0.375 0.5 0.125 0.1875 Half-sib 0.5 0.5 0 0.125 Grandparent-grandchild 0.5 0.5 0 0.125 Avuncular 0.5 0.5 0 0.125 First-cousin 0.75 0.25 0 0.0625 Half-avuncular 0.75 0.25 0 0.0625 Half-first-cousins 0.875 0.125 0 0.003125 Unrelated 1 0 0 0

It cannot be overemphasized that the underlying assumption of multivariate

normality is justified (see the discussion in Lynch and Walsh, 1998: 26-27). The

fundamental importance of the MVN is a direct consequence of the much-celebrated

Central Limit Theorem. From Cramér (1946: 213-218), the Central Limit Theorem holds

that the sum:

n21 ... ξ++ξ+ξ=ξ , Eq. 88

of n independent random variables, denoted by iξ ( )n,...,2,1i = , is approximately

distributed as a normal distribution as n becomes large and the approximation becomes

increasingly better as ∞→n . The theorem has been proven to hold in regard to the

MVN in general (Cramér, 1946: 316-317; 1970: ch. 10; Feller, 1957: 252-259). Under

restrictive conditions, Lange (1978) proved that the MVN Central Limit Theorem holds

for quantitative traits that are distributed in human pedigrees. The Central Limit

Theorem can be seen as being related to another important but perhaps lesser-known

theorem on the addition of independent, normally distributed random variables, which

77

can be called the Addition Theorem (sensu Cramér, 1946: 212-213, 1970: chs. 5-6). The

Addition Theorem holds that the sum:

...21 +η+η=η , Eq. 89

of any number of normally distributed random variables, denoted by iη ( ),...2,1i = , is

itself normally distributed (Cramér, 1946: 212). Note that the Addition Theorem holds

for any number of normally distributed random variables whereas the Central Limit

Theorem requires n to become large, which implies that the normal approximation may

not hold for small n. The Addition Theorem is important not merely for the distinction

just made but also because of its implications. In particular, Cramér (1946: 213) noted

that the Addition Theorem implies that linear functions of normally distributed random

variables are also normally distributed and, conversely, that if a linear function of random

variables is normally distributed, then its components are also normally distributed. It

was further noted by Cramér (1946: 316; cf. 1970: ch. 10) that the Addition Theorem

holds for the MVN as well. Taken together, these two theorems put the multivariate

mixed linear and polygenic models on strong theoretical grounds. Firstly, the Central

Limit Theorem underwrites the fundamental assumption that phenotypes are MVN

distributed. Secondly, the Addition Theorem underwrites the notion that MVN

phenotypes may be expressed as a linear function, where its components are also MVN.

Methods II: Theory and Model of Genotype × Environment Interaction

It will be convenient to review the mathematical definitions and relations of the

terms variance, standard deviation, covariance, and correlation coefficient because the

genotype × environment (G × E) interaction model is most easily derived from said

definitions. The following discussion will be based on material that can be found in most

78

textbooks on statistics and probability. Highly recommended sources include Cramér

(1946), Feller (1957), Parzen (1960) and Anderson (1984). Wackerly et al. (1996) and

Ross (2003) provide more current treatments. The following definitions will be made

with respect to the random variables Y and Z.

The definition of the variance of Y is:

( ) [ ]( )[ ] ( )[ ] [ ][ ] [ ] [ ][ ] [ ] [ ]( ) . YEYEYE

2YEYE2YE

Y2YEYEYEYEYVar

222Y

2

2Y

2Y

22YY

2

2YY

22Y

22Y

−=μ−=

μ+μ−=μ+μ−=

μ+μ−=μ−=−=σ≡

Eq. 90

The definition of the covariance of Y and Z is:

( ) [ ]( ) [ ]( )[ ]( ) [ ] [ ] [ ] [ ][ ]( )[ ]

[ ] [ ] [ ][ ][ ] [ ] [ ] [ ] . ZE YEYZEYZE YZE

ZE YEYZE ZYYZE

ZE YEZ YEZE YYZE ZEZYEYEZ,YCov

ZY

ZYZYZY

ZYYZ

ZYYZ

Z,Y

−=μμ−=μμ+μμ−μμ−=

μμ+μ−μ−=μμ+μ−μ−=+−−=

−−=σ≡

Eq. 91

A useful identity follows from these definitions. For the case of random variable Y say:

( ) [ ]( ) [ ]( )[ ][ ]( )[ ] ( ) . YVarYEYE

YEYYEYEY,YCov2 =−=

−−= Eq. 92

That is, the covariance of a random variable with itself is simply the variance. The

standard deviation is defined as the positive square root of the variance:

Y2Y σ+=σ . Eq. 93

The correlation coefficient is defined as:

ZY

Z,Y

2Z

2Y

Z,Y

2Z

2Y

Z,YZ,Y

σσ

σ=

σσ

σ=

σσ

σ=ρ , Eq. 94

from whence, we obtain the relations:

79

12Y

2Y

YY

Y,YY,Y =

σ

σ=

σσ

σ=ρ , Eq. 95

Y,Z

YZ

Y,Z

ZY

Z,YZ,Y ρ=

σσ

σ=

σσ

σ=ρ , Eq. 96

and an alternative expression for the covariance of Y and Z:

ZYZ,YZ,Y σσρ=σ . Eq. 97

Further, the correlation coefficient takes values in the closed interval [ ]1,1 +− . That is,

11 Z,Y +≤ρ≤− . Eq. 98

With slight modifications to Anderson’s (1984: 22) definition of a bivariate normal

covariance matrix, the above relations can be compactly illustrated as follows:

Y,ZZ,Y

Z,ZY,Y

2ZY,Z

Z,Y2Y

ZZZ,ZYZY,Z

ZYZ,YYYY,Y 1 ;

ρ=ρ

=ρ=ρ∀

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

σσ

σσ=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

σσρσσρ

σσρσσρ=Σ , Eq. 99

where the modifications only involve being explicit about the correlation coefficients.

Finally, there is a special relation, derivable from the above definitions, on the variance

of the difference of two random variables:

( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]

⇓⇓⇓⇓⇓⇓

−+−−−−=

−−−=−

ZYEZYEZY2ZYE

ZYEZYEZYVar22

2

, Eq. 100a

where ⇓ , ⇓⇓ , and ⇓⇓⇓ indicate terms that will be taken separately.

⇓

( )[ ] ( )( )[ ] [ ][ ] [ ] [ ] , ZEYZE2YE

ZYZ2YEZYZYEZYE22

222

+−=−−=−−=−

⇓⇓

80

( ) [ ] ( ) ( ) ( )[ ] ( )( )[ ]

[ ][ ] [ ] [ ] [ ]

, 242

ZE2ZE2YE2YE2 ZZYYE2

ZY2E ZEYEZYE2ZYEZYE2

2ZZY

2Y

ZYZY

ZYZY

ZY

μ−μμ+μ−=

μ−μ+μ+μ−=μ+μ−μ−μ−=

μ−μ−−=−−−=−−−

⇓⇓⇓

( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ] ( ) ( )[ ]

[ ], 2

2E

E ZEYE ZEYEEZYEZYEE

2ZZY

2Y

2ZZY

2Y

ZYZY

μ+μμ−μ=

μ+μμ−μ=

μ−μμ−μ=−−=−−

where summing the terms under ⇓⇓ and ⇓⇓⇓ yields the quantity: 2ZZY

2Y 2 μ−μμ+μ− , to

which we add the terms under ⇓ to complete the main expectation:

( )[ ] ( )[ ] ( )[ ]

( ) ( ) ( ) . ZY,2Cov ZVar YVar

μμYZE2μZEμYE ZY2Z

22Y

2

−+↓↓↓

−−−+−

We have just shown the following result:

( ) ( ) ( ) ( ), 2

Z,YCov2ZVarYVarZYVar

ZYZ,Y2Z

2Y σσρ−σ+σ=

−+=− Eq. 100b

With the previous section and the above discussion, we now have all that is needed to

proceed with the mathematical theory of G × E interaction.

A variance component model designed to detect G × E interaction can be used to

address the potentially dynamic gene expression network (GEN) that is reflective of the

behavior of the IGF-I axis along the age continuum. The foundations for this approach

trace back to Haldane’s (1946) early ideas on the importance of G × E interaction for the

determination of quantitative phenotypes, and to Falconer’s (1952) idea of treating trait

states in different environments as different traits. An operational definition of G × E

81

interaction may be taken as the environmental dependency or sensitivity of genotype

expression in the process of phenotype determination (Haldane, 1946; Falconer, 1952,

1960a&b, 1989, 1990; Lynch and Walsh, 1998). In this connection, it is common to

speak of the trait response to a change in environment (Falconer, 1989; Lynch and Walsh,

1998). To motivate the theory, consider the simplest case of a trait measured in two

different environments. In this case, the additive genetic variance in trait response to the

change in environment, denoted as 2gΔσ , can be written as (Robertson, 1959; Blangero,

1993; Wu, 1998):

( )⎪⎩

⎪⎨

⎧

σ=σ=σ∀ρ−σ

σ≠σ∀σσρ−σ+σ

=σ Δ

, ; 12

, ; 2

2g

22g

21gG

2g

22g

21g2g1gG

22g

21g

2g Eq. 101

where 21gσ and 2

2gσ are the additive genetic variances of the trait in environments 1 and

2, and Gρ is the genetic correlation of the traits between environments. Incidentally,

Robertson (1959) derived his version of Equation 101 by taking expectations as in

Equation 100a&b to get a slightly more complicated equation, which includes Equation

101. However, if we start with Falconer’s (1952) idea of treating the trait states in

different environments as different traits, then Equation 101 is seen to be merely the

statistical genetic version of Equation 100a&b. That is, on treating trait states in two

different environments as two different random variables, the above formulation follows

directly from the definition of the variance of the difference of two random variables.

There is no G × E interaction when 02g =σ Δ (Robertson, 1959; Blangero, 1993).

Nonzero G × E interaction is comprised of two components, a component due to

heteroscedasticity (also known as variance heterogeneity or unstable variance) and

82

another component due to the genetic correlation (Robertson, 1959; Dickerson, 1962;

Yamada, 1962; Eisen and Saxton, 1983; Yamada et al., 1988; Falconer, 1990; Itoh and

Yamada, 1990; Blangero, 1993; Wu, 1998). A useful theorem that makes the preceding

statements a little more rigorous will now be proven. Equation 101 specifies two main

outcomes, one holding under heteroscedasticity ( 22g

21g σ≠σ ) and the other under

homoscedasticity ( 2g

22g

21g σ=σ=σ ). These outcomes each give in turn yet three more,

general outcomes, under conditions specified for the genetic correlation that are

representative of full positive or negative correlation and of zero correlation. Assuming

22g

21g σ≠σ , Equation 101 gives:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=ρ∀σσ+σ+σ

=ρ∀σ+σ

=ρ∀σσ−σ+σ

=σσρ−σ+σ=σ Δ

.1 ; 2

.0 ;

.1 ; 2

2

G2g1g2

2g21g

G2

2g21g

G2g1g2

2g21g

2g1gG2

2g21g

2g Eq. 102

Assuming 2g

22g

21g σ=σ=σ , Equation 101 gives:

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=ρ∀σ

=ρ∀σ

=ρ∀

=ρ−σ=σ Δ

.1 ; 4

.0 ; 2

.1 ; 0

12

G2g

G2g

G

G2g

2g Eq. 103

Let nonzero G × E interaction be defined as 02g ≠σ Δ . The cases will be discussed

in relation to the conditions giving rise to 02g ≠σ Δ , and will be taken in descending order

from top to bottom for Equation 102 and then for Equation 103. With little loss in

generality, it will be assumed that 21gσ , 2

2gσ , and 2gσ are non-zero for we shall never be

83

interested in traits that show no variation. For the top case of Equation 102, setting

02g =σ Δ gives a quadratic equation in the variables, namely:

( )( )2g1g

2g1g2g1g

22g2g1g

21g

20

σ=σ⇒

σ−σσ−σ=

σ+σσ−σ=

. Eq. 104

However, 22g

21g σ≠σ by assumption and so Equation 104 amounts to a contradiction.

Therefore, for the top most case, 02g ≠σ Δ even when 1G =ρ . The middle case of

Equation 102 arises for completely uncorrelated random variables. To preclude

confusion, recall from Feller (1957: 215-216, 221-222) that independence of random

variables implies 0 covariance and, of course, 0 correlation. However, the converse, as

noted by Feller (1957: 222), is not true. That is, 0 correlation can say nothing about

whether or not the random variables of interest are independent. At any rate, the

important point here is that 02g ≠σ Δ by the definition of the variance as the expected

squared deviations from the mean (Eq. 90). For the bottom case of Equation 102, that

02g ≠σ Δ follows immediately from the definitions of the variance and standard deviation

(Eqs. 90 and 93), for a sum of positive terms is itself positive. Equation 103 is easier to

interpret. As Gρ goes from +1 to –1, 2gΔσ goes from 0 to 2

g4σ . Clearly, 02g =σ Δ for

1G =ρ . Therefore, whenever 1G <ρ , we can conclude that 02g ≠σ Δ (not including the

trivial case for 02g =σ ). The following theorem has just been proven. There is no G × E

interaction, i.e., 02g =σ Δ , if and only if 2

g2

2g21g σ=σ=σ and 1G =ρ are simultaneously

satisfied. In all other non-trivial cases, 02g ≠σ Δ . The condition of 02

g =σ Δ is now to be

84

understood as a null hypothesis for G × E interaction. It follows from the theorem that it

is sufficient to reject 2g

22g

21g σ=σ=σ , 1G =ρ , or both in order to reject the null hypothesis

that 02g =σ Δ . That is, by the simultaneity condition as required under the theorem,

rejection of just one of the stated conditions amounts to a rejection of 02g =σ Δ . An

alternative proof of the theorem is provided in Appendix A. Moreover, it can be seen that

under homoscedasticity or heteroscedasticity, as the correlation continuum is traversed

from complete positive correlation to zero correlation to complete negative correlation,

the magnitude of 2gΔσ increases monotonically to its maximum (Fig. 27).

0

1

2

3

4

5

6

0 0.5 1 1.5 2Genetic Correlation + 1

Var

ianc

e

heteroscedasticity homoscedasticity

Figure 27. G × E interaction variance under heteroscedasticity and homoscedasticity. The interaction variances were computed from Equations 102 and 103 (see text). Under heteroscedasticity, the trait variances in environments 1 and 2 were assigned the values 1 and 2, respectively. Under homoscedasticity, the trait variance was assigned the value 1. The lower limit of G × E interaction is clearly set by the homoscedasticity case whereas the upper limit is dependent on the magnitude of heteroscedasticity.

85

The theorem on G × E interaction can be generalized to multiple environments in

a straightforward manner using matrix algebra. Let there be n environments with n

corresponding trait states. The additive genetic covariance matrix for the n trait states, of

dimensions nn × , is given as:

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

σσσρσσρσσρ

σσρσσσρσσρ

σσρσσρσσσρ

σσρσσρσσρσ

=

−−

−−−−−−−

−−

−−

2n1nn1n,n2n2,n1n1,n

n1nn,1n2

1n21n2,1n11n1,1n

n2n,21n21n,222121,2

n1n,11n11n,1212,121

L

L

MMOMM

L

L

G , Eq. 105

where the elements are understood to be genetic parameters and subscripts indicate the

environment. By the above theorem, G × E interaction can be evaluated for the set of

hypotheses on the additive genetic variances across environments:

n , . . . j,i, ; 2j

2i ∀σ=σ ,

and for the set of hypotheses on the genetic correlations across environments:

n , . . . j,i, ; 1j,i ∀=ρ .

Blangero and colleagues (Blangero et al., 1987, 1988, 1989, 1990a&b; Blangero and

Konigsberg, 1991; Blangero, 1993) developed a similar model for the detection of G × E

interaction under a complex segregation analysis approach. The problem, however, is

that there is an “explosion” in the number of parameters that need to be estimated (Meyer

and Hill, 1997; Pletcher and Geyer, 1999; Meyer, 2001). For a covariance matrix of

nn × dimensions, the number of parameters, denoted by θN , is given as:

86

( )2

1nnNθ

+= . Eq. 106

For 5 environments say, there are 15 parameters in the additive genetic covariance matrix

alone. For the polygenic model, this is in addition to the number of environmental

variance parameters, which is given by n. Further, for the full multivariate linear mixed

model, the sum nNθ + is added to the numbers of parameters for the environment-

specific means, which is also given by n, and for the environment-specific covariate

effects estimates, which is given by n times the number of covariates. The simplest full

model with no covariates would still give 25 parameters in all to be estimated. At this

level of model complexity, serious problems arise in maximum likelihood estimation and

the sampling variances of the parameter estimates tend to be prohibitively large (Searle et

al., 1992; Meyer and Hill, 1997; Pletcher and Geyer, 1999; Meyer, 2001). Clearly,

another approach is needed that can circumvent the problems arising from a model

overburdened in parameters. Towards this end, 2gσ and Gρ can be modeled as functions

of the environment of interest, provided the environment is continuous (Blangero, 1993;

Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002).

This amounts to a generalization of the above ideas on G × E interaction from discrete to

continuous environments (Kirkpatrick et al., 1994).

For the present study, the continuous environment of interest is the age

continuum. To model the null hypothesis of 02g =σ Δ , 2

gσ and Gρ are parameterized as

continuous functions of age:

( )[ ] ℜ⊂=∈∀−γ+α=σ T ;ageageT p ; agepexp .max.mingg2g K ; Eq. 107

( ) Tqp, ; qpexpG ∈∀−λ−=ρ , Eq. 108

87

where 2gσ is modeled as an exponential function to ensure a positive variance (Blangero,

1993; Pletcher and Geyer, 1999), p denotes the age of an individual which belongs to the

index set of ages, denoted by T , T ranges from the minimum age, .minage , to the

maximum age, .maxage , in the sample population and is a positive, finite subset of the real

line, and the average age is that for the sample population; Gρ is modeled as the

correlation function of an Ornstein-Uhlenbeck stochastic process (see Appendix B), p

and q are any two ages in T and α , γ , and λ are parameters to be estimated.

On taking the natural logarithm of the additive genetic variance function, we will

have a log-linear function in the variance:

( )agepln gg2g −γ+α=σ , Eq. 109

which is just the equation of a line on the logarithmic scale. Thus, genotype × age

interaction obtains for a nonzero slope on the logarithmic scale of the additive genetic

variance function; that is, for 0g ≠γ . Similarly, genotype × age interaction obtains for

0≠λ in the genetic correlation function, where the null hypothesis is satisfied for 0=λ

because 1e0 = . Taking the natural logarithm of the genetic correlation function also

gives the equation of a line on the logarithmic scale:

qpln G −λ−=ρ . Eq. 110

As for the additive genetic variance function, genotype × age interaction obtains for a

non-zero slope on the logarithmic scale of the genetic correlation. The environmental

variance component of the response is modeled in similar fashion to 2gσ but there can be

88

no corresponding environmental correlation term because of the assumption that g and e

are distributed as mutually independent MVNs.

There is one more component needed to build the full genotype × age interaction

model. To allow for a covariance formulation, and in keeping with the definition of the

standard deviation (Eq. 93), let:

( )[ ] eg, ; agepexp 21

xx2

x =υ∀−γ+α=σ=σ υυυυ , Eq. 111

where this formulation holds for any individual, x , in the sample, but when taking

covariances this can be indicated with individual specific subscripts, as in x and z for

the generic case. The full genotype × age interaction model is a decomposition of the

total phenotypic variance similar to Equation 83 and so the variance and covariance

components are similarly subscripted. Taking Equations 107, 108, and 111 together and

recalling the fundamental relations detailed earlier in this section, the phenotypic

covariance may be written as:

( ) ( ) ( )[ ] ( )[ ]

( )[ ] ( )[ ] , ageqexp agepexp

ageqexp agepexp qpexpφ2y,yCov

2

1

zee2

1

xeexz

2

1

zgg2

1

xggzxxzzx

−γ+α−γ+αδ+

−γ+α−γ+α−λ−=

Eq. 112

where all the previous definitions hold. Note that because variances can always be

expressed in terms of covariances and covariances can be defined in terms of the

correlation coefficient and standard deviations, we effectively inherit a flexible means for

formulating a variance/covariance relation. This is a cross-sectional model that applies

generally to three types of pairwise comparisons of individuals. In one type, let zx =

while qp = . Equation 112 gives the variances in this situation, in accord with the

polygenic model. In a second type, it may be such that zx ≠ while qp = , and, in a third

89

type, it may be such that zx ≠ while qp ≠ . Note that none of these three types are

longitudinal comparisons, which would be the case where zx = while qp ≠ (i.e., the

same individual is measured at different ages). In the former two cases, where qp = , the

genetic correlation function, written as a function of age differences, cannot play a role in

genotype × age interaction because for this case the function equals 1. For the case

where different individuals of different ages are compared ( zx ≠ while qp ≠ ), the

variance and genetic correlation functions can both contribute to potential genotype × age

interaction. Thus, an optimal data set for the discovery of genotype × age interaction

under the above approach will have large extended pedigrees—this is because the genetic

covariance is still also a function of relatedness—whose constituents are of widely

varying ages. Taking all of these considerations together, we can rewrite Equation 112 to

explicitly cover the three types of conditions just discussed as follows:

( )

( )[ ] ( )[ ]

( )

( )[ ] ( )[ ] ⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=δ≠∀

−γ+α−γ+α

×−λ−

=δ==∀

−γ+α+−γ+α

=

. 0,zx

; ageqexp agepexp

qpexpφ2

. 1,1φ2,zx

; agepexpagepexp

y,yCov

xz

21

zgg21

xgg

zxxz

xzxz

xeexgg

zx Eq. 113

The bottom form on the right hand side covers both cases where different individuals are

of the same age or of different ages. Note that the assumption that g and e are

distributed as mutually independent MVNs is still in operation (i.e., there is no

environmental covariance term). Using the properties of the exponential function,

Equation 113 can be written so that the genetic components are represented in one

exponential function for the bottom form.

90

( )

( )[ ] ( )[ ]

( )⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

=δ≠∀

⎥⎦

⎤⎢⎣

⎡ −λ−−+γ

+α

=δ==∀

−γ+α+−γ+α

=

. 0,zx

; qpage2qp2

expφ2

. 1,1φ2,zx

; agepexpagepexp

y,yCov

xz

zxzxg

gxz

xzxz

xeexgg

zx Eq. 114

Equations 112-114 are completely analogous to Equation 83.

To begin to write the matrix model, we may use the equivalence relations

regarding age and individual identity to determine the elements of the matrix specifying

the two genetic outcomes in Equation 114 (sensu Lange, 1986). It is significant that the

equivalence relations regarding age and individual identity specify mutually exclusive

conditions that exhaust all possibilities in a cross-sectional design. Moreover, because

there is only one outcome with respect to the environmental component, it is as if the

variance component, 2eσ , is merely reparameterized (in fact, all the variance components

are reparameterized). Let there be a new matrix, ija=A . The elements in this new

matrix are specified as follows:

( )[ ]

( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≠∀⎥⎦

⎤⎢⎣

⎡−λ−−+

γ+α

=∀−γ+α

=

. ji ; qpage2qp2

exp

. ji ; agepexp

a

jijig

g

igg

ij Eq. 115

Also, let there be a diagonal matrix ijb=B , where the diagonal elements are given by:

( )[ ]agepexp xee −γ+α . All together, the matrix formulation for the genotype × age

interaction model may be given as follows:

91

, 2 BAΦΣ += o Eq. 116

where o is the Hadamard product operator (Horn and Johnson, 1991: ch. 5). Equation

116 is completely analogous to Equation 84.

That the (co)variance is being modeled as a function of some environmental

variable of interest represents a departure from traditional quantitative genetics, as is now

explained and justified. In their comprehensive discussion of G × E interaction, Lynch

and Walsh (1998: 663) noted that G × E interaction may exist even when 1G =ρ , but then

they suggested that a variance-stabilizing transformation would remove such effects.

However, Bulmer (1980: 25) and Falconer (1989: 296) both pointed out that such

transformations may not always be successful at removing interaction effects. It is

notable in this regard that D. S. Falconer, the founder of the genetic correlation approach,

emphasizes the genetic correlation in diagnosing G × E interaction in all editions of his

widely-used textbook on quantitative genetics (e.g., Falconer, 1989: 322-326; but see

Falconer (1990) for a treatment of variances) and that Robertson (1959), who originally

derived Equation 101, ultimately deferred to Falconer’s method. Intuitively, not

accounting for variance heterogeneity in a model of G × E interaction, when it is known a

priori to have an effect, leads to biased estimates of the genetic correlation and,

consequently, to necessary corrections for this bias (Robertson, 1959; Eisen and Saxton,

1983; Fernando et al., 1984; Yamada et al., 1988; Itoh and Yamada, 1990; Dutilleul and

Potvin, 1995). However, statisticians and some statistical geneticists have pointed out

that modeling (co)variance heterogeneity (Carroll and Rupert, 1982, 1988; Aitkin, 1987;

Davidian and Carroll, 1987; Blangero, 1993; Verbyla, 1993; Denis et al., 1997; Frensham

et al., 1997; Carroll, 2003), as is being done under the genotype × age interaction model,

92

is in many cases more desirable and powerful than the traditional approach of seeking a

variance-stabilizing transformation or a correction to this effect (Bartlett and Kendall,

1946; Bartlett, 1947; Box and Cox, 1964; Cox, 1984). Further, both the variance and

correlation functions can be shown to have a rigorous mathematical foundation in the

theory of stationary Gaussian stochastic processes (Kirkpatrick and Heckman, 1989;

Kirkpatrick and Lofsvold, 1989; Kirkpatrick et al., 1990, 1994; Pletcher and Geyer, 1999;

Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002; Appendix B). Thus, under the

model espoused here, equal weight is accorded to the variance and genetic correlation

functions in the search for genotype × age interaction (but their interpretations will be

different).

93

Chapter 7

Methods III: Likelihood Theory and Maximum Likelihood Estimation

As mentioned in the first section on models in the preceding chapter, Fisher is

credited with the development of the linear model in general and variance components

models in particular. It is remarkable that he (Fisher, 1912, 1922, 1925, 1934a&b, 1935,

1990) is also credited with the development of the theory of likelihood and maximum

likelihood estimation (see Edwards, 1992). These concepts will be of prime importance

in the three sections of this chapter on estimation, inference, and power.

The statistical genetics software SOLAR (Almasy and Blangero, 1998) was used

for all model analyses. In particular, SOLAR employs standard numerical computation

algorithms to compute: 1) the ln-likelihood of a statistical model, 2) the maximum

likelihood estimates of parameters under a model, and 3) the standard errors of the

maximum likelihood estimates. These will be referred to as goals. The underlying

theory is reviewed herein. It should be held in mind that for all three goals, the end result

is a scalar. Thus, SOLAR can be thought of as a fancy calculator that is used to compute

the above scalar values. The three goals will be taken in turn.

In general, the likelihood function for a population sample comprised of z

pedigrees, given a single-parameter, probability model is given as:

( ) ( ) 0c ; yfcdataθLz

1ii >∀= ∏

=

, Eq. 117

where θ and ( )yf denote the parameter and model, respectively; the likelihood of θ

conditional on the data is given by a multiplicative function of ( )yf , which holds up to a

multiplicative constant, 0c > ; and multiplication is carried out across pedigrees, the

94

constituents of which have measurements, y , that are distributed within pedigrees

according to ( )yf . Notice that on taking logarithms we will have, by a property of

logarithms, an additive function:

( ) ( ) 0c ; yflogclogdataθLlogz

1ii >∀+= ∑

=

, Eq. 118

where we may now more conveniently sum across pedigrees to obtain the sample log

likelihood. Keeping these general points in mind, we may now take the case for say a

single pedigree specifically in regard to the multivariate mixed linear and polygenic

models. Under the assumption that the trait of interest is MVN within pedigrees, and

using Equations 82-84 as an example that may be generalized to more complex models,

the likelihood function for a single pedigree of N individuals is given as:

( ) ( ) ( ) 0c ; 2

1exp2 cfc,,,L 1212N2

e2g >∀

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ ′−π=⋅=σσ −−− ΔΣΔΣyXyβ , Eq. 119

where the parameters under the multivariate mixed linear and polygenic models, namely,

β , 2gσ , and 2

eσ , are expressed as the hypothesis that they are proportional to the MVN up

to an arbitrary, multiplicative constant, 0c > , conditional on the vector of trait values, y ,

and the covariates matrix, X , and [ ] XβyyyΔ −=−= E (Blangero et al., 2001).

Traditionally, it is assumed that 1c = (Rohatgi, 1984). Taking natural logarithms yields:

( ) [ ]ΔΣΔΣXyβ 12e

2g ln)2ln(N

2

1,,,Lln −′++π−=σσ . Eq. 120

The ln-likelihood for the population sample comprised of z pedigrees is then computed

by the following additive function:

( ) [ ]∑=

−′++π−=σσz

1ii

1iiii

2e

2gz ln)2ln(N

2

1,,,Lln ΔΣΔΣXyβ . Eq. 121

95

The right hand sides of Equations 120 and 121 each have three scalar terms in the

brackets. The latter two of these terms are perhaps not so clearly seen as scalars. The

second term involves the determinant of a matrix, which is always a scalar. As for the

last term, a row vector post-multiplied by a matrix gives a row vector still, which when

post-multiplied by a column vector gives a scalar.

Parameter estimation is carried out under standard maximum likelihood

estimation procedures (Lange, 1997; Lynch and Walsh, 1998; Thompson, 2000; Thomas,

2004). Let [ ] ′σσ= ,, 2e

2gβθ denote a parameter vector. Maximum likelihood estimation

gives the parameter estimates in θ that make the ln-likelihood function (Equations 120 or

121) a maximum. To this end, there are multivariable generalizations of techniques in

univariable calculus for the identification of local maxima and minima, which are the first

and second derivative tests. According to the univariable method, a local maximum

exists where the function of interest, evaluated at the first derivative set equal to 0, is

concave down, which obtains only when the second derivative is negative in sign. For

the multivariable case, the first requirement is that the vector of first partial derivatives,

called the score vector and denoted by ( )θS , equals 0 (Lange, 1997; Magnus and

Neudecker, 1999). On simplifying the notation for the ln-likelihood function and taking

conditionality as understood, we require that:

( ) ( ) ( ) ( ) 0θθβθθ =

′

⎥⎦

⎤⎢⎣

⎡

σ∂∂

σ∂∂

∂∂= 2

e2g ˆ

ˆLln,ˆ

ˆLln,ˆˆLlnˆS , Eq. 122

where estimates are indicated by a carat. The second requirement involves the Hessian

matrix, denoted by H , which is defined as the matrix of second partial derivatives

96

evaluated at ( ) 0θ =ˆS , and, for a 1n × column vector, is of dimensions nn × (Magnus and

Neudecker, 1999). That is:

( )( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

σ∂σ∂

∂

σ∂σ∂

∂

σ∂∂

∂

σ∂σ∂

∂

σ∂σ∂

∂

σ∂∂

∂

∂σ∂

∂

∂σ∂

∂

∂′∂

∂

=′∂∂

∂= =

2e

2e

2

2e

2g

2

2e

2

2g

2e

2

2g

2g

2

2g

2

2e

2

2g

22

ˆS

2

ˆˆ

ˆLln

ˆˆ

ˆLln

ˆˆ

ˆLln

ˆˆ

ˆLln

ˆˆ

ˆLln

ˆˆ

ˆLln

ˆˆ

ˆLlnˆˆ

ˆLlnˆˆ

ˆLln

ˆˆ

ˆLln

θθ

β

θ

θθ

β

θ

β

θ

β

θ

ββ

θ

θθ

θH 0θ . Eq. 123

In general, for a multivariable function ( )y,xf , a theorem from differential calculus

holds true, provided ( )y,xf is continuous and differentiable (Horn and Johnson, 1985:

167, 392; Widder, 1989: 52-53; Magnus and Neudecker, 1999: 105-106):

( ) ( )xy

y,xf

yx

y,xf 22

∂∂

∂=

∂∂

∂. Eq. 124

The theorem states that the order in which the second partial derivatives of ( )y,xf are

obtained (that is, on differentiating with respect to x and then y or vice versa) is

inconsequential for the two partial derivatives are equal. The theorem generalizes to all

multivariable functions, ( ),...y,xf , and applies to all second partial derivatives.

Therefore, the Hessian matrix obtained from a multivariable scalar function, as in the ln-

likelihood function, is always a symmetric matrix by the theorem since the off-diagonals

are correspondingly equal. That is, for the matrix ijf , ji ij ff = for all i and j . This point

will become relevant below. The second requirement for ( )θLln to be a local maximum

is that H is negative definite, which is defined just below. That is, if H is negative

definite, then ( )θLln is taken to be a local maximum and the values in θ are taken to be

97

the maximum likelihood estimates (MLEs) (Magnus and Neudecker, 1999). Given a

matrix F and a column vector x , F is negative definite if its corresponding quadratic

form, Fxx′ , is negative definite, which holds for (Horn and Johnson, 1985: 396-397):

0xFxx ≠∀<′ ; 0 , Eq. 125

where the end-result is always a scalar quadratic function in the elements in x (recall that

a row vector post-multiplied by a matrix gives a row vector which is post-multiplied by a

column vector to give a scalar). To check if H is in fact negative definite, we can use the

second-order Taylor expansion of ( )θLln about some nearby point in the parameter

space, say θ~ , to obtain (cf. Horn and Johnson, 1985: 391-392; Stengel, 1994: 33-34):

( ) ( ) ( ) ( ) ( ) ( )θθHθθθθθθθ θθ~ˆ~ˆ

2

1 ˆS~ˆ~LlnˆLln ~ˆ −

′−+

′−+= = . Eq. 126

If ( ) ( ) 0~ˆ~ˆ <−′

− θθHθθ , where ( ) ( )θθHθθ ~ˆ~ˆ −′

− is the quadratic form, then H is negative

definite. There are other methods to determine if ( )θLln is a maximum that require ( )θS

and H (Tracy and Dwyer, 1969; Magnus and Neudecker, 1999) but the above method is

sufficient to illustrate the principles involved. Thus far, the principles underlying the

likelihood function and maximum likelihood estimation have been discussed. The

computation of the standard errors of the parameter estimates may now be addressed.

These are derived from the sampling covariance matrix of the parameter estimates, which

in turn is derived from the Fisher information matrix.

The expected Fisher information matrix, denoted by IF , is found by taking the

negative of the expectation of H (Lehmann, 1983: 126; Edwards, 1992: 146; Searle et

al., 1992: 472-474; White, 1994: 94; Shao, 1999: 136):

[ ]HF EI −= . Eq. 127

98

Similarly, the observed Fisher information matrix, denoted by irF , is the negative of H

(Efron and Hinkley, 1978). The reason for taking the negative of [ ]HE or H is that we

are working towards the sampling covariance matrix for θ . Therefore, since a proper

covariance matrix has to be positive semidefinite, one achieves this by simply taking the

negative of [ ]HE or H . Similar to the definition of a negative definite matrix, F is

positive semidefinite if (Horn and Johnson, 1985: 396-397):

n ; 0 ℜ∈∀≥′ xFxx . Eq. 128

That we are working towards the covariance matrix for θ also explains the relevance of

the point made above that H is a symmetric matrix, for a proper covariance matrix, in

addition to being positive semidefinite, must also be symmetric such that i,jj,i σ=σ for

all ji ≠ (Magnus and Neudecker, 1999: 246). Efron and Hinkley (1978) argued for

using irF in favor of IF in statistical inference (see also Skovgaard, 1985; Lindsay and Li,

1997). However, Huzurbazar (1949) showed that for fairly simple likelihood functions

IF and irF are in fact identical (see also Edwards, 1992: 150-151).

The elements of ( )θS and IF under the polygenic model have been derived and

are reported in Blangero et al. (2001). They are reported here with some slight

modifications to notation. Under the polygenic model, the elements of ( )θS are given by:

( ) ( )( ) n, . . . 1, ,0i ; β

ˆ

β

ˆLln 1ni

1

ii

=′′=′′

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂=

∂∂ −− ΔΣXeΔΣXβθ , Eq. 129

( ) ( ) ΔΦΣΣΔΦΣθ

111

2g

Trˆ

ˆLln−−− ′+=

σ∂

∂, Eq. 130

99

( ) ( ) ΔΣΣΔΣθ 111

2e 2

1Tr

21

ˆ

ˆLln −−− ′−−=σ∂

∂, Eq. 131

where ( )nie is an elementary 1n × column vector, with a “1” at the ith position and a “0” at

all other positions, and where the trace operator, ( )⋅Tr , is defined below. For large

samples, the MLEs are themselves MVN distributed, and the expected covariance of the

effects in β and the variance components is 0 (Tracy and Dwyer, 1969; Cox and Reid,

1987; Lange, 1997; Blangero et al., 2001; McCulloch and Searle, 2001):

( ) ( ) ( )eg, ;n , . . . 1, ,0i ; 0

β

ˆLlnE

ˆLlnβ

ˆLlnE 2

i

2

2i

=υ=∀=⎥⎥⎦

⎤

⎢⎢⎣

⎡

σ∂∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

σ∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

υυ

θθθ. Eq. 132

Recall that [ ] XβyyyΔ −=−= E . On taking its expectation, we have:

[ ] [ ] [ ] [ ] [ ] [ ] . EEEEEE 1nx0yyXβyXβyΔ =−=−=−= Eq. 133

Therefore, after evaluating the second partial derivatives, all terms involving [ ]ΔE

vanish. All together, we therefore have:

( ) ( )( ) ( ) n , . . . j, i, β

ˆ

β

ˆ

ββ

ˆLlnE ni

1nj

i

1

jji

2

∀′′=∂∂′

′

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂∂=

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂∂∂− −− XeΣXeβXΣXβθ . Eq. 134

( )eg, ;n i,..., ; 0

ˆβ

ˆLlnE

2i

2

=υ∀=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

σ∂∂

∂−

υ

θ. Eq. 135

( ) ( )ΦΦΣΣθ 11

2g

2g

2

Tr2ˆˆ

ˆLlnE −−=⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

σ∂σ∂

∂− . Eq. 136

( ) ( )11

2e

2e

2

Tr2

1

ˆˆ

ˆLlnE −−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

σ∂σ∂

∂− ΣΣ

θ. Eq. 137

100

( ) ( )11

2e

2g

2

Trˆˆ

ˆLlnE −−=⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

σ∂σ∂

∂− ΦΣΣ

θ. Eq. 138

Equations 129-131 and 134-138 are all scalar-valued functions. The trace of a matrix,

( )⋅Tr , is a special summation operator, which sums the diagonal elements of a matrix.

The outcomes at Equations 129 and 134 and the right most terms of Equations 135 and

136 are ultimately instances of a quadratic form, which we have seen to be a scalar

function. Equations 134-138 fully specify the elements in IF under the polygenic model.

It will be shown in Appendix C how the elements in the score vector and the expected

Fisher information matrix are derived once the ln-likelihood function is known.

Inversion of IF gives the covariance matrix for θ , denoted by θ

Σ ˆ (Lehmann,

1983: 427-430; Edwards, 1992: 159; Searle et al., 1992: 472-474; White, 1994: 94-95):

θΣF ˆ1

I =− , Eq. 139

which can be used to give the standard errors of the parameter estimates in θ . A

geometric interpretation of the relation between IF and θ

Σ ˆ is provided in Appendix D.

On writing θ

Σ ˆ in partitioned form (after Lange, 1997), we have:

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡σσ

σσ⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

σσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσσ

=××

××

2262

2666

2VeVg,Ve

Ve,Vg2Vg

25β5β4β5β3β5β2β5β1β5βμ

4β5β24β4β3β4β2β4β1β4βμ

3β5β3β4β23β3β2β3β1β3βμ

2β5β2β4β2β3β22β2β1β2βμ

1β5β1β4β1β3β1β2β21β1βμ

μ5βμ4βμ3βμ2βμ1β2μ

ˆ

000000000000

000000000000

Ω00Μ

Σθ ,

Eq. 140

101

where the sampling variances in the parameter estimates (including 5 covariates as in a

typical analysis) under the polygenic model lie along the diagonals of the block matrices

66×M and 22×Ω ( 2gˆVg σ= and 2

eˆVe σ= ). In standard matrix form, θ

Σ ˆ is a proper

covariance matrix in that it is symmetric—inherited from the Hessian—and positive

semidefinite. The standard errors, denoted by SE, of the parameter estimates are then

obtained by taking the square roots of the sampling variances along the diagonal to give

SE± . Under general regularity conditions—e.g., that the likelihood function is at least

twice differentiable—(for a full listing, see Cramér, 1946: 478-479), we have the

following theorem on the second-order efficiency of parameter estimation. For an

unbiased estimator of parameters in a parameter vector, denoted by θ~ , the Cramér-Rao

Inequality is given as (Stuart and Ord, 1991: 615-616; Shao, 1999: 251; named after Rao,

1945,1947; Cramér, 1946: 478-482):

( ) ( ) 12

ˆˆˆLlnE~Var

−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

′∂∂∂−≥

θθθθ , Eq. 141

which states that the variance of θ~ can be no less than the inverse of the Fisher

information matrix for a parameter estimate vector, θ . Note that the right hand side of

the Cramér-Rao Inequality is θ

Σ ˆ . This Cramér-Rao lower bound means that the

maximum likelihood estimates are the best estimates.

Methods III: Hypotheses and Statistical Inference

Under the polygenic model, the genetic hypothesis of interest is that the

heritability of a trait is significant. The heritability, denoted by 2h , is defined as

(Falconer, 1989):

102

2p

2g

2h σσ= , Eq. 142

which is simply the ratio of the additive genetic variance to the phenotypic variance.

Thus, the statistical null hypothesis under the polygenic model is that:

02g =σ .

Rejection of 02g =σ is taken as evidence of significant heritability.

Under the genotype × age interaction model, the genetic hypotheses for no

genotype × age interaction are that the variance is homoscedastic across the age

continuum and the genetic correlation equals 1 across any age increment. For the

example of two environments, these respectively hold that:

2g

22g

21g σ=σ=σ ,

and

1G =ρ ,

which correspond respectively, for the more general, continuous case, to the statistical

null hypotheses that:

0g =γ ,

and

0=λ .

By the arguments given in the preceding chapter, rejection of 0g =γ or 0=λ or both is

taken as evidence of significant genotype × age interaction.

On finding the maximum likelihood estimates, inferences are then made by

consideration of the likelihood ratio statistic (Wilks, 1938; Wald, 1943):

103

( )( ) ( ) ( )[ ]AN

A

N ˆLlnˆLln2ˆL

ˆLln2 θθ

θθ

−−=⎥⎦

⎤⎢⎣

⎡−=Λ , Eq. 143

where the null hypothesis, NH (parameter constrained to 0, Nθ ), is compared to the

alternative hypothesis, AH (parameter estimated, Aθ ). It should be pointed out that the

likelihood ratio test and similar such tests (see below) developed out of the Neyman-

Pearson school of thought (Neyman and Pearson, 1928a&b, 1933; Lehmann, 1950,

1959). Classically, Λ is distributed as a central chi-square random variable, denoted

by 2χ , with degrees of freedom (d.f.) equal to the difference in the number of parameters

under the null (or restricted) and alternative (or general) hypotheses (for an excellent

exposition of the d.f. concept in relation to Λ and in general, see Good, 1967, 1973). If

the null hypothesis lies on a boundary of the admissible parameter space, the asymptotic

distribution of Λ is given by a mixture of 2νχ random variables, where ν denotes the d.f.

and where the mixture may include 0=ν (Chernoff, 1954; Miller, 1977; Self and Liang,

1987). Let the p-value obtained for Λ , evaluated as a 2νχ with the appropriate d.f. or as a

mixture thereof, be denoted by ( )Λp . Then, significance is achieved for: ( ) α≤Λp ,

where we may take 05.0=α to be our nominal significance level (White, 1994: 178).

The appropriate mixture of 2νχ random variables will be derived below (for a rigorous

treatment of the derivation of the appropriate mixture of 2νχ random variables under

related models, see Shapiro, 1985, 1988). Before this is done, however, the concept of

nested model analyses needs to be introduced.

The comparisons of the full polygenic and genotype × age interaction models with

their constrained alternatives are examples of nested model analyses, where the

104

appropriate d.f. of the 2χ -tests are dictated by the difference in parameters (Thomas,

2004). For the present context, define standard conditions as cases where the null

hypothesis is not on a boundary of the admissible parameter space. This is only one of

several criteria, all termed regularity conditions, that enable a rigorous derivation of the

distribution of Λ (see Chernoff, 1954; Cox and Hinkley, 1974: 281; for a recent

discussion of what these are, see Cheng and Traylor, 1995: Sect. 2). Under standard

conditions, it may happen that parameters are significant by themselves, as would be

indicated under their respective 1-d.f. 2χ -tests, or that parameters are significant only

when considered jointly, as can be determined by carrying out their respective 1-d.f. 2χ -

tests and 2-d.f. 2χ -tests. Consider the scenario for a hypothetical 3-parameter model with

parameters a, b, and c say. We can evaluate whether a, b, and c are significant when

considered singly by carrying out their 1-d.f. 2χ -tests. However, it may happen that none

of these turn out to be significant when considered singly. At this point, we can still

carry out 2-d.f. 2χ -tests to evaluate the possibility that parameters need to be considered

jointly in order to uncover their significance. Thus, we can constrain say parameters a

and b, compare this model to the full 3-parameter model for a 2-d.f. 2χ -test, find that the

p-value indicates significance and conclude that parameters a and b are important only

when considered jointly. It turns out that this example for a hypothetical 3-parameter

model is a good description of the 5-parameter genotype × age interaction model for the

variance components. There are 4 parameters (intercept and slope parameters on the

logarithmic scale) for the additive genetic and environmental variance functions plus 1

parameter for the genetic correlation function. Whereas constraining the intercept

105

parameters ( gα and eα ) while allowing the slope parameters ( gγ and eγ ) to be estimated

is nonsensical, the reverse scenario of constraining the slope parameters while “floating”

the intercept parameters is plausible. Moreover, upon demonstrating that the polygenic

model is significantly better than the so-called sporadic model (the model in which the

phenotypic variance is not decomposed), it is no longer necessary to assess the possible

significance of the intercept parameters. In fact, floating the intercept parameters while

constraining the other 3 parameters produces a model with the exact same ln-likelihood

as the polygenic model (analyses not shown). This merely reflects the principle that

likelihoods (and ln-likelihoods) for models with continuous parameters are invariant

under reparameterization (Edwards, 1992: 28). On reparameterizing the polygenic model

in terms of the the genotype × age interaction model, we will have:

( ) ( ) ( )( ) ( )

( )⎪⎪⎩

⎪⎪⎨

⎧

α

α+α=σ

=αδ+α=, expφ2

; expexp

expexpφ2y,yCov

gxz

eg2y

exzgxzzx Eq. 144

where the previous definitions hold. Accordingly, for cases where the genotype × age

interaction model is significantly better than the polygenic model, we effectively have a

3-parameter model in terms of the kinds of 2χ -tests that are plausible. That is, we can

ask whether gγ , eγ , or λ (for the genetic correlation function) are significant when

considered singly or jointly. All together, we can carry out 3 1-d.f. 2χ -tests for each

parameter considered singly and 3 2-d.f. 2χ -tests for the possible permutations. Again,

this discussion holds for standard conditions. The situation is more complicated when the

null hypothesis is in fact on a boundary of the admissible parameter space.

106

The exact mixture of 2νχ random variables or a conservative approximation

thereof for the cases to be considered under an analysis of the genotype × age interaction

model can now be derived. There are three cases that need to be considered. These cases

are for the appropriate mixtures when comparing: 1) the polygenic model to the

genotype × age interaction model, 2) the genotype × age interaction model with one

parameter constrained to 0 to the full genotype × age interaction model, and 3) the

genotype × age interaction model with two parameters constrained to 0 to the full

genotype × age interaction model. These cases will be taken in order. It should be noted,

however, that the traditional criterion (i.e., difference in parameters) is conservative

(Stram and Lee, 1994, 1995; Almasy et al., 2001).

On finding significant heritability under the polygenic model, the intercept

parameters gα and eα of the variance functions under the genotype × age interaction

model may be dropped from further consideration because they can be thought of as

reparameterized versions of 2gσ and 2

eσ (Eq. 144). Now, the slope parameters gγ and eγ

of the variance functions may take values in the interval ( )∞∞− , ; i.e., any point on the

real line ℜ . This can be demonstrated with the following inequality:

( )[ ]( )[ ]

. 0e

0e

e

0agepexpe

0agepexp

i

i

p

age

pi

i

>⇒

>⇒

>−γ⇒

>−γ+α

⋅γ

⋅γ

⋅γ

α

Eq. 145

Because of the restriction of individual age ip to the index set T, which is a positive,

finite subset of the real line ( ℜ⊂=∈ T ;ageageT p .max.min K ), without loss in generality,

107

age ip can be assumed to be 1. Whereas γe always maintains positivity, γ can take any

value in the interval ( )∞∞− , and the inequality will always hold (Fig. 28). Therefore,

the null hypothesis cannot lie on a boundary of the parameter space because the range of

admissible values for the slope parameters is unbounded. From this fact, it is inferred

that gγ and eγ each give rise to a 21χ random variable (this satisfies the standard

condition as defined above). By contrast, the null hypothesis with respect to the genetic

correlation function, which is 0=λ , does in fact lie on the boundary of the genetic

correlation function because 1e0G ==ρ (Fig. 28). Therefore, by arguments first

developed by Chernoff (1954) and reiterated by Self and Liang (1987), λ gives rise to

the mixture ⎟⎠

⎞⎜⎝

⎛ χ+χ 21

20 2

121

. Let ⋅χ 2M denote the appropriate mixture of 2

νχ random

variables. Note that the d.f.’s are additive with respect to independent 2νχ random

variables and that the weighting frequencies of the 2νχ random variables must sum to 1

(Shapiro, 1985, 1988). On comparing the polygenic model to the full genotype × age

interaction model, we find that Λ is distributed as follows:

( )

. 21

21

21

21

21

21

egeg

egeg

,,

23

22,

22

21

20

,21

21

21

20,,

2M

γλγγγ

λ

γγλ

γλγ

⎟⎠⎞

⎜⎝⎛ χ+χ=χ+⎟

⎠⎞

⎜⎝⎛ χ+χ=

χ+χ+⎟⎠⎞

⎜⎝⎛ χ+χ=χ

Eq. 146

Thus, Λ is approximately distributed as a 50:50 mixture of 22χ and 2

3χ random variables

and ( )Λp is determined accordingly. That this is an approximation to the exact

distribution is demanded by the fact that gγ and λ are non-independent and so their

108

exp(x)

012345678

-4 -2 0 2x

f(x)

= e

xp(x

)

exp(-x)

00.10.20.30.40.50.60.70.80.9

1

0 2 4 6 8x

f(x)

= e

xp(-

x)

Figure 28. Graphical representation of exponential functions. Top panel: values of x can be any value in the interval ( )∞∞− , , whereas ( ) )xexp(xf = maintains positivity for all

ℜ∈x . Bottom panel: the exponential decay function is restricted to the closed interval [ ]1,0 , whereas x now takes values, under the restriction of ( ) )xexp(xf −= to [ ]1,0 , in the half-open interval [ )∞,0 ; that is, x now has a boundary to the left at 0x = because

1e0 = , but is unbounded to the right (values approaching infinity are legitimate).

109

mixture of 2νχ random variables is not simply additive (but note that both gγ and λ are

independent in respect to eγ ). Indeed, the exact mixture would have to somehow account

for the covariance of gγ and λ . However, one can argue that this is a conservative

approximation (cf. Stram and Lee, 1994, 1995; Almasy et al., 2001).

The second case was indirectly discussed just above. For either of the slope

parameters of the variance functions, the exact distribution for Λ is given as a 21χ

random variable. For the situation where only the genetic correlation parameter λ is

constrained to 0, the exact distribution for Λ is given by:

λλ ⎟

⎠⎞

⎜⎝⎛ χ+χ=χ

21

21

21

20

2M . Eq. 147

Suppose now it is desired to ascertain whether parameters are significant when

considered jointly. For the situation where the slope parameters of the variance functions

are jointly constrained to 0, the appropriate distribution for Λ is given as a sum of 21χ

random variables, which is just given as a 22χ random variable. For the situation where

either of the slope parameters of the variance functions and the genetic correlation

parameter λ are jointly constrained to 0, we have the following mixtures:

λγλγλγ ⎟

⎠⎞

⎜⎝⎛ χ+χ=⎟

⎠⎞

⎜⎝⎛ χ+χ+χ=χ

,

22

21

21

20

21,

2M

ggg

21

21

21

21

; Eq. 148

e

ee,

22

21

21

21

20,

2M

21

21

21

21

γλ

γλ

γλ ⎟⎠⎞

⎜⎝⎛ χ+χ=χ+⎟

⎠⎞

⎜⎝⎛ χ+χ=χ , Eq. 149

where, by the above arguments, the first is a conservative approximation and the second

is exact.

110

The preceding theory derives from the classical result that Λ is distributed as a

central chi-square 2νχ . This is all enabled by a more fundamental result of mathematical

statistics, which is the fact that maximum likelihood estimates (MLEs) are asymptotically

normally distributed (Cramér, 1946: ch. 33; Cox and Hinkley, 1974: ch. 9; Stuart and

Ord, 1991: ch. 18). From this earlier result on the asympototic normality of MLEs, we

have conservative “tests” for one-tailed and two-tailed hypotheses (Fig. 29). For a one-

tailed test and a significance level at 0.05, MLEs should be greater than roughly 2 times

their standard error (Fig. 29: Top panel). For a two-tailed test and a significance level at

0.05, MLEs should be greater than roughly 2.35 times their standard error (Fig. 29:

Bottom panel). By the arguments given earlier on admissible parameter values, testing

for 0=λ corresponds to a one-tailed test. Therefore, the MLE of λ should be greater

than 2 times its standard error. Similarly, testing either 0g =γ or 0e =γ corresponds to a

two-tailed test each time, and so their respective MLEs should be greater than 2.35 times

their respective standard errors.

111

One-tailed Test

0-4 -3 -2 -1 0 1 2 3 4

Two-tailed Test

0-4 -3 -2 -1 0 1 2 3 4

Figure 29. One- and two-tailed tests on the assumption that maximum likelihood estimates (MLEs) are normally distributed ( )1,0N 2 =σ=μ . Top panel: one-tailed test of the hypothesis that a MLE is greater than zero. Bottom panel: two-tailed test of the hypothesis that a MLE is nonzero.

~2 SD

~2.35 SD~2.35 SD

112

Methods III: Power and Alternative Test Statistics

In mathematical statistics, the central chi-square 2νχ is known to be a special case

of the more general noncentral chi-square, denoted by ( )ζνχ′ ,2 , where the two

parameters of its distribution are the d.f. given by ν and the noncentrality parameter,

denoted by ζ (Johnson et al., 1995). Indeed, from Johnson et al. (1995), the central 2νχ

can be written in terms of the noncentral ( )ζνχ′ ,2 as ( ) ( )0,0, 22 νχ′==ζνχ′ . Given this

relation, it is perhaps not too unexpected that the likelihood ratio statistic Λ , which is

distributed as a central 2νχ , can be understood in terms of the noncentral ( )ζνχ′ ,2 . This

vague intuition was given rigorous form by Wald (1943), who showed that Λ is

asymptotically distributed as a noncentral ( )ζνχ′ ,2 , with noncentrality parameter given

by (see also Anderson, 1984: 75-77; Stuart and Ord, 1991: §§ 23.4-23.8; Williams and

Blangero, 1999a&b; Blangero et al., 2001):

( ) ( )NAINAˆˆˆˆ θθFθθ −

′−=ζ . Eq. 150

Power is strictly defined as the probability that a test will correctly reject a false null

hypothesis (Blangero et al., 2001). An expression for the power of the likelihood ratio

test, denoted by ( )ΛP , is given by the integral across the region of the noncentral

( )ζνχ′ ,2 distribution with a lower limit of integration set by the ( )α−1100 percentage

point of the central 2νχ distribution, where this lower limit is denoted by ( )0,2 νχ′α (Stuart

and Ord, 1991: §§ 23.4-23.8; Williams and Blangero, 1999a&b; Blangero et al., 2001):

( ) ( )( )∫

∞

νχ′αζνχ′=Λ

0,

22 ,dP . Eq. 151

113

The upper limit of integration (at ∞ ) in Equation 151 does not represent a computational

problem by virtue of the fact that:

( ) 1,d

0

2 =ζνχ′∫∞

. Eq. 152

Therefore, a more convenient form for numerical integration is given as:

( ) ( )( )

( ) ( )( )

( )( ). ,d1

,d,d,dP

0,

0

2

0,

0

2

0

2

0,

2

2

2

2

∫

∫∫∫

νχ′

νχ′∞∞

νχ′

α

α

α

ζνχ′−=

ζνχ′−ζνχ′=ζνχ′=Λ

Eq. 153

In the Appendix C, the elements in IF will be derived. As will be seen there, the

elements in IF involve computationally-intensive matrix equations. The GaussTM

software package (Aptech Systems, Inc.) will be used for these analyses. By Equation

150, ζ can be easily determined if we know IF . Equation 153 is then evaluated with

respect to the noncentral ( )ζνχ′ ,2 distribution also using the GaussTM software. For

completeness, the noncentral ( )ζνχ′ ,2 distribution is given here following Johnson et al.

(1995). The central 2νχ distribution is given first as:

( ) ( ) [ ]( )

( )

0 x ;2x

22xexp

21

xpxp12

0,0, >∀⎟⎠

⎞⎜⎝

⎛νΓ−

==−ν

ν=ζν , Eq. 154

where ( )⋅Γ is the gamma function, defined as:

( ) [ ]( ) ℜ∈∀−=νΓ −ν∞

∫ u ; duuexp2 12

0 . Eq. 155

The noncentral ( )ζνχ′ ,2 distribution is then given as:

( ) [ ] ( ) 0 ,0 x ; xp2!n

2expxp 0,n2

n

0n, ≥ζ>∀⎟

⎠

⎞⎜⎝

⎛ ζζ−= +ν

∞

=ζν ∑ . Eq. 156

114

Two alternatives to the likelihood ratio statistic Λ are the Wald-type statistic

(after Wald, 1943), denoted by W , and Rao’s score statistic (Rao, 1948), denoted by sR

(following Bera and Bilias, 2001). Rao’s score statistic is also known in the

econometrics literature as the Lagrange multiplier statistic (after Aitchison and Silvey,

1958, 1960; Silvey, 1959). The three statistics provide asymptotically optimal tests

(Moran, 1970; Peers, 1971; Cox and Hinkley, 1974: ch.9; Buse, 1982; Engle, 1984;

Rayner, 1997; Shao, 1999: 386-387; Greene, 2003: ch. 17). Moreover, the three statistics

are equivalent asymptotically and are distributed as a 2νχ random variable according to

the theory just reviewed (Cox and Hinkley, 1974: ch.9; Buse, 1982; Engle, 1984; Rayner,

1997; Shao, 1999: 386-387; Blangero et al., 2001; Greene, 2003: ch. 17). The Wald-type

statistic W is given as:

θΣθθFθ θ1

I−′=′=W . Eq. 157

For a single scalar parameter, iθ say, this expression reduces to:

( )2θ

2 i

i

θσ

=W . Eq. 158

Rao’s score statistic sR is given as:

( ) ( ) ( ) ( )θFθθΣθ θ SSSS 1I−′=′=sR . Eq. 159

For the scalar parameter case, this expression reduces to:

( ) 2θ

2 i iθS σ=sR . Eq. 160

These statistics utilize different features of the curvature about the maximum

likelihood estimates to allow inferences to be made about the likelihood ratio of a null

versus an alternative hypothesis (Buse, 1982; Engle, 1984; Greene, 2003: ch. 17).

115

Although the sources of geometrical information (cf. Fig. D1 in Appendix D) underlying

these statistics are different, the fact that they are descriptions of the same ln-likelihood

topography about the maximum suggests equivalence in large samples. Further, the three

statistics impart to researchers the flexibility of using the most feasible statistical test

given their research design (Shao, 1999: 387; Blangero et al., 2001). For example, sR is

the least computationally-intensive because it requires estimation only under the null

hypothesis, whereas Λ is the most intensive because it requires estimation under both the

null and alternative hypotheses. It turns out, however, that W is the easiest to compute

for preliminary investigations of the statistical power properties of the genotype × age

interaction model.

116

Chapter 8

Results

Statistical Behavior of the Phenotypes

Consistent with the studies reviewed in the background section on senescence and

the IGF-I axis, circulating IGF-I levels exhibit a progressive decline starting after

adolescence and plateaus in late adulthood (Figs. 30 and 31). In contrast, IGFBP-1 levels

seem to rise at advanced ages (Figs. 32 and 33). Similar to the IGF-I pattern, circulating

IGFBP-3 levels (Figs. 34 and 35) and Ratio3 (Figs. 36 and 37) exhibit declines from

post-adolescence to late adulthood.

Model Results

Heritabilities for log IGF-I, log IGFBP-1, log IGFBP-3, and log Ratio3 are

reported in Table 5. All the traits are significantly heritable. The genotype × age

interaction model is significantly better than the polygenic model for log IGF-I, log

IGFBP-3, and log Ratio3, but not for log IGFBP-1 (Table 6). At this point, all that can be

said is that the genotype × age interaction model is more supported by the data than the

polygenic model for the traits just mentioned. In order to answer the question of whether

or not genotype × age interaction as strictly defined in chapter 6 is important, the full

genotype × age interaction model was compared to its various constrained alternatives for

log IGF-I, log IGFBP-3, and log Ratio3 (Tables 7-9, respectively). Three values are

reported in these tables, the maximum likelihood parameter estimates, their standard

errors, and the p-values under the appropriate tests (1 d.f., 2 d.f., or their equivalents). As

discussed in chapter 7, the MLEs for gγ and eγ should be greater than 2.35 times their

standard error, which correspond to a conservative two-tailed significance test each time,

117

0

50

100

150

200

250

300

350

400

>15-20 >20-25 >25-30 >30-40 >40-50 >50

Age Intervals

Mea

n IG

F-I L

evel

s

0

5000

10000

15000

20000

25000

30000

35000

40000

>15-20 >20-25 >25-30 >30-40 >40-50 >50

Age Intervals

Var

ianc

e in

IGF-

I Lev

els

Figure 30. Age-specific means and variances in IGF-I levels (ng/ml).

118

0

2

4

6

8

10

12

10 20 30 40 50 60 70 80 90

Age (years)

Log

IGF-

I

0

2

4

6

8

10

12

10 20 30 40 50 60

Body Mass Index

Log

IGF-

I

Figure 31. IGF-I versus age and BMI. Top panel: log IGF-I versus age. Bottom panel: log IGF-I versus BMI.

119

05

101520253035404550

>15-20 >20-25 >25-30 >30-40 >40-50 >50

Age Intervals (years)

Mea

n IG

FBP-

1 L

evel

s

0

200

400

600

800

1000

1200

1400

>15-20 >20-25 >25-30 >30-40 >40-50 >50

Age Intervals (years)

Var

ianc

e in

IGFB

P-1

Lev

els

Figure 32. Age-specific means and variances in IGFBP-1 levels (ng/ml).

120

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100

Age (years)

Log

IGFB

P-1

0

50

100

150

200

250

300

0 20 40 60 80

Body Mass Index

Log

IGFB

P-1

Figure 33. IGFBP-1 versus age and BMI. Top panel: Log IGFBP-1 versus age. Bottom panel: Log IGFBP-1 versus BMI.

121

0500

100015002000250030003500400045005000

>15-20 >20-25 >25-30 >30-40 >40-50 >50

Age Intervals

Mea

n IG

FBP3

Lev

els

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

>15-20 >20-25 >25-30 >30-40 >40-50 >50Age Intervals

Var

ianc

e in

IGFB

P3 L

evel

s

Figure 34. Age-specific means and variances in IGFBP-3 levels (ng/ml).

122

12

14

16

18

20

22

24

10 20 30 40 50 60 70 80 90 100

Age (years)

Log

IGFB

P-3

12

14

16

18

20

22

24

10 20 30 40 50 60

Body Mass Index

Log

IGFB

P-3

Figure 35. IGFBP-3 versus age and BMI. Top panel: Log IGFBP-3 versus age. Bottom panel: Log IGFBP-3 versus BMI.

123

0

0.02

0.04

0.06

0.08

0.1

0.12

>15-20 >20-25 >25-30 >30-40 >40-50 >50

Age Intervals (years)

Mea

n R

atio

3 L

evel

s

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

>15-20 >20-25 >25-30 >30-40 >40-50 >50

Age Intervals (years)

Var

ianc

e in

Rat

io3

Lev

els

Figure 36. Age-specific means and variances in Ratio3.

124

-14

-12

-10

-8

-6

-4

-2

00 20 40 60 80 100

Age (years)

Log

Rat

io3

-14

-12

-10

-8

-6

-4

-2

010 20 30 40 50 60

Body Mass Index

Log

Rat

io3

Figure 37. Ratio3 versus age and BMI. Top panel: Log Ratio3 versus age. Bottom panel: Log Ratio3 versus BMI.

125

Table 5. Trait Heritabilities of the IGF-I Axis Components in the SAFHS

Trait

Heritability (± SE) Covariates‡ N

Log IGF-I

0.28** (0.07) age, age2, sex × age, and BMI 681

Log IGFBP-1

0.27** (0.07) age, sex, and BMI 678

Log IGFBP-3

0.31** (0.07) age, sex, and BMI 699

Log Ratio3

0.26** (0.07) age, sex, age2, sex × age, and BMI 667

* p-value < 0.01 **p-value < 0.001 ‡Screened for significance

Table 6. Models: Polygenic versus Genotype × Age Interaction

Ln-likelihood Trait Polygenic Genotype × age

Λ †

( )ΛP ‡ at eg ,,

2M γλγχ

Log IGF-I

-299.7655 -289.7816 19.96774 0.00011

Log IGFBP-1

-304.4637 -303.8422 1.24299 0.63993

Log IGFBP-3

-335.7971 -330.7277 10.13884 0.01190

Log Ratio3

-315.0087 -311.1454 7.72652 0.03650

† ( ) ( )[ ]A0 HLlnHLln2 −−=Λ , where 0H and AH are the null (or restricted) and alternative (or general) hypotheses, respectively (see Equation 143 and the supporting text in chapter 7 of this dissertation). ‡ ( )ΛP is the p-value obtained by evaluating Λ at

eg ,,2M γλγχ . See Equation 146 and the

supporting text in chapter 7 of this dissertation.

126

Table 7. Model Fitting for Log IGF-I under the Genotype × Age Interaction Model

( )ΛP ‡ for

Model Parameters

Maximum Likelihood Estimates

± Standard Error

1 d.f. test or equivalent

2 d.f. test or equivalant

gα

-0.16339 0.06033 NN NN

gγ

0.01857 0.00399 1.15E-06 NN

G E N

λ

0.33675 0.13786 0.16417 1.08E-31

eα

-24.15996

11.93879

NN NN E N V eγ

-1.10072 0.55613 0.00204 NN

‡see chapter 7 (pp. 104-109) for explanation. NN – Not necessary (see chapter 6, pp. 104-106). E denotes exponentiation (base 10).

Table 8. Model Fitting for Log IGFBP-3 under the Genotype × Age Interaction Model

( )ΛP for

Model Parameters

Maximum Likelihood Estimates

± Standard Error

1 d.f. test or equivalent

2 d.f. test or equivalent

gα -0.88715 0.25811 NN NN

gγ 0.01665 0.01167 0.17765 0.03620

G E N

λ 0.01862

0.01284 0.04000 NN

eα -0.55627

0.17305

NN NN E N V eγ 0.00257

0.00901 0.78185

0.03620

127

Table 9. Model Fitting for Log Ratio3 under the Genotype × Age Interaction Model

( )ΛP for

Model Parameters

Maximum Likelihood Estimates

± Standard Error

1 d.f. test or equivalent

2 d.f. test or equivalent

gα -1.90485

0.66491

NN NN

gγ -0.08033

0.03676 0.00797 NN

G E N

λ 0.00000

0.07445‡

0.50000

0.01322

eα -0.39432

0.11337

NN

NN E N V eγ 0.01867

0.00629

0.01060

NN

‡ – Computed by the method of “gridding” in SOLAR

and the MLE for λ should be greater than 2 times its standard error, which corresponds

to a one-tailed significance test. For log IGF-I, Table 7 reveals that the null hypotheses

as regards gγ and λ are significantly rejected. Therefore, there is significant genotype ×

age interaction for log IGF-I. The elements of this inference are illustrated in Figures 38

and 39. While the likelihood ratio test indicates significance in respect to eγ for log IGF-

I, the conservative requirement that the eγ estimate be greater than 2.35 times its

standard error urges caution. However, since the environmental variance is decreasing

( eγ is negative) while the additive genetic variance is increasing, it appears that genotype

× age interaction was becoming more and more important with increasing age. For log

IGFBP-3, the relatively large standard errors of the respective parameter estimates would

128

Log IGF-I

0

1

2

3

4

5

6

15 35 55 75 95Age (years)

Vg

null

Log IGF-I

1E-561E-202E-203E-204E-205E-206E-207E-208E-209E-201E-19

15 35 55 75 95Age (years)

Ve

Figure 38. Genotype × age interaction for log IGF-I in SAFHS Mexican Americans. Top panel: Additive genetic variance function and its null. Bottom panel: Environmental variance function (displaced far away from its null).

129

Log IGF-I

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

Age Differences (years)

ρ G

Log IGF-I

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80

Age Differences (years)

ρ G

Figure 39. Genotype × age interaction for log IGF-I in SAFHS Mexican Americans: The same genetic correlation function is displayed on 2 scales. The line y = 1 is the null function.

130

seem to invalidate the apparent evidence of genotype × age interaction (Table 8; Fig. 40).

Lastly, as regards log Ratio3, it appears that there is evidence of genotype × age

interaction and heteroscedasticity in the environmental variance (Table 9). However, the

pattern is almost the complete reverse of that exhibited by log IGF-I. That is, the additive

genetic variance is significantly decreasing while the environmental variance is

significantly increasing (Fig. 41). It appears therefore that environmental effects were

becoming more and more important in the determination of Ratio3 levels while additive

genetic effects were becoming less and less important.

Log IGFBP-3

0

0.5

1

1.5

2

2.5

3

3.5

15 35 55 75 95

Age (years)

Var

ianc

es

VpVgVeNull-p

Figure 40. Questionable genotype × age interaction for log IGFBP-3 in SAFHS Mexican Americans: Phenotypic, additive genetic and environmental variance functions. The null function for the phenotypic variance function is indicated as Null-p. For clarity, the null functions for the additive genetic and environmental variance functions are omitted.

131

Log Ratio3

00.0050.01

0.0150.02

0.0250.03

0.0350.04

0.0450.05

15 35 55 75 95

Age (years)

Vg

Log Ratio3

00.5

11.5

22.5

33.5

44.5

5

15 35 55 75 95

Age (years)

Ve

VeNull-e

Figure 41. Apparent genotype × age interaction due to significant heteroscedasticity in the environmental variance function for log Ratio3 in the environmental variance in SAFHS Mexican Americans. Top panel: Additive genetic variance function. The curve is displaced far away from its null function (not shown). Bottom panel: Environmental variance function and its null function, Null-e.

132

Power Analyses and Results of the Genotype × Age Interaction Model

By a series of involved derivations, it can be shown that not only is the likelihood

ratio statistic Λ asymptotically distributed as a noncentral ( )ζνχ′ ,2 , but also Λ itself

gives the noncentrality parameter (Anderson, 1984: 75-77; Stuart and Ord, 1991: §§ 23.4-

23.8; Williams and Blangero, 1999a&b; Blangero et al., 2001):

( ) ( ) Λ=−′

−=ζ NAINAˆˆˆˆ θθFθθ . Eq. 161

Since Λ = W = sR asymptotically, W (or sR ) is a suitable surrogate test statistic, the

power of which can be computed according to Equation 153. For preliminary power

analyses, W (but neither Λ nor sR ) is all the more suitable because it does not require

the simulation of phenotypes. It just so happens that the computation of W does not

require phenotype values, whereas the computation of both Λ and sR ultimately require

a phenotype vector. This latter point concerning the input requirements of these statistics

can be confirmed by inspecting the components that go into their computation. The

likelihood ratio statistic Λ (Eq. 143) requires the ln-likelihood function (Eq. 120), and,

on examining Equation 120, the ln-likelihood function requires XβyΔ −= , where y is

the phenotype vector. Similarly, Rao’s score statistic sR (Eq. 159) requires the score

vector (and the sampling covariance matrix), and the elements of the score vector (Eqs.

129-131) all require Δ . As for the Wald-type statistic W , inspection of Equation 157

(and Eq. 158 for the scalar parameter case) shows that W only requires a parameter

vector, which can be specified, and IF . It will be seen in Appendix C that none of the

elements in IF require Δ . In deriving the elements in IF in Appendix C, each element

consisted in part of the form [ ] ΣΔΔ =′E , which, when pre-multiplied by its inverse,

133

effectively canceled out of the equation; i.e., IΣΣ =−1 (where I is the identity matrix of

matrix algebra).

In particular, W requires computing IF under a given set of parameter values,

which can be specified by the investigator for a plausible range to study the asymptotic

statistical behavior of the model. Proceeding with W then, one can write a program in

GaussTM to compute IF , W , and power under a range of parameter values for a fixed

pedigree structure and age distribution. At this point, the author must express his

immense debt of gratitude to Dr. Thomas D. Dyer (staff scientist at the SFBR,

Department of Genetics) not only for writing such a program in GaussTM but also for

carrying out preliminary simulations to validate the program and for teaching him some

of the basics in GaussTM. This section of the dissertation simply would not have been

possible were it not for the expert help of Dr. Dyer. Strictly speaking, the program is a

computational program that computes IF , W , and power given certain input

specifications, but it is not a simulation program.

As choices for pedigree structure and age distribution, the author requested of Dr.

Dyer that these be directly determined by the pedigree structure and age distribution for

the analyses of the log IGF-I data so that inferences concerning the empirical analyses (in

the first component) could be made. The sample size used for the current computations is

N = 690 individuals. The pedigree structure was described in the first methods chapter of

this dissertation. The age distribution of all individuals in the SAFHS with IGF-I data is

compared to the age distribution used for the present computations (Fig. 42). As can be

seen, the latter is a fairly representative sub-sample of the total given that a number of

134

Age Distribution for Log IGF-I

0

10

20

30

40

50

60

70

15.48 40.67709677 65.87419355Age

Freq

uenc

y

Simulated Age Distribution: N = 690

0

10

20

30

40

50

60

70

15.75 45.1 74.45

Ages

Freq

uenc

y

Figure 42. Top panel: age distribution in the sub-sample of the SAFHS with log IGF-I data. Bottom panel: age distribution used in the program used to compute power.

135

individuals would have been excluded from the empirical analyses for lack of covariate

data or if their data were deemed to be outliers at values greater than ± 4 SD from the

mean log IGF-I level. The ages of individuals in the sample were used to determine the

sample mean age and the elements of the “age” matrices, A , B , C , and D (see

Appendix C for the definition of C and D ). The sample mean age is 38.24 years of age.

Three analyses were carried out, one each for the effect on the power function of

variable values for gγ , λ , and eγ (Table 10). The variable parameters were varied so as

to achieve a p-value = 1 and power = 1. This ensures that the power function is filled out.

The other parameters (other than the parameter under analysis) were assigned their

corresponding MLE parameter values for log IGF-I. The results of these analyses are

reported in Figures 43-45.

Table 10. Power Analyses: Parameter Sets

Parameter Values

Parameter gγ analysis λ analysis eγ analysis

gα -0.16339 -0.16339 -0.16339

gγ variable 0.01857 0.01857

λ 0.33675 variable 0.33675

eα -24.15996 -24.15996 -24.15996

eγ -1.10072 -1.10072 variable

136

MLE >

3.841...

0

10

20

30

40

50

0.848 0.858 0.868

parameter values of Vg

Wal

d St

atis

tic

MLE p = 0.9976

p = 0.050

0.2

0.4

0.6

0.8

1

0.848 0.858 0.868

parameter values of Vg

pow

er

Figure 43. Top panel: values of W for parameter values of gγ expressed as the additive genetic variance, denoted by Vg. Bottom panel: power curve for parameter values of gγ expressed as the additive genetic variance, Vg.

137

MLE

2.705...

0

2

4

6

8

10

12

14

16

0 0.5 1

parameter values of ρG

Wal

d St

atis

tic

MLE p = 0.9334

p = 0.1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

parameter values of ρG

pow

er

Figure 44. Top panel: values of W for parameter values of λ expressed as the genetic correlation, denoted by Gρ . Bottom panel: power curve for parameter values of λ expressed as the genetic correlation, Gρ .

138

MLE<

3.841...

0

10

20

30

40

50

0.0085 0.0135 0.0185 0.0235 0.0285

parameter values of Ve x 109

Wal

d St

atis

tic

MLE p = 0.3477

p = 0.05

0

0.2

0.4

0.6

0.8

1

0.0085 0.0135 0.0185 0.0235 0.0285

parameter values of Ve x 109

pow

er

Figure 45. Top panel: values of W for parameter values of eγ expressed as the environmental variance × 109, denoted by Ve × 109. Bottom panel: power curve for parameter values of eγ expressed as Ve × 109.

139

Before interpreting the results, there are several details concerning their make-up

that should be addressed. Several interpretational guideposts that are provided in the

various graphs are in need of explanation. The significance levels, 05.0=α in the case

of gγ and eγ and 1.0=α in the case of λ , are indicated on the graphs of the power

curve. The significance level for λ is adjusted upward because the parameter gives rise

to the mixture ⎟⎠

⎞⎜⎝

⎛ χ+χ 21

20 2

121

. To see this, fix 05.0=α . Then, for parameter λ ,

significance obtains for:

( ) ( )

( )

( ) , p1.0

p21

005.0

p21

p21

05.0

21

21

21

20

χ≤

⇒χ+≤

⇒χ+χ≤

Eq. 162

where ( )2p νχ is the p-value obtained by evaluating W as a 2νχ random variable on ν

degrees of freedom. The 21χ values corresponding to 05.0=α and 1.0=α are

84146.321 ≅χ and 70554.22

1 ≅χ , respectively. The appropriate 21χ values are indicated

on the graphs of W under the three analyses. For the gγ and eγ analyses, the line at

50=W , corresponding to a power that is effectively 1 for unbounded parameters, is

indicated on their graphs.

Recalling Equation 153, let ( ) ( )WPP =Λ , where ( )WP is the power of W . For

the gγ and eγ analyses, ( )WP was computed as follows:

( ) ( )( ) ( ) 1 ; 1184146.3

0

20

0

22

=∀=′−==′−= ∫∫′

νν,ζχdν,ζχdPν,χα WWW . Eq. 163

140

For the λ analysis, ( )WP was computed as follows:

( ) ( )( ) ( ) 1 ; 11.705542

0

20

0

22

=∀=′−==′−= ∫∫′

νν,ζχdν,ζχdPν,χ α WWW . Eq. 164

Notice that the integrals differ at the upper limit of integration, where the upper limits are

given by 84146.321 ≅χ and 70554.22

1 ≅χ , respectively, which, as just noted above,

correspond to 05.0=α and 1.0=α , respectively.

It will have been noticed that the gγ , λ , and eγ parameters were respectively

expressed in terms of the additive genetic variance (Vg), genetic correlation ( Gρ ), and

environmental variance (Ve) functions. The information needed to do this is to be found

in Table 10, and the equations describing how exactly this is to be done were reviewed

earlier. For the variance functions, an age term of 1 was used. For the genetic correlation

function, the age term was also set at 1. The calculated Ve term was rescaled by

multiplying by 109 because of its extremely small values.

The likelihood ratio statistic Λ —and W or sR asymptotically in most cases—is

known to have at least two optimum properties (for ample discussion, see Das Gupta et

al., 1964; Anderson, 1984: ch. 8; Freund, 1992: ch. 12; Kuriki, 1993; Shao, 1999: ch. 8).

The first of these is known as unbiasedness of the test statistic, where a test statistic is

said to be unbiased if it achieves its minimum at the null hypothesis. The second of these

is that the corresponding power function of the test statistic is monotonic, which is said to

obtain when the power increases with increasing distance between the null and alternative

hypotheses.

As can be seen, W is unbiased and its corresponding power function is

monotonic in respect to gγ , λ , and eγ (Figs. 43-45). For the gγ analysis, W achieves

141

its minimum at the null value, which for 0g =γ is: 85.0)16339.0exp()exp( g ≈−=α (Fig.

43: Top panel). Moreover, its corresponding power function increases with increasing

distance between this null value and other point-wise alternatives (Fig. 43: Bottom

panel). For the eγ analysis, W achieves its minimum at the null value, which for 0e =γ

is: 032.010)15996.24exp(10)exp( 99e ≈×−=×α (Fig. 45: Top panel). Further, its

corresponding power function increases for point-wise alternatives that increasingly

depart from the null value (Fig. 45: Bottom panel). For the λ analysis, recall that the null

is 0=λ or 1G =ρ . Clearly, W achieves it minimum at the null value of 1G =ρ (Fig.

44: Top panel) and its corresponding power function increases for increasing departures

from the null (Fig. 44: Bottom panel).

The specific implications of the results presented here are now taken up. By now,

it is perhaps sufficiently clear that the power of a test statistic, W in the present case,

depends on the hypothesis being tested, and hence on the parameter being analyzed and

its estimated effect size (besides other factors such as the significance level given by α ,

study design and sample size). As defined earlier, power is the probability of rejecting

the null hypothesis when it is false. A complementary view holds that power is the

probability that a phenomenon exists for a given estimated effect size, where the

phenomenon is defined in contradistinction to what the null hypothesis is formulated to

negate (Cohen, 1977). Here, the phenomenon is G × E interaction, which obtains when

either or both of the null hypotheses of 0g =γ or 0=λ are rejected. In particular, if the

null hypothesis is 0g =γ , then, strictly speaking, the phenomenon is heteroscedasticity in

the additive genetic variance; its negation is homoscedasticity in the additive genetic

142

variance. Similarly, if the null hypothesis is 0=λ , then the phenomenon is

nonstationarity in the genetic correlation; its negation is correlation stationarity at 1G =ρ .

Similar to the convention of a significance level of 05.0=α , the convention for

deciding that a given set-up (to include significance level α , study design, sample size,

and the estimated parameter effect size) has adequate power seems to be a power of 0.80

(Berry et al., 1998; for statistical genetic, variance components models, cf. Williams and

Blangero, 1999a; Blangero et al., 2001, where the cited authors studied the combinations

of study design, sample size, and parameter effect sizes needed to achieve a power of

0.80). The powers with respect to the MLEs of gγ , λ , and eγ are reported in the bottom

panels of Figures 43, 44, and 45, respectively. For both gγ and λ , the power to detect

their particular MLEs was greater than 0.90, well above the 0.80 convention. Given that

both of their corresponding null hypotheses were rejected (Table 7), for a 1 d.f. and 2 d.f.

test, respectively, these high probabilities of observing the MLEs strengthen the

conclusion that genotype × age interaction was discovered at least for log IGF-I. In

contrast, the eγ analysis reveals that the set-up did not have sufficient power to detect the

phenomenon that it underlies, namely heteroscedasticity in the environmental variance.

While the null hypothesis of 0 :H e0 =γ was apparently rejected (Table 7), the finding of

a low power is consistent with the fact that the parameter estimate is less than 2.35 times

its standard error. Precisely because the environmental variance was extremely small to

begin with due to the effect size of eα and that it was declining to smaller values still due

to the negative eγ , both the power to detect eγ and the ability to measure eγ with

143

precision would have been low. Ironically, it is therefore logical to believe that there was

significant heteroscedasticity in the environmental variance in log IGF-I.

Taken together, these results indicate that genotype × age interaction was

becoming an increasingly important component of phenotype determination, specifically

in relation to log IGF-I levels. How can this latest conclusion be made to agree with the

conclusion regarding the declining influence of the environmental variance, if the age

continuum is regarded as a continuous environment? The answer to this apparent

conundrum lies in the fact that what is called the environmental variance is really just the

residual variance, after accounting for other variance components, which in the present

case are those components representative of polygenic and interaction effects. The

inference here is that genotype × age interaction, at least for the system under study, was

absorbing the variance that normally would have gone into the environmental (i.e.,

residual) variance. This is yet another reason to believe that there was significant

heteroscedasticity in the environmental variance. Thus, all of the power results exhibit an

encouraging level of internal consistency.

144

Chapter 9

Discussion

The discussion is divided into three sections. The first outlines the findings of

this study and these will be discussed in relation to the literature. The next two sections

will focus on the biomedical and evolutionary ramifications of these findings.

Statistical Genetic Findings

It will be useful to first state clearly what the statistical genetic findings are:

1) All four of the traits analyzed are significantly heritable.

2) The additive genetic variance function for log IGF-I was significantly increasing with

age. The genetic correlation function for log IGF-I significantly departed from 1G =ρ .

Taken separately or together, these findings indicate that the determination of IGF-I

levels is affected by genotype × age interaction.

3) There was more than adequate power to detect an increasing additive genetic variance

function and a changing genetic correlation function whereas there was not adequate

power to detect a decreasing environmental variance function. The power results

strengthen the conclusion that genotype × age interaction is important for IGF-I.

4) IGFBP-1 showed no evidence of genotype × age interaction.

5) IGFBP-3 showed some evidence of genotype × age interaction by the likelihood ratio

test, but this inference was not supported by the conservative tests.

6) Ratio3 initially showed evidence of genotype × age interaction, but on further analysis

it appeared that the signal was due to heteroscedasticity in the environmental variance.

Thus far, there are four examples of the G × E interaction model developed by

Blangero (1993). These will be discussed in the order they were published. The first

145

example comes from Blangero (1993). In a study on captive bred baboons at the SFBR,

Blangero (1993) found significant G × E interaction for serum levels of apolipoprotein B

(apo B). For this case, the continuous environment was the temperature at which apo B

was measured. This is relevant to CVD because apo B is a major component of LDL,

which is a major CVD risk factor. Blangero (1993) interpreted this result in relation to

seasonal variation in lipoprotein levels. He argued that ambient temperatures, through

their effects on enzymatic activity, might bring about variation in lipoprotein levels. In

the second study employing the G × E interaction model and the first for genotype × age

interaction, Jaquish et al. (1997) analyzed ultrasound fetal morphometric measurements

of 438 male and 454 female baboon fetuses at the SFBR. They found significant

genotype × age interaction for biparietal diameter and femur length (Fig. 46). This study

demonstrated that genotype × age interaction is manifest during the critical intrauterine

period of development. The third example is provided by the work of Duggirala et al.

(2000), who analyzed genotype × age interaction in CVD risk factors in a Mennonite

population. Duggirala et al. (2000) discovered significant genotype × age interaction for

serum levels of high density lipoprotein-cholesterol (HDL-C) and creatinine, both of

which are important quantitative correlates of CVD (Fig. 47). This study is the first to

demonstrate genotype × age interaction effects in a human population using the model of

Blangero (1993). The fourth example is provided by the work of Diego et al. (2003). For

the Genetic Analysis Workshop 13, Diego et al. (2003) analyzed the Framingham Heart

Study data and found significant genotype × age interaction for systolic blood pressure

(SBP) and fasting glucose levels (Fig. 48) and significant quantitative trait locus (QTL) ×

age interaction for a QTL influencing SBP. Taken together with the present study, it is

146

Biparietal Diameter

10152025303540455055

60 110 160

Gestational Age (days)

Vg

Femur Length

0

10

20

30

40

50

60

60 110 160

Gestational Age (days)

Vg

Figure 46. Additive genetic variance in fetal ultrasound morphometrics in the baboon, Papio hamadryas (spp.). Top panel: biparietal diameter. Parameter estimates are provided in Jaquish et al. (1997: 837, Table 1 therein). Bottom panel: femur length. Parameter estimates are provided in Jaquish et al. (1997: 843, Table 5 therein).

147

HDL-C

20

40

60

80

100

120

140

17 37 57 77

Age (years)

Vg

Serum Creatinine

00.5

11.5

22.5

33.5

44.5

5

17 37 57 77

Age (years)

Vg x

100

Figure 47. Additive genetic variances in phenotypes associated with atherosclerosis. Top panel: High density lipoprotein-cholesterol (HDL-C). Bottom panel: serum creatinine. Parameter estimates were obtained from Duggirala et al. (2000: 93-94, Tables 10 and 11 therein, respectively).

148

0

500

1000

1500

2000

30 40 50 60 70 80

Age

Vg

SBPFG

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35

Age Difference

ρ G

SBPFG

Figure 48. Genetic parameters for atherosclerosis risk factors in the Framingham Heart Study. Top panel: variance functions for systolic blood pressure (SBP; solid line) and fasting glucose levels (FG; diamonds). Bottom panel: correlation functions for SBP and FG. Modified from Diego et al. (2003: 3, Figure 1 therein).

149

justifiable to conclude that G × E interaction is an important component of phenotype

determination and that the model of Blangero (1993) provides a feasible approach for the

study of G × E interaction in anthropological and/or biomedical settings.

Power analyses of G × E interaction models from a statistical genetic perspective

are extremely rare in the literature. After an intensive search of this literature, there

appear to be only two such reports, one by Fry (1992) and the other by Boomsma and

Martin (2002). These will be discussed in relation to the present work.

The λ analysis results compare well with those of Fry (1992). Fry (1992), using

an analysis of variance (ANOVA) approach, showed that the power to detect G × E

interaction via departures from 1G =ρ declines as the true value approaches unity, which

is classical monotonicity with respect to the null hypothesis of 1 :H G0 =ρ (Fig. 49). In

terms of achieving maximum power, Fry (1992) also showed that as the true value of the

genetic correlation coefficient approaches unity the optimal design, while keeping the

total sample number constant at 500 individuals, approaches designs that use a collection

of large extended families from designs that use a lesser number of larger extended

families. For instance, the three maximum power values in Figure 49 correspond

respectively from left to right to designs that use a collection of about 10.5 families of

about 48 individuals, about 9 families of about 56 individuals, and about 5 families of

100 individuals. This seems to imply that as the effect size in terms of departure from the

null of 1G =ρ becomes larger, requirements on the amount of genetically related

individuals relative to the total sample become more permissive. Conversely, as the null

hypothesis is approached, rejection of the null increasingly requires a greater amount of

genetic information for the same sample size, as would be provided by larger and larger

150

~ 0.56

~ 0.86~ 0.95

0

0.2

0.4

0.6

0.8

1

0.2 0.5 0.8

ρG

pow

er

Figure 49. Power to detect G × E interaction by ANOVA. The power values are the maximum power values across a range of study designs (N = 500 in all cases). Data and results are from Fry (1992: his Figure 1, p. 545).

extended families. The power analyses of the present study were carried out for a sample

size of 690 individuals who constitute a random subset of a larger sample of 1,047

individuals from 48 families. Moreover, judging from Figure 39 of this dissertation, the

effect size was rather large.

Boomsma and Martin (2002) carried out power analyses of G × E interaction

using the approach known as genetic covariance structure modeling (GCSM), which falls

under the more general approach known as structural equation modeling. GCSM is

equivalent to the variance components approach in that the total phenotypic variance is

decomposed in exactly the same way. The main difference is that GCSM tends to

employ samples of monozygotic (MZ) or dizygotic (DZ) twins or some mixture of MZ

151

and DZ twins. In fact, Boomsma and Martin (2002) carried out simulation studies on two

designs utilizing different mixtures of MZ and DZ twins. In one design, which they

denoted as N1, they simulated a sample comprised of 50% MZ and 50% DZ twin pairs.

In the other design, which they denoted as N2, they simulated a sample comprised of

40% MZ and 60% DZ twin pairs. As their measure of G × E interaction, Boomsma and

Martin (2002) used the change in heritability, here denoted by 2hΔ , on going from one

environment to another (cf. the discrete case of two environments in the second section of

Chapter 5). The results of Boomsma and Martin (2002) need to be treated with some

caution because, as shown earlier, G × E interaction depends on two components,

heteroscedasticity in the additive genetic variance and departure from a genetic

correlation of 1G =ρ . Therefore, 2hΔ will be comprised of some unknown mix of the

two components. Their simulation strategy was to vary sample size and 2hΔ for fixed

power values of 0.50, 0.80, and 0.90. Their results are reported in Figure 50. To

interpret their results, recall the well-known relationship between power and sample size:

As the sample size increases, the power increases. Figure 50 shows that as 2hΔ

increases, a smaller sample size is needed to achieve the same level of power. This

relationship implies that the power inherent in the set-up is increasing with increasing

2hΔ , which is an implication of a monotone power function. Because of the fact that

2hΔ does not allow for heteroscedasticity in the additive genetic variance and

nonstationarity in the genetic correlation, it is difficult to compare the results of

Boomsma and Martin (2002) with the results of the present study. Overall, however, the

results of the present study are consistent with those of Boomsma and Martin (2002) in

that it was shown that there is sufficient power to detect both components of G × E

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0

200

400

600

800

1000

1200

1400

0.15 0.25 0.35 0.45 0.55 0.65

Δ h2

num

ber

of tw

in p

airs

50% 80% 90%50% 80% 90%

Figure 50. Power analysis of G × E interaction in samples of twin pairs. Open symbols represent the N1 design of 50% MZ and 50% DZ twin pairs and shaded symbols represent the N2 design of 40% MZ and 60% DZ twin pairs. Data and results are from Boomsma and Martin (2002: their Table XIII.3, p. 185).

interaction. Other points can be gleaned from Figure 50. Figure 50 shows that the better

study design in all cases seems to be the N1 design, which has proportionately more MZ

twins. This indicates that more information in terms of genetic relatedness translates to

higher power. The same observation also implies that the inclusion of longitudinal

measurements in the modeling framework of the present study will increase power; that

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is, to the extent any given individual may be conceptualized as a “twin” of him- or herself

at any two longitudinal points.

While the results of the present study are in good agreement with those reported

by Fry (1992) and by Boomsma and Martin (2002), it is desirable to carry out in the near

future more intensive simulation investigations that would vary sample sizes, study

designs, parameter effect sizes and levels of power. To the best of the author’s

knowledge, Fry (1992) and Boomsma and Martin (2002) are the only reports in the

literature of power analyses of G × E interaction based on the same (or similar)

underlying theory as the one espoused in this study, namely the “Falconer-Robertson”

formulation. Further, Fry (1992) only studied the power to detect departures from 1G =ρ

(à la Falconer), which, as explained earlier, is only partly responsible for G × E

interaction, whereas Boomsma and Martin (2002) did not distinguish between the two

components of G × E interaction. Therefore, the present study is the first to carry out

power analyses of a model—formulated in the spirit of the Falconer-Robertson

formulation—that incorporates G × E interaction due to heteroscedasticity in the additive

genetic variance and to departure of the genetic correlation from 1G =ρ .

Biomedical Ramifications

Recall the 3-phase model for the behavior of the IGF-I axis throughout ontogeny:

1) the autocrine/paracrine mode predominates during late fetal development; 2) the

endocrine mode becomes increasingly important postnatally for somatic growth and is

maximally important in this regard over the course of the pubertal growth spurt; and 3)

the endocrine mode undergoes a transition from being a regulator of somatic growth to

being a regulator of metabolism and somatic maintenance over the course of adulthood.

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The present analyses address the third phase of the above model of the behavior of the

IGF-I axis throughout ontogeny.

Relation to Metabolism in Adulthood and the Metabolic Syndrome

There are at least two physiological hypotheses that can explain the discovery of

genotype × age interaction for IGF-I. Both of these relate in complex ways to the

physiology of insulin and leptin, which are secreted by β-cells of the pancreas and

adipocytes of adipose tissue, respectively. A brief account of the relevant processes is

given just following.

It is known that the condition of obesity is associated with the up-regulation of

pancreatic secretion of insulin (Polonsky et al., 1988; Polonsky, 2000) and adipose-tissue

secretion of leptin (Ahima and Flier, 2000; Baile et al., 2000; Harris, 2000). Insulin

promotes leptin secretion indirectly by promoting adipogenesis and fat mass

accumulation and directly by stimulating adipose-tissue secretion of leptin (Fried et al.,

2000; Harris, 2000; Kieffer and Habener, 2000). This contrasts with the reverse effect

that leptin has on insulin secretion, where leptin indirectly down-regulates insulin

secretion by exerting effects at the hypothalamus and negatively regulates β-cell insulin

secretion (Harris, 2000; Kieffer and Habener, 2000; Havel, 2004). The complementary

signaling systems of insulin and leptin in peripheral tissues taken together with their

complementary and overlapping actions in the hypothalamus (Porte et al., 1998, 2002;

Niswender and Schwartz, 2003; Benoit et al., 2004) has been called the “adipoinsular”

axis (Kieffer and Habener, 2000).

It was hypothesized by several investigative groups that prolonged conditions of

obesity are associated with a state of leptin resistance due to the elevated secretion of this

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hormone, which in turn contributes to defects in the leptin-specific blood-brain-barrier

(BBB) transport system, and desensitization in the signal transduction networks that are

targets of leptin action (Maffei et al., 1995; Caro et al., 1996; Considine and Carro, 1996;

Considine et al., 1996; Hassink et al., 1996; Schwartz et al., 1996). However, the issue of

whether leptin resistance is pathogenic with respect to obesity or merely pathognomonic

of same still remains to be conclusively resolved. Subsequent investigations along this

line have generally supported the hypothesis of obesity-associated leptin resistance by

defects in the leptin-specific BBB transport system to the hypothalamus and in the

components of the leptin-specific signal transduction network (reviewed in Friedman and

Halaas, 1998; Jéquier and Tappy, 1999; Friedman, 2002; Cummings and Schwartz, 2003;

Sahu, 2004). The latest extension of the hypothesis of obesity-associated leptin

resistance suggests that the sustained high levels of circulating leptin typical of prolonged

obesity may render β-cells unresponsive to the leptin signal which would in turn result

sequentially in dysregulated β-cell insulin secretion, hyperinsulinemia, β-cell exhaustion

and/or insulin resistance and the attendant sequelae of dysfunctional glucose homeostasis

including the eventual progression to frank, full-blown T2D (Seufert et al., 1999a&b;

Kieffer and Habener, 2000; Seufert, 2004). This may be referred to as the hypothesis of

dysregulation of the adipoinsular axis (Kieffer and Habener, 2000; Seufert, 2004). Work

on a rat model by Vickers et al. (2001) suggested that dysregulation of the adipoinsular

axis may originate in the fetus. Given that the studies giving rise to the dysregulated

adipoinsular hypothesis were carried out on a mouse model (Seufert et al., 1999a) and a

cell culture system of human pancreatic β-cells (Seufert et al., 1999b), it is notable that

Söderberg et al. (2002), in a study of adult men and women, have confirmed the

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prediction of a diminishing infuence of leptin on pro-insulin concentrations with

increasing obesity. It is also notable that a recent review of the pathogenesis of T2D has

suggested that β-cell hypersecretion of insulin now appears to be the fundamental defect

that initiates the metabolic derangements culminating in T2D (Cusi and DeFronzo, 2001).

On average, the SAFHS Mexican Americans are clinically obese with a

combined-sex mean BMI slightly above 30 (Comuzzie et al., 1996). By criteria for

defining hyperinsulinemia, the Mexican Americans of the San Antonio Heart Study

(SAHS)—the epidemiological precursor of the SAFHS—are known to be relatively

hyperinsulinemic (Han et al., 2002). It is therefore highly likely that the SAFHS

Mexican Americans are also relatively hyperinsulinemic. One hypothesis that can

explain the IGF-I patterns is that the relatively obese and hyperinsulinemic condition of

the SAFHS Mexican Americans results in the up-regulation of IGF-I, which is consistent

with the knowledge that insulin is a potent stimulator of liver secretion of IGF-I (Jones

and Clemmons, 1995). Another hypothesis is that because IGF-I exhibits much overlap

with insulin structure and function (Froesch and Zapf, 1985), the conditions of β-cell

exhaustion due to chronic hyperinsulinemia and of insulin resistance may result in the

mobilization of additional compensatory mechanisms such as the potentially up-regulated

IGF-I axis (the author would like to thank Dr. Anthony G. Comuzzie, who is a Scientist

at the SFBR Department of Genetics, for suggesting this hypothesis to him). Indeed, the

knowledge that the IGF-I axis (to include the actions of the IGFBPs and IGF-I receptor)

may compensate for insulin resistance provides the physiological basis for its potential

clinical uses in the control of insulin resistance and T2D (Froesch et al., 1994, 1996a&b;

Hussain and Froesch, 1995; Hussain et al., 1995, 1996; Froesch, 1997; Simpson et al.,

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1998; Holt et al., 2003). Both of these hypotheses are consistent with the findings that

the additive genetic variance function in log IGF-I significantly increases with age, the

genetic correlation function significantly departs from 1G =ρ , and the mean circulating

level of IGF-I decreases with increasing age in the SAFHS Mexican Americans, as is

depicted in Figure 51. The explanation of Figure 51 needs to be prefaced by some

caveats regarding the relation between measures of obesity and insulin secretion. While

1 2 3 4 n-1 nn-2

A

a

↓↑σ<ρ↓μ 2gG ; 1 ;

Figure 51. Schematic diagram of changes in rank and scale along n segments of a continuous environment. I. Here, A and a represent a parent population and a fraction of the same population, respectively, measured at different points along a continuous environment. The parent population measure decreases throughout while the population fraction measure decreases at a slower rate and then increases. Due to pathophysiological events—occurring at 4 in the figure—the population fraction measure increases while the parent population measure is still decreasing.

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BMI and insulin secretion are no doubt highly correlated, the relation is a nonlinear one

(Kahn et al., 2001). Furthermore, the pattern of body fat distribution and not just BMI

has significant influences on several metabolic profiles, including insulin sensitivity and

secretion indices (Wajchenberg, 2000; Kahn et al., 2001). Given that measures of

obesity, insulin secretion and leptin secretion are continuous traits that may exhibit

complex nonlinear relations, it follows that the mechanisms proposed regarding

downstream effects on IGF-I secretion would be differentially manifested in a manner

roughly reflective of their joint distribution. Thus, a scenario of jointly heterogeneous

obesity status and leptin and insulin secretion patterns would be expected to generate an

increasing additive genetic variance function and a significantly changing genetic

correlation function in IGF-I in the face of declining mean circulating levels. In Figure

51, heterogeneity is simplistically depicted by supposing that a parent population and a

fraction of the same population exhibit different behaviors in the mean and variance in

some generic measure.

Now why should Ratio3 exhibit a distinctly different behavior from IGF-I? As

discussed in the first methods chapter, Ratio3 is an index of free IGF-I, and hence a

coarse marker of the bioavailability of IGF-I at the level of the individual. At finer levels

such as the organ- and/or tissue-levels, however, IGF-I bioavailability is largely

determined by the local milieu of hormones (to include insulin, leptin, estrogen, etc.; see

below), cytokines, generically named “factors”, as well as by the suite of supporting

proteins specific to the IGF-I axis, which include the receptor, binding, phosphorylating,

and proteolytic proteins, and the acid labile sub-unit (ALS). Moreover, there is no reason

to expect a priori that the same local milieu of determinants will be present at say the

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growth plate in longitudinal bones and at the complex interface of the blood with the

smooth muscle and epithelial cells of the vascular system, although we know that both

“compartments” involve an extraordinarily complex mix of the aforementioned

components (for the local milieu at the growth plate in relation to IGF-I, see Lindahl et

al., 1996; Rosen and Donahue, 1998; Robson et al., 2002; van der Eerden et al., 2003; for

the local milieu in the vascular system in relation to IGF-I, see Bar et al., 1988; Raines

and Ross, 1995, 1996; Sowers, 1997; Delafontaine et al., 2004). Indeed, intuition

suggests the contrary. That is, we would more likely expect that the average mix of

components of the local milieu would be reflective of the biological functions that need

to be carried out, and to the extent that these biological functions differ at the tissue-

and/or organ-levels, the local milieu would also differ. Thus, the statistical genetic

findings that IGF-I exhibits significant genotype × age interaction and that Ratio3

exhibits significantly increasing environmental variation are consistent with the fact that

there is vastly more opportunity for the determinants of free IGF-I to exhibit individual-

specific or, equivalently, environmental variation than for the determinants of IGF-I

secretion, since the latter are a subset of the former and where the determinants of the

latter predominantly act at a single place in the body, namely the liver. This argument is

recast in terms of the underlying genetic architecture in the evolutionary section.

Ontogeny, Aging, and Neuroendocrine Cascades

It is tempting to speculate that periods of intense hormonal activity, as in the

normal conditions of puberty (see below) or the pathological consequences of a chronic

state of obesity, are sufficient to mobilize the IGF-I axis GEN such that we observe

signals in the variance and genetic correlation functions. In the case of puberty, it is

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known that the IGF-I axis is in fact maximally up-regulated during this period (Giustina

and Veldhuis, 1998; Müller et al., 1999; Rogol et al., 2002; Grumbach and Styne, 2003).

In the case of chronic obesity, it is highly likely that the IGF-I axis is mobilized in

response to dysregulation of the adipoinsular axis, although by precisely what

mechanisms we currently do not know. In both cases, hormones with wide ranging

physiological effects and/or tissue targets are mobilized at relatively high concentrations.

The up-regulation of the IGF-I axis is a normal physiological process in the case of

puberty and a potential, endogenous mechanism to restore metabolic homeostasis in the

case of chronic obesity.

The results of the present study and the above speculations may be related to the

hypothesis elaborated by Finch (1975, 1977, 1976, 1979, 1987, 1988, 1990, 1993; Finch

and Landfield, 1985) that the pathologies of senescence are mediated by neuroendocrine

cascades that are late-life occurrences of physiological control systems responsible for

homeostasis throughout ontogeny. Finch’s neuroendocrine hypothesis implicitly assumes

that homeostatic systems decline with age and so on this assumption the neuroendocrine

cascades may be seen as inducers of pathology or progressively inefficacious

mechanisms for restoring homeostasis. Finch’s neuroendocrine cascade hypothesis may

be understood as a more recent and refined version of the long established concept of

systemic homeostatic decline with advancing age.

The 3-phase model in general and the second phase thereof in particular is

consistent with the widely-held belief that the dramatic increase in GH and endocrine

IGF-I secretion during puberty is causally related to the adolescent growth spurt in

humans (Martha and Reiter, 1991; Clark and Rogol, 1996; Bogin, 1998; Hibi and

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Tanaka, 1998; Mauras, 2001; Rogol et al., 2002; Grumbach and Styne, 2003; Styne,

2003; van der Eerden et al., 2003) and nonhuman primates (Copeland et al., 1981, 1982,

1985; Liu et al., 1991; Styne, 1991; Crawford and Handelsman, 1996; Crawford et al.,

1997; Suzuki et al., 2003). It will be instructive to briefly review the main hormonal

determinants of pubertal growth. Pubertal growth is a consequence of the concerted

actions of the gonadotropin/sex steroid hormone and GH/IGF-I axes. At the onset of

puberty, elevated pituitary secretion of gonadotropin brings about elevated gonadal

secretion of the sex hormones, which are estrogen in females and androgens in males

(Terasaw and Fernandez, 2001; Grumbach, 2002), and the gene encoding the intracellular

enzyme aromatase, which carries out biosynthesis of estrogen from steroid precursors,

exhibits increased expression in the ovaries and testes (Grumbach and Auchus, 1999;

Alonso and Rosenfield, 2002). The aromatase gene (CYP19) is expressed in non-gonadal

tissues as well, most notably adipose and bone tissues (Simpson, 2000; Simpson et al.,

2002). Therefore, the total estrogen in circulation derives from endocrine and

autocrine/intracrine sources, which is somewhat similar to the case for IGF-I. In both

sexes, the rise in estrogen synthesis shortly after the onset of puberty eventually promotes

increased secretion of GH and then this of course brings about increased circulation

levels of IGF-I (Martha and Reiter, 1991; Clark and Rogol, 1996; Caufriez, 1997;

Grumbach and Auchus, 1999; Mauras, 2001; Rogol et al., 2002; Grumbach and Styne,

2003; Styne, 2003; Veldhuis, 2003). But the role of estrogen is not limited to the

elevation of GH and IGF-I levels. In fact, skeletal growth is a function of the additive

actions of estrogen on the one hand and GH and IGF-I on the other and of their

synergistic interactions (Martha and Reiter, 1991; Clark and Rogol, 1996; Caufriez, 1997;

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Grumbach and Auchus, 1999; Grumbach, 2000; Soyka et al., 2000; Mauras, 2001; Riggs

et al., 2002; Rogol et al., 2002; Grumbach and Styne, 2003; Styne, 2003; Veldhuis,

2003). Estrogen and IGF-I also synergistically interact at the level of the central nervous

system (CNS) in the regulation of reproductive physiology (Melcangi et al., 2002) and

their interactions may even confer neuroprotective effects on brain tissue (Cardona-

Gómez et al., 2001, 2003). The latter point is supported by independent lines of research

on the neuroprotective effects of IGF-I (D’Ercole et al., 1996, 2002; Trejo et al., 2004)

and estrogen (Garcia-Segura et al., 2003; Norbury et al., 2003; Maggi et al., 2004). The

effects of the physiological upheaval during puberty on the IGF-I axis GEN are

theoretically depicted in Figure 52.

The similarities between the hormonal regulation of puberty and adult metabolic

pathophysiology run deeper than their common ground in the IGF-I axis. Although the

gonadotropin/sex steroid hormone and GH/IGF-I axes are most important for pubertal

growth regulation, leptin and insulin are also active and of some importance. Similarly,

estrogen is not restricted to pubertal and reproductive endocrinology, as it is thought to be

involved in numerous aspects of the metabolic syndrome. To establish these points, the

roles of leptin and insulin in puberty and, conversely, the role of estrogen in the

metabolic syndrome will be reviewed.

Leptin is thought to be a permissive factor that contributes to the suite of complex

signals in the CNS initiating the onset of puberty (Grumbach, 2002; Margetic et al., 2002;

Grumbach and Styne, 2003; Shalitin and Phillip, 2003; Styne, 2003; Veldhuis, 2003).

Ong et al. (1999) suggested that leptin’s role in regard to weight regulation and

163

1 2 3 4 n-1 nn-2

A

B

1G <ρ

↓↑σ2g

↓σ=ρ 2gG ; 1

Figure 52. Schematic diagram of changes in rank and scale along n segments of a continuous environment. II. Here, A and B are genotypes measured at different points along a continuous environment. The change in environment from 1 to 2 elicits a change in both rank and scale and is reflected by genetic correlation less than 1 and changing additive genetic variance (first decreasing then increasing). Subsequently, similar incremental changes in environment elicit only changes in scale and are reflected by a decreasing additive genetic variance.

maintenance contributes to the physiological mechanism underlying the positive

association between adiposity levels and the onset of puberty (Frisch, 1985, 1987).

Further, Maor et al. (2002) recently reported that leptin has skeletal growth factor

properties in a mouse model of endochondral ossification. They hypothesized that leptin

may help to explain the accelerated growth in obese adolescents relative to nonobese

adolescents. Given that the study of leptin biology has only recently been emphasized in

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biomedical research, it appears likely that leptin will be found to play more roles in

pubertal growth regulation.

Studies have shown that normal puberty (i.e., defined by the absence of endocrine

disorders) is associated with increased insulin resistance and compensatory

hyperinsulinemia in non-Hispanic Whites and Hispanic children (Bloch et al., 1987;

Caprio et al., 1989, 1993, 1994a&b; Amiel et al., 1991; Savage et al., 1992; Cook et al.,

1993; Caprio and Tamborlane, 1994; Potau et al., 1997; Moran et al., 1999; Goran and

Gower, 2001). There are important population differences, however. Arslanian and

colleagues have shown that elevated compensatory β-cell insulin secretion in African

American adolescents does not occur despite an increase in insulin resistance relative to

their prepubertal counterparts, which is in contradistinction to what occurs in White

American adolescents (Arslanian and Suprasongsin, 1996; Arslanian and Danadian,

1998; Arslanian, 1998, 2002; Saad et al., 2002). However, several studies have found

that African American adolescents seem to compensate for insulin resistance by

reduction in the rate of hepatic insulin extraction, which renders β-cell compensation

unnecessary (Jiang et al., 1996; Goran et al., 2002; Gower et al., 2002; cf. Mittelman et

al., 2000 for the same mechanism in dogs). This finding is likely to be robust because

studies comparing adults of African ancestry to White American, Mexican American and

White European adults have reported reduced hepatic insulin extraction rates in the

foremost group (Cruickshank et al., 1991; Osei and Schuster, 1994; Osei et al., 1997;

Harris et al., 2002). Therefore, hyperinsulinemia may arise by either of two mechanisms,

by β-cell compensation (Kahn, 1996) or reduction in the rate of hepatic insulin extraction

(Goran et al., 2002; Gower et al., 2002).

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The effects of pubertal insulin resistance (PIR) (sensu Goran et al., 2003) are

largely restricted to carbohydrate metabolism and seem not to be manifested in protein or

lipid metabolism (Amiel et al., 1991; Caprio et al., 1993, 1994a&b; Caprio and

Tamborlane, 1994; Arslanian and Kalhan, 1994). However, as regards protein

metabolism in particular, Arslanian and Kalhan (1996) found that PIR and the resultant

hyperinsulinemia seemed to suppress proteolysis. It was hypothesized that the

hyperinsulinemia following PIR would therefore help to promote protein anabolism

during pubertal growth (Amiel et al., 1991; Caprio et al., 1993, 1994a; Caprio and

Tamborlane, 1994). This hypothesis is consistent with the knowledge that insulin, GH

and IGF-I exert coordinated, anabolic actions on muscle tissues (Fryburg and Barrett,

1995; Liu and Barrett, 2002). Moreover, one of insulin’s more important roles in protein

anabolism is to inhibit protein degradation (Fryburg and Barrett, 1995; Wolfe and Volpi,

2001), which is consistent with the finding of Arslanian and Kalhan (1996) mentioned

above. Caprio (1999a&b) further suggested that hyperinsulinemia suppresses circulating

levels of IGFBP-1 and this would in turn increase circulating levels of free IGF-I. But

recall that insulin up-regulates liver secretion of IGF-I (Jones and Clemmons, 1995) and

so this too may play a role.

There are now numerous studies implicating estrogen as a major player in the

pathophysiology of the metabolic syndrome and these fall roughly into three classes: 1)

studies on postmenopausal women, 2) studies on the effect of estrogen or aromatase

deficiency in men and 3) animal models of estrogen or aromatase deficiency. These

studies have demonstrated that estrogen deficiency is associated with insulin resistance

and impaired glucose tolerance in adults and that estrogen treatment, usually involving

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estrogen replacement therapy (ERT), tends to ameliorate dysfunction in carbohydrate

metabolism (see reviews by Sharp and Diamond, 1993; Gaspard et al., 1995; Faustini-

Fustini et al., 1999; Meinhardt and Mullis, 2002; Rochira et al., 2002; Murata et al.,

2002). Estrogen also has demonstrable cardiovascular protective effects, such as

associated reductions in lipid levels and suppression of the vascular response to chronic

inflammatory stress (Gaspard et al., 1995; Farhat et al., 1996; Nathan and Chaudhuri,

1997; Mendelsohn and Karas, 2001; Mendelsohn, 2002; Baker et al., 2003). It should be

noted, however, that there is still considerable controversy surrounding estrogen’s role in

cardiovascular protection (Barrett-Connor and Grady, 1998; Mendelsohn and Karas,

2001; Mikkola and Clarkson, 2002; Pradhan and Sumpio, 2004).

Taken all together, the two main features of Finch’s neuroendocrine cascade

hypothesis seem to be upheld. The pathologies of senescence, such as those associated

with the metabolic syndrome, do in fact involve neuroendocrine cascades. Further, these

neuroendocrine cascades are late-life occurrences of homeostatic mechanisms that

operate in coordinated fashion during developmentally critical periods in ontogeny well

before the onset of senescence.

Evolutionary Ramifications: Relation to the Evolution of Senescence

As indicated in the background chapter on senescence and the IGF-I axis, the

evolution of senescence can be explained by the disposable soma (DS) theory. However,

the DS theory—in its present formulation at least—can say nothing about the statistical

genetic expectations regarding senescence. On the other hand, there are two other

theories of the evolution of senescence that are formulated in terms of population

genetics and that make statistical genetic predictions (these should be viewed as being

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complementary to the DS theory; cf. Kirkwood and Rose, 1991). These are the mutation

accumulation (MA) and antagonistic pleiotropy (AP) theories of the evolution of

senescence (Medawar, 1952; Williams, 1957; Rose, 1991; Charlesworth, 1994a&b). In

its modern form, the MA model posits that senescence evolves as the result of age-

structured mutation-selection balance, where the mutation rate across age classes is

assumed uniform and the sensitivity of fitness or, equivalently, the selection intensity can

be shown to decline with increasing age (Charlesworth, 1994a&b, 2001). The AP model

posits that senescence evolves as the result of an age related tradeoff in the beneficial and

detrimental effects of genes (Williams, 1957; Rose, 1991; Charlesworth, 1994a&b).

Under AP theory, genes that confer beneficial fitness effects early in the lifespan are

maintained by selection, but later in the lifespan, when selection intensity declines

significantly, it may happen that the very same genes confer detrimental effects. This

model is conceptually similar to the so-called “hitchhiking” population genetic models

that explain the higher-than-expected frequencies of neutral alleles as the result of the

linkage of a neutral locus to a locus that is selectively advantageous. In the case of the

AP model, alleles that are harmful late in the lifespan (but effectively neutral with respect

to fitness effects in the evolutionary sense) can evolve to higher-than-expected

frequencies if they are selectively advantageous early in the lifespan.

Charlesworth and Hughes (1996) showed that both the MA and AP theories

predict that the additive genetic variance in life history traits increases with increasing

age. This prediction has been upheld in a number of studies (e.g., Charlesworth and

Hughes, 1996; Snoke and Promislow, 2003; see also the most recent review by Hughes

and Reynolds, 2005), but there is an important, observed deviation from expectations

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discussed below. It should be noted that there are other testable predictions that are

derivable from MA and AP theory (see reviews by Zwaan, 1999; Kirkwood and Austad,

2000; Partridge and Gems, 2002; Hughes and Reynolds, 2005). However, only the

prediction of an increase in the additive genetic variance of life history traits will be

addressed in relation to the present study.

While the IGF-I axis is not a classical life history trait, it is ostensibly one of the

more important “endophenotypes” of such for it significantly affects growth rate, size at

maturity, fecundity and mortality, all of which are classical life history traits. Therefore,

the IGF-I axis should be taken as a “microcosm” for testing the MA and AP theories.

Another reason for using the IGF-I axis as a microcosm of life history traits is that for

traits such as fecundity and mortality there is always additional statistical error arising

from the fact that such traits have to be estimated rather than directly measured (Shaw et

al., 1999). Therefore, a focus on endophenotypes of life history traits is likely to be less

hindered by the introduction of additional error.

The overall results of the present study are not consistent with the MA and AP

models in that their joint prediction of increasing additive genetic variance is not born

out. Only IGF-I exhibits significantly increasing additive genetic variance, but IGFBP-1

and perhaps IGFBP-3 show a stable additive genetic variance with increasing age and

Ratio3 exhibits a significantly declining additive genetic variance with increasing age.

This overall disagreement of results with theory is similar to what was reported for the

additive genetic variance in log or ln mortality in Drosophila melanogaster by the

laboratory and colleagues of J. W. Curtsinger (Curtsinger et al., 1995; Promislow et al.,

1996; Shaw et al., 1999). Their group reported that the additive genetic variance first

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increased and then started to decline at the most advanced age groups. In a reanalysis of

the data analyzed by Charlesworth and Hughes (1996) and by Promislow et al. (1996),

Shaw et al. (1999) found that, whereas both studies exhibit the pattern first discovered by

Curtsinger et al. (1995), only the study by Promislow et al. (1996) had enough power to

detect the declining additive genetic variance at the oldest age groups (Fig. 53).

As discussed in the previous section, IGF-I secretion is largely a consequence of

multiple determinants acting at a single site, namely the liver, whereas the levels of free

IGF-I throughout the body will always include the determinants of IGF-I secretion and, in

addition to these, all of the tissue- and/or organ-specific determinants. Moreover, liver

IGF-I secretion is a classical endocrine response in that several to many signals converge

at a site to elicit a common response, which is usually the increased or decreased

expression of certain genes. In the case of liver IGF-I secretion, many signals, which are

mainly hormonal and/or nutritional or the two acting together (Clemmons and

Underwood, 1991; Corpas et al., 1993; Thissen et al., 1994; Jones and Clemmons, 1995;

Ketelslegers et al., 1995; Giustina and Veldhuis, 1998; Müller et al., 1999), converge at

the liver to effectuate increased liver expression of the gene encoding IGF-I and the

secretion of these gene products. Recalling Figure 11 in the background chapter on

senescence and the IGF-I axis, the increasing additive genetic variance in IGF-I and the

changing genetic correlation coefficient may have been reflective of the process of

increasing mobilization of the genetic elements of the IGF-I axis GEN. In contrast, the

compartment-wise determinants of free IGF-I will have exhibited environmental

variation across individuals. This idea explains why Ratio3 should exhibit significantly

increasing environmental variance in the face of the IGF-I pattern.

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Figure 53. Additive genetic variance in ln mortality in Drosophila melanogaster. Top panel: The figure here is modified from a reanalysis by Shaw et al. (1999: 559, Figure 1 therein) of the data from Charlesworth and Hughes (1996). The general increase in additive genetic variance is significant whereas the decline exhibited at the ends of the trajectories are not significant due to lack of power (Shaw et al., 1999). Bottom panel: The figure here is modified from Shaw et al. (1999: 560, Figure 3 therein) as well. Additive genetic variance in ln mortality for females (top curve) and males (bottom curve). All parts of the trajectories are significant.

171

In response to the lack of agreement between the population genetic theories of

senescence and data, a number of investigators have called for revisions in the way the

evolution of senescence is conceptualized and modeled (Promislow et al., 1996; Pletcher

et al., 1998; Promislow and Tatar, 1998; Mangel, 2001; Promislow and Pletcher, 2002).

Indeed, Promislow and Pletcher (2002) argued that the over-reliance on classical models

of senescence has been a hindrance to advances in understanding the evolution of

senescence. In line with this appeal, the present study supports the following two

suggestions: 1) Evolutionary models of senescence need to be conceptualized so that they

are in closer agreement with the underlying physiological processes of senescence. This

may be achieved by conceptualizing a model that unites the DS theory with the

neuroendocrine cascades theory. In this regard, an attractive approach that links life

history evolution with physiological processes is provided by the reliability models

reviewed earlier. 2) The statistical genetic approach advocated herein allows one to draw

inferences and biological interpretations that would be useful in the conceptualization of

such a united model.

172

Chapter 10

Conclusions

This last chapter is divided into three sections, caveats, prospectus and

conclusions. The first section emphasizes the limitations of this study. The second

section presents an extension of the genotype × age interaction model to accommodate

the theories of oxidative stress and mitochondrial dysfunction in senescence. The last

section summarizes the conclusions of this dissertation research.

Caveats

As with all studies, there are limitations that need to be recognized. A main

limitation of the present study is related to the way in which the “environment” is

accounted for. Strictly speaking, variance components models do not account for the

environment but rather relegate all factors that cannot be accounted for in genetic terms

to the environment. This at once confounds numerous aspects of the environment that

are ostensibly important. Moreover, the random environmental term may even include

non-additive genetic factors, such as dominance and/or epistasis. Therefore, this study

must be regarded as being rather preliminary. Indeed, there is much more that needs to

be done in terms of adding to the genotype × age interaction model.

Prospectus

The genotype × age interaction model is easily extended to the existing

framework for a statistical genetic analysis of mitochondrial effects. Maternal effects

sensu stricto as opposed to maternally-inherited cytoplasmic factors (i.e., mitochondria in

animals and mitochondria and chloroplasts in plants) can be easily distinguished within

the framework of the multivariate mixed linear model (Beavis et al., 1987; Schork and

173

Guo, 1993; Zhu and Weir, 1994, 1997; Czerwinski et al., 2001; Kent et al., in press;

Lease et al., in press).

It should be recalled that mitochondrial effects exhibit strong age dependencies

(Shoffner and Wallace, 1992; Wallace, 1992a&b, 1995, 1999; Ozawa, 1995, 1997, 1998,

1999; Melov et al., 1999; Kokoszka et al., 2001; Shoffner, 2001; Wallace et al., 2001). In

particular, reactive oxygen species (ROS) that are generated largely as a result of

oxidative phosphorylation (OXPHOS), which takes place in mitochondria, contribute in a

cumulative manner to the total cellular and intracellular damage incurred over the life

span (Wallace, 1992a&b, 1995, 1999; Ames et al., 1993; Shigenaga et al., 1994; Ozawa,

1995, 1997, 1998, 1999; Sohal and Weindruch, 1996; Nagley and Wei, 1998; Wei, 1998;

Wei et al., 1998; Ashok and Ali, 1999; Cortopassi and Wong, 1999; Esposito et al., 1999;

Finkel and Holbrook, 2000; Van Remmen and Richardson, 2001; Shoffner, 2001;

Wallace et al., 2001; Sastre et al., 2003). Perhaps the most important consequence of an

age-related ROS load is the high rate of mutation in the mitochondrial DNA (mtDNA),

which in turn is strongly associated with an age-related decline in OXPHOS capacity

(Wallace, 1992a&b, 1995, 1999; Ames et al., 1993; Shigenaga et al., 1994; Lee et al.,

1997; Lenaz, 1998; Nagley and Wei, 1998; Wei, 1998; Wei et al., 1998; DiMauro and

Schon, 2001, 2003; Kokoszka et al., 2001; Shoffner, 2001; Lenaz et al., 2002; Pak et al.,

2003). It has been pointed out that the age-related processes of increasing ROS and

somatic mutation loads and of decreasing OXPHOS capacity are inherently stochastic

within individuals (Wallace, 1999; Stadtman, 2002).

These considerations lead to the prediction that, across individuals at the

population level, the variance in mitochondrial effects is itself age-dependent. Therefore,

174

on strong biochemical and physiological grounds, the variance in mitochondrial effects is

expected a priori to be heteroscedastic across the age continuum. The high mtDNA

mutation rate gives rise to a more subtle age dependency, which has important effects on

the correlation structure (of mitochondrial effects) inherent in a given population of

relatives. This other type of age dependency is due to the phenomenon known as

“replicative segregation”, which refers to the mutation-driven departure from

homoplasmy (mitochondrial genome comprised of wild-type mtDNA) towards increasing

heteroplasmy (mitochondrial genome comprised of mutant mtDNA) (Shoffner and

Wallace, 1992; Wallace, 1992a&b, 1995, 1999; Lightowlers et al., 1997; Ozawa, 1997;

DiMauro and Schon, 2001, 2003; Shoffner, 2001; Wallace et al., 2001). This process has

been modeled as a genetic drift process and the model behavior seems to be consistent

with data (Chinnery and Samuels, 1999; Chinnery et al., 2002; Elson et al., 2001).

Because of this genetic drift type process, the correlation structure in mitochondrial

effects is expected a priori to decay with increasing age differences between any two

individuals belonging to the same maternal lineage. In terms of assumptions, the

preliminary mitochondrial model assumes homoscedasticity in the mitochondrial

variance and a stationary correlation structure at complete, positive correlation for all

individuals belonging to the same maternal lineage. Both of these assumptions can be

relaxed by modeling the mitochondrial variance and correlation in mitochondrial effects

as functions of age and age differences, respectively, in the same manner as under the

genotype × age interaction model.

It will be interesting to fully develop and analyze these models in relation to the

IGF-I axis. The IGF-I axis seems to be a universal regulator of senescence, as the axis

175

and its homologs have been studied in relation to senescence in yeast, nematodes, fruit

flies, and mammals (Ghigo et al., 1996, 2000; Arvat et al., 1999, 2000; Guarente and

Kenyon, 2000; Kenyon, 2000; Finch and Ruvkun, 2001; Gems and Partridge, 2001;

Longo and Finch, 2002, 2003; Barbieri et al., 2003; Tatar et al., 2003; Browner et al.,

2004). The IGF-I axis has been integrated with oxidative stress and mitochondrial

dysfunction in relation to senescence. The most supported physiological model along the

lines of combining oxidative stress, mitochondrial dysfunction, and neuroendocrine

factors holds that the decline in IGF-I axis activity over the life span in turn decreases

metabolic activity and hence oxidative stress and mitochondrial dysfunction (Carter et al.,

2002a&b, Bartke et al., 2003; Brown-Borg, 2003; Brown-Borg and Harman, 2003;

Hursting et al., 2003; Holzenberger, 2004). Work on several murine models has

demonstrated that the decline in IGF-I axis activity in conjunction with caloric restriction

is a significant determinant of life span extension (Shimokawa et al., 2002, 2003; Tirosh

et al., 2003, 2004; Al-Regaiey et al., 2005; Miskin et al., 2005). In one of these murine

models, interaction of IGF-I signalling pathways with mitochondrial function was

thought to be important (Tirosh et al., 2003, 2004; Miskin et al., 2005).

Conclusions

In multicellular organisms, the IGF-I axis is central to processes that are

fundamental to life, including development, growth, somatic maintenance and

metabolism. The importance of the IGF-I axis holds for most of the duration of

ontogeny, although its precise roles may vary dramatically over the lifespan. It was

hypothesized that this dynamic endocrine system is reflected by a GEN and, hence, it

would be an ideal system to study genotype × age interaction in humans. Convincing

176

evidence of genotype × age interaction was presented. Specifically, it was found that in

Mexican Americans in the San Antonio Family Heart Study, the additive genetic variance

and genetic correlation functions change significantly with age for IGF-I. These findings

were discussed in terms of their implications for the pathophysiology of the metabolic

syndrome, the neuroendocrine cascade hypothesis of senescence, and evolutionary

theories of senescence. The idea that the IGF-I axis is mobilized as an integral

component of neuroendocrine cascades, that are age-specific in the case of puberty and

age-associated in the case of obesity, is consistent with the treatment of the age

continuum as a continuous index of the range of environments experienced by organisms

(Hegele, 1992; Zerba and Sing, 1992; Zerba et al., 1996, 2000). The results of the

present study justify the belief that the genotype × age interaction model can detect cases

where the expression of genotype is highly dependent on environment, which, for the

IGF-I axis, includes the prevailing hormonal milieu.

The IGF-I axis is only one kind of complex trait. Similarly, age is only one kind

of continuous environment that is likely to have important influences in the determination

of complex traits. The genotype × age interaction model is a specific version of a general

G × E interaction model that can be applied to virtually any other complex trait and any

other continuous environment. Further, this model has been extended to the level of

quantitative trait loci (QTLs) and can be used for QTL × environment interaction

analyses (Almasy et al., 2001; Diego et al., 2003). Further still, these models fall under

the general class of variance components models (Blangero et al., 2000, 2001), which

also includes multivariate models that can assess pleiotropy at both the polygenic and

linkage levels (Comuzzie et al., 1996; Almasy et al., 1997; Williams et al., 1999).

177

Indeed, there is some evidence of pleiotropy with respect to metabolic syndrome traits

and the IGF-I axis (Comuzzie et al., 1996). There is an emerging picture of how complex

physiological networks are modulated by dynamic modulation in their critical regulatory

factors, such as the IGF-I axis (Finch and Ruvkun, 2001; Gems and Partridge, 2001;

Longo and Finch, 2002). Therefore, it is justifiable to conclude that analyses using these

models hold much promise for understanding the biology of dynamic, complex traits in

general and of senescence in particular.

178

Appendix A: A Geometric Proof of the G × E Interaction Theorem

Mathematical concepts emanating from one branch of mathematics—if they are

ultimately derived from some deep underlying set of truths—may often be translated into

the language of another branch. Thus, theorems and their respective proofs may often be

(and have often been) delivered in the different languages of mathematics. It turns out

that this is the case in regard to the theorem on G × E interaction and its proof, for now a

geometric representation of the theorem and a proof of its validity can be constructed.

The following constitutes more than an independent proof of the theorem, however, for

while offering an independent proof, it contributes a novel perspective on G × E

interaction that may one day prove useful.

The restatement and proof require certain definitions from vector space geometry.

In particular, the proof will be confined to the vector space in the plane, 2ℜ , or 2-space,

but the underlying theory is easily generalized to nℜ for n arbitrarily large because all

that is required is that two vectors lie in the same plane. Hausner (1965) provides a

useful reference for the vector space approach to geometry. Extensions of this

perspective to multivariate statistics are presented in Dempster (1969) and Wickens

(1995). Further, the insightful articles by Herr (1980) and Bryant (1984) inspired the

current approach. Some axioms and definitions from Euclidean geometry are also

needed but these can be mentioned as the exposition unfolds.

The following can be found in linear algebra and calculus texts that cover vector

space. Let there be two vectors ⎥⎦

⎤⎢⎣

⎡=2

1

uu

u and ⎥⎦

⎤⎢⎣

⎡=2

1

vv

v . The Euclidean norm (also

known as the length, magnitude, or absolute value) of u , denoted by u , is given as:

179

( ) 22

21

2

1n

1i

2i uuu

2

1

+=⎟⎠⎞

⎜⎝⎛=′= ∑

=

uuu , Eq. A1

and similarly for v . For now, let the inner product or dot product of u and v ,

denoted by vu ⋅ , be defined as:

( )22112

1

2

1 vuvuvv

uu

+=⎥⎦⎤

⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=⋅ vu , Eq. A2

and so it follows that:

( ) ( )[ ] 2 2 22

21

22

21

2

1

2

1 uuuuuu

uu

uuu =+=+=⎥⎦⎤

⎢⎣⎡⎥⎦⎤

⎢⎣⎡=⋅ . Eq. A3

Both operations give rise to scalars. Suppose that u and v are centered at the origin in

2ℜ as depicted in Figure A1. Addition and subtraction hold as in the general case for

vectors but note now the geometric meaning of vector subtraction, vu − , in Figure A1.

vu −

uv

θ, =∠ vu

θ

Figure A1. Schematic Representation of Vector Space in 2ℜ . Ideally, the vector, vu − , should not be offset but its direction is more clearly seen this way. That is, the three vectors form a triangle. The angle between u and v , from u to v is θ .

180

An important law holds here, namely the Law of Cosines, which is given as:

θ cos 2 2 2 2 vuvuvu −+=− . Eq. A4

At this point, we may begin to see where all of this is leading for the Law of Cosines

evokes a familiar form. Now, by the property of inner products implied by Equation A3,

the vector given by vu − has the following inner product:

( ) ( )vuvuvu −⋅−=− 2 . Eq. A5

Further, inner products are distributive, associative, and commutative as in scalar algebra,

and their like products are additive. Thus, Equation A5 may be rewritten to yield:

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ). 2

vuvvuu

vvuvvuuuvuvvuuvuvu

⋅−⋅+⋅=⋅+⋅−⋅−⋅=

−⋅−−⋅=−⋅− Eq. A6

Invoking the property implied by Equation A3 once more gives:

( ) ( ) ( ) vuvuvuvvuu ⋅−+=⋅−⋅+⋅ 2 2 2 2 . Eq. A7

Equating the right hand side of Equation A7 with the right hand side of Equation A4 in

this order and discarding like terms gives a new definition for the inner product:

θ cos vuvu =⋅ . Eq. A8

On rearranging Equation A8, we have that:

θ cos

vu

vu ⋅= , Eq. A9

which leads to the theorem that two vectors are orthogonal (perpendicular) if and only if:

0=⋅ vu , Eq. A10

which holds when u and v are at a right angle to each other. Angle θ is restricted to the

range πθ0 ≤≤ . While the cosine function is periodic on π2 , it declines monotonically

181

from 0θ = to πθ = , taking values from +1 to –1 so that we have for the range of θ cos ,

under the restriction, the closed interval [ ]1,1 +− , or 1θ cos1 +≤≤− . The range of θ cos

in vector space follows directly from the Cauchy-Schwarz Inequality (Halmos, 1958:

125-126; Horn and Johnson, 1985: 15), which holds that:

. 1

11

1

+≤⋅≤−⇒≤⋅⇒

≤⋅

⇒≤⋅

vuvu

vuvu

vuvu

vuvu Eq. A11

The Cauchy-Schwarz Inequality plays fundamental roles in vector space geometry

(Hausner, 1965), matrix analysis (Horn and Johnson, 1985), and in probability theory

(Parzen, 1960). In terms of random variables Y and Z, the probabilistic version of the

Cauchy-Schwarz Inequality is given as (Parzen, 1960: 363):

, 111

1

ZY

Z,Y

ZY

Z,Y

ZY

Z,YZYZ,Y

+≤σσ

σ≤−⇒≤

σσσ

⇒

≤σσ

σ⇒σσ≤σ

Eq. A12

which may be immediately recognized as the correlation coefficient (Eq. 94 in the text).

We are almost in a position to restate the theorem. It is a common practice in

multivariate statistics to express random variables or statistical parameters as vectors

endowed with the properties of such in vector space (Dempster, 1969; Herr, 1980;

Bryant, 1984; Wickens, 1995). Following this tradition and on comparing Equations A11

and A12 element by element, let now Euclidean norms u and v be understood as

metrics in vector space of 1gσ and 2gσ , respectively. As immediate consequences, we

find that the squared Euclidean norms 2 u , 2 v , and 2 vu − become metrics of

182

21gσ , 2

2gσ , and 2gΔσ , respectively. As the analog of Gρ in vector space, we have θ cos ,

where θ is restricted to the closed interval [ ]π,0 . In the language of vector space, G × E

interaction holds for 0 2 ≠− vu . The theorem on G × E interaction may now be

restated. There is no G × E interaction, i.e., 0 2 =− vu , if and only if 2 2 vu =

and 1θ cos = . Similar to the algebraic proof in the text, the trivial cases corresponding to

0 2 2 == vu will not be considered below. But again, there is little loss in

generality with these concessions.

It can now be seen that the fundamental equation for G × E interaction arises from

the Law of Cosines, which may now be rewritten to yield:

( )⎪⎩

⎪⎨

⎧

=∀−

≠∀−+=−

. ; θ cos1 2

. ; θ cos 2

2 2 2

2 2 2 2

2

vuu

vuvuvuvu Eq. A13

Further still, by specifying for θ cos the values 1, 0, and –1 (corresponding to o0 , o90 ,

and o180 , respectively), we recover the six cases under the algebraic approach in Chapter

6 in the text, but this time under the assumptions that 2 2 vu ≠ and 2 2 vu = .

Assuming 2 2 vu ≠ , Equation A13 gives:

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=∀++

=∀+

=∀−+

=−+=−

1.θ cos ; 2

0.θ cos ;

1.θ cos ; 2

θ cos 2

2 2

2 2

2 2

2 2 2

vuvu

vu

vuvu

vuvuvu

Eq. A14

183

Assuming 2 2 vu = , Equation A13 gives:

( )

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−=∀

=∀

=∀

=−=−

. 1θ cos ; 4

. 0θ cos ; 2

. 1θ cos ; 0

θ cos1 2

2

2 2 2

u

uuvu Eq. A15

These cases will be taken in turn as before, but this time with a focus on the geometry of

the situation. In fact, the proof requires six figures to treat each case (Figs. A2-A7).

These are treated from top to bottom for Equation A14 and then for Equation A15.

The geometric systems depicted in Figure A2 on the left hand side are known as

degenerate triangles (Pedoe, 1970). The axioms and derived theorems of Euclidean

geometry are satisfied, the main theorem for the present case being:

( ) ( ) o180 , , , =−∠+−∠+∠ vuvvuuvu , Eq. A16

where the theorem is tailored to the present circumstances. For Figure A2, the angles are:

o0, =∠ vu , o180, =−∠ vuu , and o0, =−∠ vuu . Clearly, these sum to o180 . The

important point here is that the vector, ( )vu − , is nonzero and so 2 vu − is nonzero by

definition (Equations A1 and A3).

For the second case, Figure A3 amounts to an illustration of the fact that the

Pythagorean theorem, namely:

2 2 2 vuvu +=− , Eq. A17

is merely a special case of the Law of Cosines. It is the Law of Cosines for o90θ = ; that

is, for right triangles. As regards the theorem of G × E interaction, 2 vu − is clearly

nonzero for the squared magnitudes of u and v are nonzero.

184

2 2 vu ≠ 1 θ cos ; 0 θ == o

=

u

v

vu −

=

u

v

vu −

• • •

• • •

vu, ∠

vuu , −∠

vuu , −∠

vu, ∠

vuu , −∠

vuu , −∠

Figure A2. Geometry of Heteroscedasticity I: The Degenerate Triangle in Vector Space. The geometric systems on the left hand side arise from vector subtraction under the stated conditions.

u

vu −v

v

u

vu −

2 2 vu < 2 2 vu >

0 θ cos ; 90 θ == o

Figure A3. Geometry of Heteroscedasticity II: The Law of Pythagoras in Vector Space.

185

One of the more surprising results herein arises from the geometric system

presented in Figure A4 (cf. the corresponding homoscedastic system). It turns out that all

three vectors constitute the side opposite o180θ = , which is the vertex of u and v (the

other two sides are given by zero vectors; see below). This result represents the

maximum squared magnitude that the vector ( )vu − can attain. Therefore, 0 2 ≠− vu .

The geometric system depicted in Figure A5 is the crucial case under the theorem

of G × E interaction, for here arises the geometric lower limit on 2 vu − . The system

is not a triangle in vector space. In fact, the system degenerates even further to a 0 angle

and a point represented by the zero vector in vector space. That is, ( )vu − does exist but

it is the zero vector. For when uv = , the vector ( )vu − is:

• • •

2 2 vu ≠ 1 θ cos ; 180 θ −== o

=

vu −

uvvuv , −∠ vuu , −∠

vu, ∠

Figure A4. Geometry of Heteroscedasticity III: The Degenerate Triangle in Vector Space. The vectors u and v are separated by a line to indicate their relative magnitudes.

186

122

1

2

1

00

uu

uu

×=⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡−⎥⎦

⎤⎢⎣⎡=− 0uu . Eq. A18

Further, the zero vector has a defined magnitude, given by:

0000 2212 ==+=×0 . Eq. A19

Moreover, the zero vector extends in all directions in vector space (Hausner, 1960). It is

a proper vector. As regards the theorem of G × E interaction, 0 2 =− vu in this case.

• =v

u

vu −•

2 2 vu = 1 θ cos ; 0 θ == o

Figure A5. Geometry of Homoscedasticity I: The Zero Vector in Vector Space.

The next two cases depicted in Figures A6 and A7 need little comment since

similar arguments to the ones given for their corresponding cases under

heteroscedasticity apply just as well under homoscedasticity. By those arguments, the

cases depicted in Figures A6 and A7 give rise to 0 2 ≠− vu .

187

2 2 vu =

v

u

vu −

0 θ cos ; 90 θ == o

Figure A6. Geometry of Homoscedasticity II: The Law of Pythagoras in Vector Space.

• • • =

u

vu −

v

2 2 vu = 1 θ cos ; 180 θ −== o

Figure A7. Geometry of Homoscedasticity III: The Degenerate Triangle in Vector Space.

188

Appendix B: Relation to the Gaussian and Ornstein-Uhlenbeck Stochastic Processes

The derivation of genotype × age interaction model has its counterpart in the

theory of Gaussian stationary stochastic processes. It will be instructive to briefly discuss

such processes in relation to the genotype × age interaction model to draw out common

themes. Gaussian stationary stochastic processes can be shown to be covariance

stationary (sensu Parzen, 1962) in translation along some environmental continuum of

interest. Moreover, according to Karlin and Taylor (1975: 446), “For covariance

stationary processes, the crux of the matter . . . is whether or not the covariance function

converges to zero as the time difference [age difference in the genotype × age interaction

model] . . . becomes large, and if it does so vanish, the rate at which this convergence

takes place has relevance.” By a limit equation presented below, the overall covariance

function vanishes exponentially for large age differences under the genotype × age

interaction model. For phenotypes, the Gaussian nature of the stochastic process comes

from the assumption that a given phenotype at any point along the age continuum follows

a Gaussian or multivariate normal distribution. The process of phenotype determination

along the age continuum may therefore be conceptualized as a Gaussian covariance

stationary stochastic process. In fact, several investigators have developed the stochastic

process approach as a model of phenotype determination independently of the concept of

G × E interaction (Kirkpatrick and Heckman, 1989; Kirkpatrick and Lofsvold, 1989;

Kirkpatrick et al., 1990, 1994; Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000;

Pletcher and Jaffrézic, 2002).

189

The Gaussian covariance stationary stochastic process model is a surprisingly

straightforward extension of Equations 83-84. Let ( )ty denote a phenotype function in

time, t , where ℜ⊂∈Tt . Analogous to the classical case, the linear model for ( )ty is:

( ) ( ) ( ) ( )tetgtμty ++= , Eq. B1

where ( )tμ is the mean function, ( )tg and ( )te are independent Gaussian processes, and

the following expectations hold: ( )[ ] ( )ttyE μ= , and ( )[ ] ( )[ ] 0teEtgE == . By the

assumptions of independent Gaussian processes and of additivity in the random effects

functions, the phenotypic covariance function, denoted by ( )2

tyσ , can be decomposed as

follows (Kirkpatrick and Heckman, 1989; Pletcher and Geyer, 1999; Jaffrézic and

Pletcher, 2000; Pletcher and Jaffrézic, 2002):

( ) ( ) ( )2

te2

tg2

ty σ+σ=σ , Eq. B2

where ( )2

tgσ and ( )2

teσ are the genetic and environmental covariance functions,

respectively. From general treatments of stochastic processes (Parzen, 1962; Karlin and

Taylor, 1975), covariance stationarity requires that:

( ) ( ) ( ) ( ) ℜ⊂∈=υ∀σ=σ=σ==σ υυ−υυ Tts, ; eg, ; ... 2t

2s

21s

20 , Eq. B3

which means that the variance is stationary in translation along the time axis. It is also

required that the correlation function is a function in absolute time differences (Parzen,

1962; Karlin and Taylor, 1975). Doob (1942) had pointed this out for the Ornstein-

Uhlenbeck stochastic process, which is a special case of Gaussian stochastic processes

and is the inspiration of Equation 108 (see below). For the process of phenotype

determination along the time (or age) continuum, we can restrict the correlation

requirement to the genetic correlation function, denoted by ( ) ( )tstg −ρ , by the assumption

190

that the genetic and environmental effects are independent Gaussian processes.

Therefore, the phenotypic covariance function may be written as:

( ) ( ) ( ) ( ) ( )2

te2

tgtg2

ty ts σ+σ−ρ=σ . Eq. B4

Significantly, the variance stationarity requirement can be relaxed by modeling variance

heterogeneity with any suitable parametric model as long as it maintains positivity

(Pletcher and Geyer, 1999; Jaffrézic and Pletcher, 2000; Pletcher and Jaffrézic, 2002).

Thus, the parameterizations in Equations 107 and 108 for the variance and correlation

functions, respectively, are both acceptable even under the theory of Gaussian stationary

stochastic processes. It should be emphasized that the idea of environmental insensitivity

or invariance under translation is taken as the null process in both the stochastic process

and G × E interaction concepts. That is, no G × E interaction or insensitivity to

environmental change is equivalent to stationarity or invariance under translation and

both concepts are defined with respect to a specific environmental continuum.

The Ornstein-Uhlenbeck (O-U) stochastic process is clearly fundamental to the

current formulation of the G × E interaction model for this is where the genetic

correlation function comes from. The question of the origin of the correlation function of

Equation 112 will now be addressed. The ensuing is in no way an exhaustive account or

a rigorous derivation of the O-U stochastic process. Lange (1986) provides a rigorous

development of the theory in relation to statistical genetics. General treatments of the O-

U stochastic process in the light of modern stochastic calculus can be found in Karatzas

and Shreve (1991), Durrett (1996), and Krylov (2002).

The standard derivation of the O-U stochastic process starts with Langevin’s

Equation (cited in Chandrasekhar, 1943; Karatz and Shreve, 1991; see also Doob, 1942;

191

Nelson, 2001). Incidentally, Langevin’s Equation is thought to mark the origin of the

theory of stochastic differential equations (Karatz and Shreve, 1991; Nelson, 2001). We

will start with the original treatment by Uhlenbeck and Ornstein (1930) and take it to the

point where the formal integration machinery developed by Doob (1942) and others

applies. For clarity of exposition, we will pretend for the moment that the context is

known and that all terms have been defined. After the crucial manipulations have been

carried out, we will then describe the context and terms. Also, the Leibniz and prime

notations will be used simultaneously, as this will be advantageous. The following is

known as Langevin’s Equation:

( ) ( ) ( )tAtu

dt

tdu=λ+ . Eq. B5

Langevin’s Equation is a first-order, linear differential equation. As such, it will fall to

the integration factor method. That is, we seek an integration factor, ( )th say. Recall the

Product Rule of differential calculus for the product of two functions, ( )tu and ( )th :

( ) ( )[ ] ( ) ( ) ( ) ( )thtuthtuthtu ′⋅+⋅′=′⋅ . Eq. B6

Multiplying ( )th across Equation B5, temporarily suppressing the function notation, and

switching to the Leibniz notation, we have:

Ahhuhu =λ+′ . Eq. B7

Now we have two equations and one unknown so that we may solve for ( )th . To do so,

we equate the right hand side of Equation B6 to the left hand side of Equation B7. Then,

on dropping redundant terms, we will have:

λ=′

⇒λ=′h

hhh , Eq. B8

192

which is a first-order, separable differential equation and is immediately integrable.

Recalling that we have functions in t, integration with respect to t gives:

tehthlndth

hλ=⇒λ=⇒λ=

′∫∫ . Eq. B9

Having solved for ( )th , we rewrite Equation B7 accordingly to give:

ttt Aeeueu λλλ =λ+′ . Eq. B10

But the left hand side of Equation B10 is merely the derivative of ⎥⎦⎤

⎢⎣⎡ ⋅ λteu , by the

Product Rule. Therefore, Equation B10 may be rewritten as:

[ ] tt Aeeu λλ =′⋅ , Eq. B11

which is also immediately integrable. Integrating across the interval from 0t = to t , and

no longer suppressing the function notation in t , we find:

( )[ ] ( )

( ) ( ) , dsesAuetu

dsesAesu

t

0

s0

t

t

0

s t

0

s

∫

∫∫

λλ

λλ

⋅=−⋅

⇒⋅=′⋅

Eq. B12

which gives the formal solution as:

( ) ( )∫ λλ−λ− ⋅+⋅=t

0

stt0 dsesAeeutu . Eq. B13

We are now at the point where the details can be filled in. Langevin’s Equation is

an ingenious variant of Newton’s Second Law (after dividing by the mass): maF = ,

where F is force, m is mass, and a is acceleration. Rearranging gives:

( ) ( ) ( )tAtudt

tdu+λ−= .

193

Now, ( )tu gives the velocity of a particle, the term ( )tuλ− gives a deterministic frictional

effect on ( )tu and the term ( )tA gives the residual effects on ( )tu , which are assumed to

be stochastic (Doob, 1942, 1953; Nelson, 2001). Hence, Langevin’s Equation was a

rather bold hypothesis for it claimed that the rate of change in ( )tu , ( )

dt

tdu, is given by a

linear combination of deterministic and stochastic effects.

Doob (1942) noted that Langevin’s Equation caused much controversy. Indeed,

the equation must have railed against the dogma of deterministic theories in theoretical

physics (on the reign of deterministic philosophy in theoretical physics up until the rise of

quantum mechanics in the late 1920s see Popper, 1977: ch. 6). Now, the solution given

by Equation B13 is a formal solution. However, the integral involving the stochastic

term did not at the time of Uhlenbeck and Ornstein (1930) admit a straightforward

solution. The reason for that state of affairs is obvious in retrospect. The formal

apparatus of measure theory for probabilistic phenomena had not yet been laid down.

Indeed, the probability calculus would not receive its fundamental enunciations until the

1930s (see Doob, 1941, 1953: Supplement and Appendix, 1996). Moreover, not until

after this period do we observe the formalization of the theory of stochastic integrals by

the pioneering work of Doob (1942, 1953) and K. Itô (Itô’s early works in the 1940s were

published in Japanese journals but they are considered fundamental in stochastic

calculus; see Karatzas and Shreve, 1991; Durrett, 1996; Brzeźniak and Zastawniak,

1999). The formal apparatus of the stochastic calculus is quite intricate and lies beyond

the scope of this section (for general treatments, see Karatzas and Shreve, 1991; Durrett,

194

1996; Brzeźniak and Zastawniak, 1999). At this point, we wish only to report the

fundamental results of the stochastic calculus applied to Equation B13.

It is sufficient to point out that it has been proven that the O-U stochastic process

is multivariate Gaussian across the time continuum (using the methods of statistical

physics, see Uhlenbeck and Ornstein, 1930; Chandrasekar, 1943; using the methods of

stochastic calculus, see Doob, 1942; Durrett, 1996). Further, Doob (1942) showed that

for a standardized O-U stochastic process, we have for the mean:

( )[ ] ( ) μ=μ= t tuE , Eq. B14

and, for the covariance function:

( )[ ] ( )[ ] ( ) ℜ∈∀−λ−σ=μ−μ− ts, ; stexpsu tuE 2 , Eq. B15

where the exponential term is the correlation function. Limits at infinity and zero, for

fixed 0≠λ , provide boundaries on the correlation function as follows:

[ ] 0stexp lims-t

=−λ−∞→

; Eq. B16

[ ] 1stexp lim0s-t

=−λ−→

. Eq. B17

Equations B15-B17 have a simple interpretation: 1) The covariance function may

ultimately be expressed as a function in increments in time (or whatever the continuum

may be) and, by the limits imposed, the covariance function is 2) sufficiently stationary

for small increments or approximates stationarity exponentially fast or 3) approaches 0

exponentially fast for large increments.

So finally, where does the correlation function come from? Apparently, the

correlation function comes from the integration factor method for the formal solution of

195

Langevin’s Equation. Further, on rearranging Equation B13, we see that the solution is

obtained by computing the stochastic integral, as Chandrasekhar (1943) noted:

( ) ( )∫ λλ−λ− ⋅=⋅−t

0

stt0 dsesAeeutu .

By certain assumptions under the theory of stochastic calculus, the right hand side can be

rewritten as (for a similar form of this particular stochastic integral under rigorous

definitions, see Karatzas and Shreve, 1991: 358; Krylov, 2002: 106):

( ) ( )

( ) ( )( )

( ) ( ) . dsesA

dseesA

dseesAeutu

t

0

st

t

0

st

t

0

stt0

∫

∫

∫

−λ−

−λ−λ−

λλ−λ−

⋅=

⋅=

⋅=⋅−

Eq. B18

One last point should be made before leaving this appendix. It should be

reiterated that the O-U stochastic process is a special case of Gaussian covariance

stationary stochastic processes. As such, it is immediately applicable to Gaussian

phenomena manifest along a continuum as a model of their probabilistic behavior. In

nature, environments will more often than not exhibit continuous rather than discrete

variation. This is particularly true of the age continuum. Hence, any phenotype manifest

along the age continuum, or any other continuous environment of interest, can be

modeled using the approach discussed herein.

196

Appendix C: Derivation of the Elements in the Expected Fisher Information Matrix

To compute θ

Σ ˆ for the genotype × age interaction model, we need to first write

the ln-likelihood function for the model. Taking the case of a single pedigree, the ln-

likelihood function of the genotype × age interaction model is given as:

( ) [ ]ΔΣΔΣXyβ 1eegg ln)2ln(N

2

1,,,,,,Lln −′++π−=γαλγα . Eq. C1

Let the parameter vector under the genotype × age interaction model be denoted by:

[ ] ′γαλγα= ,,,,, eeggβθ , where the carats have been dropped for easier notation. The

partial derivatives of ( )θLln with respect to effects in β will not have changed under the

genotype × age interaction model. Note that on taking the first partial derivative of

( )θLln with respect to any parameter θ in θ , the right hand side will always involve the

derivative of a constant, which is always 0, thus leaving only two terms to differentiate:

( )

. θθ

ln21

θθln

021

θθln

θ)2ln(N

21

θLln

1

11

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂′∂

+∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂′∂

+∂

∂+−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂′∂

+∂

∂+

∂π∂

−=∂

∂

−

−−

ΔΣΔΣ

ΔΣΔΣΔΣΔΣθ

Eq. C2

Recall that the genetic covariance function in Equation 112 is really just one function:

( )[ ] ( )[ ] ( )

( ) ( ) ( )

( ) ( )[ ] ( )zxzxg

g

zxzgg

xgg

zx21

zgg21

xgg

qpexpageqagep2

exp

qpexpageq22

expagep22

exp

qpexp ageqexp agepexp

−λ−⎥⎦

⎤⎢⎣

⎡−+−

γ+α=

−λ−⎥⎦

⎤⎢⎣

⎡−

γ+

α⎥⎦

⎤⎢⎣

⎡−

γ+

α=

−λ−−γ+α−γ+α

197

( ) ( )

( ) . qpage2qp2

exp

qpexpage2qp2

exp

zxzxg

g

zxzxg

g

⎥⎦

⎤⎢⎣

⎡−λ−−+

γ+α=

−λ−⎥⎦

⎤⎢⎣

⎡−+

γ+α=

Further, for zx = , the covariance function just gives the variance function:

( ) ( )

( ) ( )[ ] . agepexp02

agep2exp

0age2p22

expppage2pp2

exp

xggxg

g

xg

gxxxxg

g

−γ+α=⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−γ+α=

⎥⎦

⎤⎢⎣

⎡λ−−

γ+α=⎥

⎦

⎤⎢⎣

⎡−λ−−+

γ+α

It will simplify matters to put: ( )

2

age2qpc ji −+

= and ji qpd −= . Let there be matrices

of ages, ijnn c=×C , and of age differences, ijnn d=×D , n, . . . ,1j ; n, . . . ,1i == , where:

( ) ( ) ( )

( )⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

≠∀−+

=∀−=−

=−

=

, ji ; 2

age2qp

ji ; agep2

agep2

2

age2p2

c

ji

iii

ij Eq. C3

and

⎪⎪⎩

⎪⎪⎨

⎧

≠∀−

=∀==−=

. ji ; qp

ji ; 00ppd

ji

ii

ij Eq. C4

In finding the partial derivatives, the Product Rule will be invoked at various stages. In

general, the Product Rule applies to products of functions. For matrices, the Product

Rule applies for a product of matrices of variables, since matrices can be viewed as

198

matrix-valued functions. This fact holds for standard or Hadamard matrix multiplication,

even under trace operations because the trace operation is linear. Having made these

remarks, it should be pointed out that Φ , C , and D are matrices of constants and so the

Product Rule does not apply to products involving them. This can be seen by taking their

derivatives, which will be nn×0 (compare the case for the derivatives of scalar constants).

Lastly, a fact that will prove useful in evaluating the second partial derivatives in the

ensuing is that a second partial derivative is merely the partial derivative of a first partial

derivative.

To compute the partial derivative of ( )θLln with respect to gα , we write:

( ) [ ] [ ] [ ]g

1

gg

1

g 21ln

21ln)2ln(N

21lnL

α∂′∂−

α∂∂

−=α∂

′++π∂−=

α∂∂ −− ΔΣΔΣΔΣΔΣθ , Eq. C5

where the right hand side makes explicit the fact that we can differentiate the remainder

term-by-term, which follows from the linearity of differential operators. Now,

[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂−=

α∂∂

− −

g

1

g

Tr21ln

21 ΣΣ

Σ, Eq. C6

and, on recalling that BAΦΣ += o2 , we have:

[ ] ( ). 22

22

gggggg α∂∂

=+α∂

∂=

α∂∂

+α∂

∂=

α∂+∂

=α∂∂ A

Φ0A

ΦBAΦBAΦΣ

oooo

Eq. C7

The partial derivative of a matrix with respect to a scalar parameter is the matrix of the

partial derivatives of its elements with respect to the parameter, whereby differentiation is

carried out element-by-element (Cullen, 1990: 265; Horn and Johnson, 1991: 490; Lange,

1997: 125). All of the elements in A are given by the covariance function:

⎥⎦⎤

⎢⎣⎡ λ−γ+α ijijgg dcexp ,

199

and so their partials with respect to gα are given by:

⎥⎦⎤

⎢⎣⎡ λ−γ+α=

α∂

⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

ijijgg

g

ijijgg

dcexpdcexp

. Eq. C8

In terms of a matrix 22×A say, we have:

.

dcexpdcexp

dcexpdcexp

dcexpdcexp

dcexpdcexp

2222gg2121gg

1212gg1111gg

g

2222gg

g

2121gg

g

1212gg

g

1111gg

g

A

A

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡ λ−γ+α⎥⎦

⎤⎢⎣⎡ λ−γ+α

⎥⎦⎤

⎢⎣⎡ λ−γ+α⎥⎦

⎤⎢⎣⎡ λ−γ+α

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

α∂⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

α∂⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

α∂⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

α∂⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

=α∂

∂

Eq. C9

Thus, in this case and this case only, it happens that:

AΦA

Φ oo 22g

=α∂

∂, Eq. C10

and so

[ ] ( )[ ] ( )[ ]AΦΣAΦΣΣ

oo11

g

Tr2Tr21ln

21 −− −=−=

α∂∂

− . Eq. C11

Now to the remainding term. We have:

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂′−=

α∂′∂

−−−

ΔΣ

ΔΔΣΔ

g

1

g

1

21

21

, Eq. C12

and, by the above evaluation of gα∂

∂Σ , we find that:

200

( ) 111

g

1

g

1

2 −−−−−

−=α∂∂

−=α∂

∂ΣAΦΣΣ

ΣΣ

Σo , Eq. C13

thereby giving:

[ ] ( ) [ ] ( ) . 221

21 1111

g

1

ΔΣAΦΣΔΔΣAΦΣΔΔΣΔ −−−−

−

′=−′−=α∂′∂

− oo Eq. C14

Combining results, we find that:

( ) ( )[ ] ( ) ΔΣAΦΣΔAΦΣθ 111

g

TrLln −−− ′+−=α∂

∂oo . Eq. C15

For the other two variance components terms, we will need:

⎥⎦⎤

⎢⎣⎡ λ−γ+α=

γ∂⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

ijijggijg

ijijggdcexpc

dcexp, Eq. C16

and

⎥⎦⎤

⎢⎣⎡ λ−γ+α−=

λ∂⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

ijijggij

ijijggdcexpd

dcexp. Eq. C17

Notice that the differences in these cases in comparison to g

ijijgg dcexp

α∂⎥⎦⎤

⎢⎣⎡ λ−γ+α∂

are

given by ijc and ijd− , respectively. Indeed, we find for 22×A :

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ],

dcexpcdcexpc

dcexpcdcexpc

dcexpdcexp

dcexpdcexp

2222gg222121gg21

1212gg121111gg11

g

2222gg

g

2121gg

g

1212gg

g

1111gg

g

AC

A

o=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

λ−γ+αλ−γ+α

λ−γ+αλ−γ+α=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

γ∂λ−γ+α∂

γ∂λ−γ+α∂

γ∂λ−γ+α∂

γ∂λ−γ+α∂

=γ∂

∂

Eq. C18

201

and similarly, we find:

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ ]

( )[ ] [ ]

[ ] [ ].

dcexpddcexpd

dcexpddcexpd 1

dcexpddcexpd

dcexpddcexpd

dcexpdcexp

dcexpdcexp

2222gg222121gg21

1212gg121111gg11

2222gg222121gg21

1212gg121111gg11

2222gg2121gg

1212gg1111gg

AD

A

o−=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

λ−γ+αλ−γ+α

λ−γ+αλ−γ+α−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

λ−γ+α−λ−γ+α−

λ−γ+α−λ−γ+α−=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

λ∂λ−γ+α∂

λ∂λ−γ+α∂

λ∂λ−γ+α∂

λ∂λ−γ+α∂

=λ∂

∂

Eq. C19

That is, the same differences hold for differentiation of the matrix A . Therefore, the

elements of the score vector involving the two other variance components parameters are:

( ) ( ) ( )[ ]ACΦΣΔΣACΦΣΔθ

oooo111

g

TrLln −−− −′=γ∂

∂, Eq. C20

and

( ) ( )[ ] ( ) ΔΣADΦΣΔADΦΣθ 111Tr

Lln −−− ′−=λ∂

∂oooo . Eq. C21

The partials with respect to the environmental parameters are similarly computed.

For the first of these, we have:

( ) [ ] [ ]e

1

ee 21ln

21Lln

α∂′∂−

α∂∂

−=α∂

∂ − ΔΣΔΣθ . Eq. C22

Starting with the first term, we find that:

[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂−=α∂

∂− −

e

1

e

Tr21ln

21 ΣΣ

Σ, Eq. C23

and that:

202

eeeee

2α∂

∂=

α∂∂

+=α∂

∂+

α∂∂

=α∂

∂ BB0

BAΦΣ o. Eq. C24

For say 22×B , we have:

,

cexp0

0cexp

cexp0

0cexp

22ee

11ee

e

22ee

e

11ee

e

B

B

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎦⎤

⎢⎣⎡ γ+α

⎥⎦⎤

⎢⎣⎡ γ+α

=

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

α∂⎥⎦⎤

⎢⎣⎡ γ+α∂

α∂⎥⎦⎤

⎢⎣⎡ γ+α∂

=α∂

∂

Eq. C25

and so the first term reduces to:

[ ] ( )BΣΣ 1

e

Tr21ln

21 −−=

α∂∂

− . Eq. C26

The second term is given by:

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂′−=

α∂′∂

−−−

ΔΣ

ΔΔΣΔ

e

1

e

1

21

21

, Eq. C27

and, by the above evaluation of eα∂

∂Σ , we also have:

111

e

1

e

1−−−−

−

−=α∂

∂−=

α∂∂

BΣΣΣΣ

ΣΣ

, Eq. C28

which gives:

[ ] ( )[ ] ΔBΣΣΔΔBΣΣΔΔΣΔ 1111

e

1

21

21

21 −−−−

−

′=−′−=α∂′∂− . Eq. C29

203

Combining results yields:

( ) ( )[ ] Tr21Lln 111

e

BΣΔBΣΣΔθ −−− −′=α∂

∂ . Eq. C30

It will come as no surprise to find that eγ∂

∂B is similar to gγ∂

∂A in final form. To wit:

[ ]

[ ]

[ ]

[ ].

cexpc0

0cexpc

cexp0

0cexp

22ee22

11ee11

e

22ee

e

11ee

e

BC

B

o=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

γ+α

γ+α=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

γ∂γ+α∂

γ∂γ+α∂

=γ∂

∂

Eq. C31

Therefore, we have for the last element of the score vector:

( ) ( ) ( ) [ ] Tr21Lln 111

e

BCΣΔΣBCΣΔθ

oo−−− −′=

γ∂∂

. Eq. C32

The elements in IF can now be derived. The following fact will be crucial in all

the derivations (Searle, 1982: 27; McCulloch and Searle, 2001: 309):

( ) ℜ∈∀= z ; zzTr , Eq. C33

which states that a scalar is equal to its own trace. Now consider the generic quadratic

form ZΔΔ′ , where XβyΔ −= and Z is some matrix of dimensions nn × . As pointed

out earlier, ZΔΔ′ is a scalar quadratic function. So the above fact can be used in the

following theorem (Magnus and Neudecker, 1999: 247; cf. Lange, 1997: 127; McCulloch

and Searle, 2001: 309):

( ) ( ) nnnn ; TrTr ×

× ℜ∈∀′=′=′ ZΔZΔZΔΔZΔΔ . Eq. C34

204

It will be useful to note some facts regarding the expectation operator (Magnus and

Neudecker, 1999: 244; cf. Lange, 1997: 127; McCulloch and Searle, 2001: 309):

[ ] ℜ∈∀= z ; zzE ; [ ] n ; E ℜ∈∀= zzz ; [ ] nnnn ; E ×

× ℜ∈∀= ZZZ , Eq. C35

which respectively state that the expectation of a scalar is the scalar itself, the expectation

of vector of constants is the vector itself, and the expectation of a matrix of constants is

the matrix itself. For say a matrix of constants and a vector-valued function, we have:

[ ] [ ]( ) nnnn

n ; ; EE ×× ℜ∈ℜ∈∀= ZzzZZz , Eq. C36

and

( )[ ] ( )[ ] ( )[ ] nnnn

n ; ; ETrETrTrE ×× ℜ∈ℜ∈∀′=′=′ ZΔΔΔZΔZΔΔZΔ . Eq. C37

Further, we also have:

[ ] ( )( ) [ ]( ) [ ]( ) ΣyyyyXβyXβyΔΔ =⎥⎦⎤

⎢⎣⎡ ′−−=⎥⎦

⎤⎢⎣⎡ ′−−=′ EEEEE , Eq. C38

which is the standard definition of Σ (Magnus and Neudecker, 1999: 246).

For the diagonal element in IF for parameter gα , we have:

( ) ( )

( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ]. E TrE

Tr

E

Tr

E

Lln

ELln

E

g

11

g

1

g

111

g

111

gggg

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′+−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

α∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

−

−−−

−−−

−−−

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

θθ

oo

oo

oo

Eq. C39

205

It will be convenient to evaluate the resultant terms separately. Taking the first term:

( ) ( )

( ) ( )

( )

( ) ( ) ( )[ ]

( )[ ] ( ) . 2TrTr

2Tr

Tr

Tr

TrTrE

2 11

111

1

g

1

g

1

g

1

g

1

g

1

g

1

⎥⎦⎤

⎢⎣⎡−=

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

α∂∂

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

α∂∂

+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

α∂∂

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

α∂∂

−−

−−−

−−−

−−

−−

AΦΣAΦΣ

AΦΣAΦΣAΦΣ

AΦΣΣ

ΣA

ΦΣ

AΦΣAΦ

Σ

AΦΣAΦΣ

oo

ooo

oo

oo

oo

Eq. C40

For the second term, we find:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( ) ( )

( ) ( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−α∂

∂+

α∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

α∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

−−−−−

−−−−

−

−−−

−

−−−−

−−−−

2 Tr

Tr

Tr

ETr TrE

E E

111

g

1

g

1

1

g

11

g

1

g

1

g

111

g

1

g

11

g

11

g

11

g

11

IAΦΣAΦΣAΦΣΣ

ΣA

ΦΣ

ΣΣΣ

ΣAΦΣAΦΣAΦ

Σ

ΣΣ

AΦΣΣAΦΣ

ΔΔΣAΦΣ

ΔΣAΦΣ

Δ

ΔΣAΦΣ

ΔΔΣAΦΣΔ

oooo

ooo

oo

oo

oo

Eq. C41

206

( ) ( ) ( ) ( ) ( )[ ]( )

( ) ( ) ( ) ( ) ( )[ ]( )

( )[ ] ( ) . Tr4Tr

22 Tr

22 Tr

2 11

11111

11111

⎥⎦⎤

⎢⎣⎡+−=

−−−=

−−−=

−−

−−−−−

−−−−−

AΦΣAΦΣ

AΦΣAΦΣAΦΣAΦΣAΦΣ

AΦΣAΦΣAΦΣAΦΣAΦΣ

oo

ooooo

ooooo

Summing terms gives:

( ) ( )[ ] ( )

( )[ ] ( )

( ) . Tr2

4TrTr

Tr2TrLln

E

2 1

2 11

2 11

gg

2

⎥⎦⎤

⎢⎣⎡=

⎥⎦⎤

⎢⎣⎡+−

⎥⎦⎤

⎢⎣⎡−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

−

−

−−

−−

AΦΣ

AΦΣAΦΣ

AΦΣAΦΣθ

o

oo

oo

Eq. C42

For the diagonal element in IF for parameter gγ , we find:

( ) ( )

( ) ( )[ ]

( ) [ ] ( )

( ) ( ) . E TrE

Tr

E

Tr

E

Lln

ELln

E

g

11

g

1

g

11

g

1

g

111

gggg

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+γ∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′+−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

γ∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂γ∂∂

−

−−−

−−−

−−−

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

θθ

oooo

oooo

oooo

Eq. C43

We find for the first term:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂ −−

γ

Tr γ

TrEg

1

g

1 ACΦΣACΦΣ oooo Eq. C44

207

( ) ( )

( )

( ) ( ) ( )[ ]

( )[ ] ( ) . Tr2 Tr

2Tr

γγTr

γγTr

2 11

111

1

g

1

g

1

g

1

g

1

⎥⎦⎤

⎢⎣⎡−=

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

+∂

∂=

−−

−−−

−−−

−−

ACΦΣACCΦΣ

ACΦΣACΦΣACCΦΣ

ACΦΣΣ

ΣA

CΦΣ

ACΦΣACΦ

Σ

ooooo

ooooooo

oooo

oooo

For the second term, we have:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( ) ( )( ) ( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

γ∂∂

−

γ∂∂

+γ∂

∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

γ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

−−

−−−

−−

−−−

−

−−

−−

−−−

−

−−−−

−−−−

2

2 Tr

2

Tr

Tr

Tr

ETr TrE

E E

11

111

11

1

g

1

g

1

1

g

11

g

1

g

1

g

111

g

1

g

11

g

11

g

11

g

11

ACΦΣACΦΣ

ACΦΣACΦΣACCΦΣ

I

ACΦΣACΦΣ

ACΦΣΣ

ΣA

CΦΣ

ΣΣΣ

ΣACΦΣ

ACΦΣACΦ

Σ

ΣΣ

ACΦΣΣACΦΣ

ΔΔΣACΦΣ

ΔΣACΦΣ

Δ

ΔΣACΦΣ

ΔΔΣACΦΣΔ

ooooK

Kooooooo

ooooK

Koooo

ooK

Koooo

oooo

oooo

oooo

Eq. C45

208

( ) ( )

( )[ ] ( ) . Tr4Tr

4 Tr

2 11

2 11

⎥⎦⎤

⎢⎣⎡+−=

⎟⎠⎞⎜

⎝⎛

⎥⎦⎤

⎢⎣⎡ −−=

−−

−−

ACΦΣACCΦΣ

ACΦΣACCΦΣ

ooooo

ooooo

Summing terms gives:

( ) ( )[ ] ( )

( )[ ] ( )

( ) . Tr2

4TrTr

Tr2TrLln

E

2 1

2 11

2 11

gg

2

⎥⎦⎤

⎢⎣⎡=

⎥⎦⎤

⎢⎣⎡+−

⎥⎦⎤

⎢⎣⎡−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂γ∂∂

−

−

−−

−−

ACΦΣ

ACΦΣACCΦΣ

ACΦΣACCΦΣθ

oo

ooooo

ooooo

Eq. C46

The diagonal element in IF for parameter λ is similarly computed as follows:

( ) ( )

( ) ( ) [ ]

( ) [ ] ( )[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′∂

+λ∂

−∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂+′−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡λ∂

∂λ∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂λ∂∂

−

−−−

−−−

Tr

E

Tr

E

Lln

ELln

E

111

111

2

ΔΣADΦΣΔADΦΣ

ADΦΣΔΣADΦΣΔ

θθ

oooo

oooo Eq. C47

( ) ( ) . E TrE

111

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

−=−−− ΔΣADΦΣΔADΦΣ oooo

We find for the first term:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂

−−−

Tr TrE11 ADΦΣADΦΣ oooo

Eq. C48

209

( ) ( )

( )

( ) ( ) ( )[ ]

( )[ ] ( ) . Tr2 Tr

2Tr

Tr

Tr

2 11

111

111

11

⎥⎦⎤

⎢⎣⎡−=

−−−−=

⎥⎦

⎤⎢⎣

⎡

λ∂∂

−⎭⎬⎫

⎩⎨⎧

λ∂∂

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

+λ∂

∂−=

−−

−−−

−−−

−−

ADΦΣADDΦΣ

ADΦΣADΦΣADDΦΣ

ADΦΣΣ

ΣA

DΦΣ

ADΦΣADΦ

Σ

ooooo

ooooooo

oooo

oooo

For the second term, we have:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( )[ ] ( )

( ) ( )

( ) ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

λ∂∂

−

λ∂∂

+λ∂

∂

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

+λ∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

λ∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂′=

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂′=⎟

⎟⎠

⎞⎜⎜⎝

⎛

λ∂′∂

−

−−

−−

−−−

−

−−−−

−−−−

Tr

Tr

ETr TrE

E E

1

11

11

111

1

1111

1111

ΣΣΣ

ΣADΦΣ

ADΦΣADΦ

Σ

ΣΣ

ADΦΣΣADΦΣ

ΔΔΣADΦΣ

ΔΣADΦΣ

Δ

ΔΣADΦΣ

ΔΔΣADΦΣΔ

ooK

Koooo

oooo

oooo

oooo

Eq. C49

( )

( ) ( )

( ) ( ) ( )( ) ( ) ⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

−−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

λ∂∂

−⎟⎠

⎞⎜⎝

⎛λ∂

∂=

−−

−−−

−−

−−−

2

2 Tr

2

Tr

11

111

11

111

ADΦΣADΦΣ

ADΦΣADΦΣADDΦΣ

I

ADΦΣADΦΣ

ADΦΣΣ

ΣA

DΦΣ

ooooK

Kooooooo

ooooK

Koooo

210

( ) ( )

( )[ ] ( ) . Tr4Tr

4 Tr

2 11

2 11

⎥⎦⎤

⎢⎣⎡+−=

⎟⎠⎞⎜

⎝⎛

⎥⎦⎤

⎢⎣⎡ +−=

−−

−−

ADΦΣADDΦΣ

ADΦΣADDΦΣ

ooooo

ooooo

Summing terms gives:

( ) ( )[ ] ( )

( )[ ] ( )

( ) . Tr2

4TrTr

Tr2TrLln

E

2 1

2 11

2 112

⎥⎦⎤

⎢⎣⎡=

⎥⎦⎤

⎢⎣⎡+−

⎥⎦⎤

⎢⎣⎡−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

λ∂λ∂∂

−

−

−−

−−

ADΦΣ

ADΦΣADDΦΣ

ADΦΣADDΦΣθ

oo

ooooo

ooooo

Eq. C50

We now look to the environmental component of the model. For the diagonal

element in IF for parameter eα , we find:

( ) ( ) ( )[ ]

( )[ ] [ ] [ ].

21

E Tr21

E 21Tr

21

E

Tr

21

E Lln

ELln

E

e

11

e

1

e

11

e

1

e

111

eeee

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

+α∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′+−∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡α∂

∂α∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

−

−−−−−−

−−−

ΔBΣΣΔBΣΔBΣΣΔBΣ

ΔBΣΣΔBΣθθ

Eq. C51

For the first term, we find:

[ ] ( )

[ ] [ ] ( ) . Tr21

Tr21

Tr21

Tr21

Tr21

Tr21

Tr21

E

2 111111

e

11

e

1

e

1

e

1

e

1

⎥⎦⎤

⎢⎣⎡−=−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂

−−−−−−−−

−−

−−

BΣBΣBBΣΣBΣBΣΣ

ΣBΣ

BΣB

ΣBΣBΣ

Eq. C52

211

For the second term, we have:

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] ( ) . TrTr21

2Tr21

Tr21

Tr21

Tr21

ETr21

Tr21

E 21

E 21

E

2 112 11

111

e

11

1

e

11

e

1

e

1

e

111

e

1

e

11

e

11

e

11

e

11

⎥⎦⎤

⎢⎣⎡+−=⎟

⎠⎞⎜

⎝⎛

⎥⎦⎤

⎢⎣⎡ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

α∂∂

−−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−α∂

∂+

α∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂−=⎟

⎟⎠

⎞⎜⎜⎝

⎛′

α∂∂

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

−−−−

−−−−−

−−−−

−

−−−

−−−

−−−−−−

BΣBΣBΣBΣ

IBBΣΣBΣΣ

ΣBΣ

ΣΣΣ

BΣΣBΣB

Σ

ΣΣ

BΣΣBΣ

ΔΔBΣΣ

ΔBΣΣ

ΔΔBΣΣ

ΔΔBΣΣΔ

Eq. C53

Summing terms gives:

( ) [ ] ( )

[ ] ( )

( ) . Tr21

TrTr21

Tr21

Tr21Lln

E

2 1

2 11

2 11

ee

2

⎥⎦⎤

⎢⎣⎡=

⎥⎦⎤

⎢⎣⎡+−

⎥⎦⎤

⎢⎣⎡−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

−

−

−−

−−

BΣ

BΣBΣ

BΣBΣθ

Eq. C54

For the diagonal element in IF for parameter eγ , we have:

( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

γ∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂γ∂∂

− Lln

ELln

Eeeee

2 θθ Eq. C55

212

( ) ( )[ ]

( ) [ ] ( )[ ]

( )[ ] ( )[ ].

21

E Tr21

E

21Tr

21

E

Tr

21

E

e

11

e

1

e

11

e

1

e

111

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+γ∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′+−∂

−=

−−−

−−−

−−−

ΔΣBCΣΔBCΣ

ΔΣBCΣΔBCΣ

ΔΣBCΣΔBCΣ

oo

oo

oo

For the first term, we have:

( )[ ] ( )[ ]

( ) ( )

( )⎥⎦

⎤⎢⎣

⎡

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛γ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂=

γ∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

−−−

−−

−−

BCΣΣ

ΣB

CΣ

BCΣBC

Σ

BCΣBCΣ

oo

oo

oo

1

e

1

e

1

e

1

e

1

e

1

e

1

Tr21

Tr21

Tr21

Tr21

E

Eq. C56

( ) ( ) ( )[ ]

( )[ ] ( ) . Tr21

Tr21

Tr21

2 11

111

⎥⎦⎤

⎢⎣⎡−=

−=

−−

−−−

BCΣBCCΣ

BCΣBCΣBCCΣ

ooo

oooo

For the second term, we find:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛′

γ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

−−−−

−−−−

ETr21

Tr21

E

21

E 21

E

e

11

e

11

e

11

e

11

ΔΔΣBCΣ

ΔΣBCΣ

Δ

ΔΣBCΣ

ΔΔΣBCΣΔ

oo

oo

Eq. C57

213

( )[ ] ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )[ ] ( ) . TrTr21

2Tr21

Tr21

Tr21

Tr21

2 11

2 11

111

e

11

1

e

11

e

1

e

1

e

111

e

1

⎥⎦⎤

⎢⎣⎡+−=

⎟⎠⎞⎜

⎝⎛

⎥⎦⎤

⎢⎣⎡ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

γ∂∂

−−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−γ∂

∂+

γ∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂−=

−−

−−

−−−−−

−−−−

−

−−−

−

BCΣBCCΣ

BCΣBCCΣ

IBCΣBCΣBCΣΣ

ΣBCCΣ

ΣΣΣ

ΣBCΣBCΣBC

Σ

ΣΣ

BCΣΣBCΣ

ooo

ooo

ooooo

ooo

oo

On summing terms, we find:

( ) ( )[ ] ( )

( )[ ] ( )

( ) . Tr21

TrTr21

Tr21

Tr21Lln

E

2 1

2 11

2 11

ee

2

⎥⎦⎤

⎢⎣⎡=

⎥⎦⎤

⎢⎣⎡+−

⎥⎦⎤

⎢⎣⎡−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂γ∂∂

−

−

−−

−−

BCΣ

BCΣBCCΣ

BCΣBCCΣθ

o

ooo

ooo

Eq. C58

By the theorem on the identity of mixed partial derivatives regardless of

differentiation order (Eq. 124), there are ten, unique, mixed partial derivatives involving

the variance components parameters and, together with the above results, these will give

the sampling variances and covariances in 55×Ω in θ

Σ ˆ after inversion of IF . By

symmetry, there will be four, three, two, and one unique, mixed partial derivatives with

respect to the first partial derivatives evaluated with respect to say gα , gγ , λ , and eα ,

respectively. Incidentally, any permutation of four of the five variables of differentiation

214

could have been taken. This order is consistent, however, with the order adhered to thus

far. All together, we will have 15 unique, second partial derivatives.

Given the above order, the first of the mixed elements in IF is ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−gg

2 LlnE

θ,

which is computed as follows:

( ) ( )

( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ]. E TrE

Tr

E

Tr

E

Lln

ELln

E

g

11

g

1

g

111

g

111

gggg

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′+−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

γ∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

−−−

−−−

−−−

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

θθ

oo

oo

oo

Eq. C59

Taking the first term, we have:

( ) ( )

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−−

−−

AΦΣAΦ

Σ

AΦΣAΦΣ

oo

oo

g

1

g

1

g

1

g

1

Tr

Tr TrE

Eq. C60

( )

( ) ( ) ( )[ ]

( )[ ] ( ) ( )[ ] . Tr2 Tr

2Tr

Tr

111

111

1

g

1

g

1

AΦΣACΦΣACΦΣ

AΦΣACΦΣACΦΣ

AΦΣΣ

ΣA

ΦΣ

ooooo

ooooo

oo

−−−

−−−

−−−

−=

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

=

215

The second term is evaluated as follows:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ]( )

( )[ ] ( ) ( )[ ] ( ) ( )[ ] . Tr2Tr2Tr

22 Tr

2 Tr

Tr

Tr

ETr TrE

E E

11111

11111

111

g

1

g

1

1

g

11

g

1

g

1

g

111

g

1

g

11

g

11

g

11

g

11

ACΦΣAΦΣAΦΣACΦΣACΦΣ

ACΦΣAΦΣAΦΣACΦΣACΦΣ

IACΦΣAΦΣAΦΣΣ

ΣA

ΦΣ

ΣΣΣ

ΣAΦΣAΦΣAΦ

Σ

ΣΣ

AΦΣΣAΦΣ

ΔΔΣAΦΣ

ΔΣAΦΣ

Δ

ΔΣAΦΣ

ΔΔΣAΦΣΔ

oooooooo

oooooooo

ooooo

ooo

oo

oo

oo

−−−−−

−−−−−

−−−−−

−−−−

−

−−−

−

−−−−

−−−−

++−=

−−−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−γ∂

∂+

γ∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

γ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

Eq. C61

On summing terms, we have:

( ) ( )[ ] ( ) ( )[ ]( )[ ] ( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ] . Tr2

Tr2

Tr2 Tr

Tr2 TrLln

E

11

11

111

111

gg

2

ACΦΣAΦΣ

ACΦΣAΦΣ

AΦΣACΦΣACΦΣ

AΦΣACΦΣACΦΣθ

ooo

oooK

Kooooo

ooooo

−−

−−

−−−

−−−

=

+

+−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

Eq. C62

216

In the stated order, the second of the mixed elements in IF is ( )

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂λ∂∂

−g

2 LlnE

θ,

which is computed as follows:

( ) ( )

( ) ( )[ ]

( ) [ ] ( )[ ]

( ) ( )[ ]. E TrE

Tr

E

Tr

E

Lln

ELln

E

111

111

111

gg

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′−∂

+λ∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′+−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

λ∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂λ∂∂

−

−−−

−−−

−−−

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

θθ

oo

oo

oo

Eq. C63

The first term is found to be:

( ) ( )

( ) ( )

( )⎥⎦

⎤⎢⎣

⎡λ∂

∂−⎟

⎠

⎞⎜⎝

⎛λ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

+λ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

−−−

−−

−−

AΦΣΣ

ΣA

ΦΣ

AΦΣAΦ

Σ

AΦΣAΦΣ

oo

oo

oo

111

11

11

Tr

Tr

Tr TrE

Eq. C64

( ) ( ) ( )[ ]

( )[ ] ( ) ( )[ ] . Tr2 Tr

2Tr

111

111

AΦΣADΦΣADΦΣ

AΦΣADΦΣADΦΣ

ooooo

ooooo

−−−

−−−

+−=

−−−=

The second term is evaluated as follows:

217

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( ) ( )

( )

( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( )[ ] ( ) ( )[ ]( ) ( )[ ] . Tr2

Tr2 Tr

2

2 Tr

2

2 Tr

2

Tr

Tr

Tr

ETr TrE

E E

11

111

11

111

11

111

11

11

11

1111

1

111

1

1111

1111

ADΦΣAΦΣ

AΦΣADΦΣADΦΣ

ADΦΣAΦΣ

AΦΣADΦΣADΦΣ

ADΦΣAΦΣ

AΦΣADΦΣADΦΣ

I

ADΦΣAΦΣ

AΦΣΣ

ΣA

ΦΣ

ΣΣΣ

ΣAΦΣAΦΣAΦ

Σ

ΣΣ

AΦΣΣAΦΣ

ΔΔΣAΦΣ

ΔΣAΦΣ

Δ

ΔΣAΦΣ

ΔΔΣAΦΣΔ

oooK

Kooooo

oooK

Kooooo

oooK

Kooooo

oooK

Koo

ooo

oo

oo

oo

−−

−−−

−−

−−−

−−

−−−

−−

−−

−−

−−−−

−

−−−

−

−−−−

−−−−

−

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

+−−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

−−−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

λ∂∂

−⎟⎠

⎞⎜⎝

⎛λ∂

∂−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

−λ∂

∂+

λ∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

+λ∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

λ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

λ∂′−∂

Eq. C65

On summing terms, we have:

( ) ( )[ ] ( ) ( )[ ]( )[ ] ( ) ( )[ ]

( ) ( )[ ]ADΦΣAΦΣ

AΦΣADΦΣADΦΣ

AΦΣADΦΣADΦΣθ

oooK

Kooooo

ooooo

11

111

111

g

2

Tr2

Tr2 Tr

Tr2 TrLln

E

−−

−−−

−−−

−

−+

+−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

α∂λ∂∂

−

Eq. C66

218

( ) ( )[ ] . Tr2 11 ADΦΣAΦΣ ooo−−−=

The third of the mixed elements in IF under the specified order is

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

−ge

2 LlnE

θ. It is computed as:

( ) ( )

( ) ( )[ ]

( ) [ ] ( )[ ]

( ) ( )[ ]. E TrE

Tr

E

Tr

E

Lln

ELln

E

e

11

e

1

e

11

e

1

e

111

gege

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

+α∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′+−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

α∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

−

−−−

−−−

−−−

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

θθ

oo

oo

oo

Eq. C67

Starting with the first term, we find:

( ) ( )

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−−

−−

AΦΣAΦ

Σ

AΦΣAΦΣ

oo

oo

e

1

e

1

e

1

e

1

Tr

Tr TrE

Eq. C68

( )

( ) ( )[ ] ( )[ ] . TrTr

Tr

11111

1

e

1

e

1

AΦBΣΣAΦBΣΣ0Σ

AΦΣΣ

ΣA

ΦΣ

oo

oo

−−−−−

−−−

−=−=

⎥⎦

⎤⎢⎣

⎡

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂=

The second term is evaluated as follows:

219

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )[ ]( )

( )[ ] ( )[ ] . Tr Tr

Tr

Tr

Tr

d

Tr

Ed

Tr d

TrE

dE

dE

1111

11111

111

e

1

e

1

1

e

11

e

1

e

1

e

111

e

1

e

11

e

11

e

11

e

11

BΣAΦΣAΦBΣΣ

BΣAΦΣAΦBΣΣ0Σ

IBΣAΦΣAΦΣΣ

ΣA

ΦΣ

ΣΣΣ

ΣAΦΣAΦΣAΦ

Σ

ΣΣ

AΦΣΣAΦΣ

ΔΔΣAΦΣ

ΔΣAΦΣ

Δ

ΔΣAΦΣ

ΔΔΣAΦΣΔ

−−−−

−−−−−

−−−−−

−−−−

−

−−−

−

−−−−

−−−−

+=

−−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−α∂

∂+

α∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

α∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂′−

oo

oo

ooo

ooo

oo

oo

oo

Eq. C69

Summing terms gives:

( ) ( )[ ]( )[ ] ( )[ ]

( )[ ] . Tr

TrTr

TrLln

E

11

1111

11

ge

2

BΣAΦΣ

BΣAΦΣAΦBΣΣ

AΦBΣΣθ

−−

−−−−

−−

=

++

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

−

o

oo

o

Eq. C70

The fourth of the mixed elements in IF under the specified order is

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−ge

2 LlnE

θ, which is found as follows:

220

( ) ( )

( ) ( )[ ]

( ) [ ] ( )[ ]

( ) ( )[ ]. E TrE

Tr

E

Tr

E

Lln

ELln

E

e

11

e

1

e

11

e

1

e

111

gege

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+γ∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′+−∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

γ∂∂

−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

α∂γ∂∂

−

−−−

−−−

−−−

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

ΔΣAΦΣΔAΦΣ

θθ

oo

oo

oo

Eq. C71

The first term is given as:

( ) ( )

( ) ( )

( )

( ) ( ) ( )[ ]

( ) ( )[ ] . Tr

Tr

Tr

Tr

Tr TrE

11

111

1

e

1

e

1

e

1

e

1

e

1

e

1

AΦΣBCΣ

AΦΣBCΣ0Σ

AΦΣΣ

ΣA

ΦΣ

AΦΣAΦ

Σ

AΦΣAΦΣ

oo

oo

oo

oo

oo

−−

−−−

−−−

−−

−−

−=

−=

⎥⎦

⎤⎢⎣

⎡

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛γ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

Eq. C72

The next term is evaluated as follows:

( )[ ] ( )[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂ −−−−

EEe

11

e

11

ΔΣAΦΣ

ΔΔΣAΦΣΔ oo

Eq. C73

221

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ]( )

( ) ( )[ ] ( ) ( )[ ] . TrTr

Tr

Tr

Tr

Tr

ETr TrE

1111

11111

111

e

1

e

1

1

e

11

e

1

e

1

e

111

e

1

e

11

e

11

BCΣAΦΣAΦΣBCΣ

BCΣAΦΣAΦΣBCΣ0Σ

IBCΣAΦΣAΦΣΣ

ΣA

ΦΣ

ΣΣΣ

ΣAΦΣAΦΣAΦ

Σ

ΣΣ

AΦΣΣAΦΣ

ΔΔΣAΦΣ

ΔΣAΦΣ

Δ

oooo

oooo

oooo

ooo

oo

oo

−−−−

−−−−−

−−−−−

−−−−

−

−−−

−

−−−−

+=

−−−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡−

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−γ∂

∂+

γ∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

γ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=

Summing terms gives:

( ) ( ) ( )[ ]( ) ( )[ ] ( ) ( )[ ]

( ) ( )[ ] . Tr

TrTr

TrLln

E

11

1111

11

ge

2

BCΣAΦΣ

BCΣAΦΣAΦΣBCΣ

AΦΣBCΣθ

oo

oooo

oo

−−

−−−−

−−

=

++

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

Eq. C74

The next set of mixed elements in IF have their first partial derivative evaluated with

respect to gγ . The first of these involves the two genetic slope parameters, gγ and λ , for

the additive genetic variance and genetic correlation functions, respectively:

( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

λ∂∂

−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

γ∂λ∂∂

− Lln

ELln

Egg

2 θθ Eq. C75

222

( )[ ] ( )

( )[ ] ( )

( ) ( ) . E TrE

Tr

E

Tr

E

111

111

111

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′−∂

+λ∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂′+−∂

−=

−−−

−−−

−−−

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

oooo

oooo

oooo

Evaluating the first term gives:

( ) ( )

( ) ( )

( )

( ) ( ) ( )[ ]

( ) ( ) ( )[ ]

( )[ ] ( ) ( )[ ] . Tr2 Tr

2Tr

2Tr

Tr

Tr

Tr TrE

111

111

111

111

11

11

ACΦΣADΦΣADCΦΣ

ACΦΣADΦΣADCΦΣ

ACΦΣADΦΣADCΦΣ

ACΦΣΣ

ΣA

CΦΣ

ACΦΣACΦ

Σ

ACΦΣACΦΣ

ooooooo

ooooooo

ooooooo

oooo

oooo

oooo

−−−

−−−

−−−

−−−

−−

−−

+−=

+−=

−−−=

⎥⎦

⎤⎢⎣

⎡λ∂

∂−⎟

⎠

⎞⎜⎝

⎛λ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

+λ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

Eq. C76

The next term is found as follows:

( )[ ] ( )[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

λ∂′−∂ −−−−

E E1111

ΔΣACΦΣ

ΔΔΣACΦΣΔ oooo

( )[ ] ( )[ ] [ ]⎟⎟⎠

⎞⎜⎜⎝

⎛′

λ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂∂′−=

−−−−

ΔΔΣACΦΣ

ΔΣACΦΣ

Δ ETr TrE1111

oooo Eq. C77

223

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( ) ( )

( )[ ] ( ) ( )[ ]( ) ( )[ ] . Tr2

Tr2Tr

2

2 Tr

2

2 Tr

2

Tr

Tr

Tr

11

111

11

111

11

111

1

111

111

111

11

1

111

1

ADΦΣACΦΣ

ACΦΣADΦΣADCΦΣ

ADΦΣACΦΣ

ACΦΣADΦΣADCΦΣ

IADΦΣACΦΣ

ACΦΣADΦΣADCΦΣ

ΣΣ

ΣADΦΣACΦΣ

ACΦΣΣ

ΣA

CΦΣ

Σ

ΣΣ

ΣACΦΣ

ΣACΦΣACΦ

Σ

ΣΣ

ACΦΣΣACΦΣ

ooooK

Kooooooo

ooooK

Kooooooo

ooooK

Kooooooo

ooooK

Koooo

ooK

Koooo

oooo

−−

−−−

−−

−−−

−−

−−−

−

−−−

−−−

−−−

−−

−

−−−

−

−

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

+−−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+

−−−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

λ∂∂

−⎟⎠

⎞⎜⎝

⎛λ∂

∂−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

λ∂∂

−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

λ∂∂

+λ∂

∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

λ∂∂

+λ∂

∂−=

Summing terms gives:

( ) ( )[ ] ( ) ( )[ ]( )[ ] ( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ] . Tr2

Tr2

Tr2Tr

Tr2TrLln

E

11

11

111

111

g

2

ADΦΣACΦΣ

ADΦΣACΦΣ

ACΦΣADΦΣADCΦΣ

ACΦΣADΦΣADCΦΣθ

oooo

ooooK

Kooooooo

ooooooo

−−

−−

−−−

−−−

−=

−

−+

+−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂λ∂∂

−

Eq. C78

The next mixed element in IF that has its first derivative evaluated with respect

224

to gγ is evaluated as follows:

( ) ( )

( )[ ] ( )

( )[ ] ( )

( ) ( ) . E TrE

Tr

E

Tr

E

Lln

ELln

E

e

11

e

1

e

11

e

1

e

111

gege

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

+α∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′+−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

α∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂α∂∂

−

−−−

−−−

−−−

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

θθ

oooo

oooo

oooo

Eq. C79

The first term is found as follows:

( ) ( )

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−−

−−

ACΦΣACΦ

Σ

ACΦΣACΦΣ

oooo

oooo

e

1

e

1

e

1

e

1

Tr

Tr TrE

Eq. C80

( )

( ) ( )[ ]

( )[ ] . Tr

Tr

Tr

11

111

1

e

1

e

1

ACΦBΣΣ

ACΦBΣΣ0Σ

ACΦΣΣ

ΣA

CΦΣ

oo

oo

oooo

−−

−−−

−−−

−=

−=

⎥⎦

⎤⎢⎣

⎡

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂=

The second term is evaluated as follows:

( )[ ] ( )[ ]⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂ −−−−

E Ee

11

e

11

ΔΣACΦΣ

ΔΔΣACΦΣΔ oooo

Eq. C81

225

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( )

( )

( )

( )

( ) ( ) ( )[ ]( )

( )[ ] ( )[ ] . TrTr

Tr

Tr

Tr

Tr

ETr TrE

1111

11111

11

1

e

1

e

1

1

e

11

e

1

e

1

e

111

e

1

e

11

e

11

BΣACΦΣACΦBΣΣ

BΣACΦΣACΦBΣΣ0Σ

I

BΣACΦΣ

ACΦΣΣ

ΣA

CΦΣ

ΣΣΣ

ΣACΦΣ

ACΦΣACΦ

Σ

ΣΣ

ACΦΣΣACΦΣ

ΔΔΣACΦΣ

ΔΣACΦΣ

Δ

−−−−

−−−−−

−−

−−−

−

−−

−−

−−−

−

−−−−

+=

−−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂

−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

α∂∂

−

α∂∂

+α∂

∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

α∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′−=

oooo

oooo

ooK

Koooo

ooK

Koooo

oooo

oooo

Summing terms gives:

( ) ( )[ ]( )[ ] ( )[ ]

( )[ ] . Tr

TrTr

TrLln

E

11

1111

11

ge

2

BΣACΦΣ

BΣACΦΣACΦBΣΣ

ACΦBΣΣθ

−−

−−−−

−−

=

++

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂α∂∂

−

oo

oooo

oo

Eq. C82

The last mixed element in IF with gγ as its first partial derivative is:

( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

γ∂∂

−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

γ∂γ∂∂

− Lln

ELln

Egege

2 θθ Eq. C83

226

( )[ ] ( )

( )[ ] ( )

( ) ( ) . E TrE

Tr

E

Tr

E

e

11

e

1

e

11

e

1

e

111

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+γ∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′+−∂

−=

−−−

−−−

−−−

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

ΔΣACΦΣΔACΦΣ

oooo

oooo

oooo

The first term gives:

( ) ( )

( ) ( )

( )

( ) ( ) ( )[ ] ( ) ( )[ ] . TrTr

Tr

Tr

Tr TrE

11111

1

e

1

e

1

e

1

e

1

e

1

e

1

ACΦΣBCΣACΦΣBCΣ0Σ

ACΦΣΣ

ΣA

CΦΣ

ACΦΣACΦ

Σ

ACΦΣACΦΣ

oooooo

oooo

oooo

oooo

−−−−−

−−−

−−

−−

−=−=

⎥⎦

⎤⎢⎣

⎡

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛γ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

Eq. C84

The second term is evaluated as follows:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( ) ( )⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

γ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

−−−

−

−−−−

−−−−

ΣΣ

ACΦΣΣACΦΣ

ΔΔΣACΦΣ

ΔΣACΦΣ

Δ

ΔΣACΦΣ

ΔΔΣACΦΣΔ

Tr

ETr TrE

E E

e

111

e

1

e

11

e

11

e

11

e

11

oooo

oooo

oooo

Eq. C85

227

( ) ( )

( )

( )

( ) ( )

( ) ( ) ( ) ( ) ( )[ ]( )

( ) ( )[ ] ( ) ( )[ ] . TrTr

Tr

Tr

Tr

1111

11111

11

1

e

1

e

1

1

e

11

e

1

e

1

BCΣACΦΣACΦΣBCΣ

BCΣACΦΣACΦΣBCΣ0Σ

I

BCΣACΦΣ

ACΦΣΣ

ΣA

CΦΣ

ΣΣΣ

ΣACΦΣ

ACΦΣACΦ

Σ

oooooo

oooooo

oooK

Koooo

ooK

Koooo

−−−−

−−−−−

−−

−−−

−

−−

−−

+=

−−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

−=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

γ∂∂

−

γ∂∂

+γ∂

∂

−=

On summing terms, we find:

( ) ( ) ( )[ ]( ) ( )[ ] ( ) ( )[ ]

( ) ( )[ ] . Tr

TrTr

TrLln

E

11

1111

11

ge

2

BCΣACΦΣ

BCΣACΦΣACΦΣBCΣ

ACΦΣBCΣθ

ooo

oooooo

ooo

−−

−−−−

−−

=

++

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂γ∂∂

−

Eq. C86

The next two mixed elements in IF have their first partial derivatives evaluated with

respect to λ . The first of these is:

( ) ( )

( )[ ] ( )

( )[ ] ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′∂

+α∂

−∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡λ∂

∂α∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂α∂∂

−

−−−

−−−

Tr

E

Tr

E

Lln

ELln

E

e

11

e

1

e

111

ee

2

ΔΣADΦΣΔADΦΣ

ΔΣADΦΣΔADΦΣ

θθ

oooo

oooo Eq. C87

228

( ) ( ) . E TrE

e

11

e

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂′∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−=−−− ΔΣADΦΣΔADΦΣ oooo

The first term is evaluated as follows:

( ) ( )

( ) ( )

( )

( ) ( )[ ]

( )[ ] . Tr

Tr

Tr

Tr

Tr TrE

11

111

1

e

1

e

1

e

1

e

1

e

1

e

1

ADΦBΣΣ

ADΦBΣΣ0Σ

ADΦΣΣ

ΣA

DΦΣ

ADΦΣADΦ

Σ

ADΦΣADΦΣ

oo

oo

oooo

oooo

oooo

−−

−−−

−−−

−−

−−

=

−−=

⎥⎦

⎤⎢⎣

⎡

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛α∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

−

Eq. C88

The second term is found to be:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]

( ) ( )

( ) ( )

( ) ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

α∂∂

−

α∂∂

+α∂

∂

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

α∂∂

+α∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

α∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′=

⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂′=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂′∂

−

−−

−−

−−−

−

−−−−

−−−−

ΣΣΣ

ΣADΦΣ

ADΦΣADΦ

Σ

ΣΣ

ADΦΣΣADΦΣ

ΔΔΣADΦΣ

ΔΣADΦΣ

Δ

ΔΣADΦΣ

ΔΔΣADΦΣΔ

1

e

11

e

1

e

1

e

111

e

1

e

11

e

11

e

11

e

11

Tr

Tr

ETr TrE

E E

ooK

Koooo

oooo

oooo

oooo

Eq. C89

229

( )

( )

( ) ( ) ( )[ ]( )

( )[ ] ( )[ ] . TrTr

Tr

Tr

1111

11111

11

1

e

1

e

1

BΣADΦΣADΦBΣΣ

BΣADΦΣADΦBΣΣ0Σ

I

BΣADΦΣ

ADΦΣΣ

ΣA

DΦΣ

−−−−

−−−−−

−−

−−−

−−=

−−=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

α∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂∂

=

oooo

oooo

ooK

Koooo

On summing terms, we find:

( ) ( )[ ]( )[ ] ( )[ ]

( )[ ] . Tr

TrTr

TrLln

E

11

1111

11

e

2

BΣADΦΣ

BΣADΦΣADΦBΣΣ

ADΦBΣΣθ

−−

−−−−

−−

−=

−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂α∂∂

−

oo

oooo

oo

Eq. C90

The next mixed element in IF that has its first partial derivative evaluated with respect to

λ is evaluated as follows:

( ) ( )

( )[ ] ( )

( )[ ] ( )

( ) ( ) . E TrE

Tr

E

Tr

E

Lln

ELln

E

e

11

e

1

e

11

e

1

e

111

ee

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′∂

+γ∂

−∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡λ∂

∂γ∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂γ∂∂

−

−−−

−−−

−−−

ΔΣADΦΣΔADΦΣ

ΔΣADΦΣΔADΦΣ

ΔΣADΦΣΔADΦΣ

θθ

oooo

oooo

oooo

Eq. C91

The first term is evaluated as follows:

230

( ) ( )

( ) ( )

( )

( ) ( ) ( )[ ]

( ) ( )[ ] . Tr

Tr

Tr

Tr

Tr TrE

11

111

1

e

1

e

1

e

1

e

1

e

1

e

1

ADΦΣBCΣ

ADΦΣBCΣ0Σ

ADΦΣΣ

ΣA

DΦΣ

ADΦΣADΦ

Σ

ADΦΣADΦΣ

ooo

ooo

oooo

oooo

oooo

−−

−−−

−−−

−−

−−

=

−−=

⎥⎦

⎤⎢⎣

⎡

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛γ∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−

Eq. C92

The second term is found to be:

( )[ ] ( )[ ]

( )[ ] ( )[ ] [ ]⎟⎟⎠

⎞⎜⎜⎝

⎛′

γ∂∂

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂′∂

−−−−

−−−−

ΔΔΣADΦΣ

ΔΣADΦΣ

Δ

ΔΣADΦΣ

ΔΔΣADΦΣΔ

ETr TrE

E E

e

11

e

11

e

11

e

11

oooo

oooo

Eq. C93

( ) ( )

( ) ( )

( )

( )

( ) ( ) ⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

γ∂∂

−⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂

=

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

γ∂∂

−

γ∂∂

+γ∂

∂

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂=

−−

−−−

−

−−

−−

−−−

−

I

BCΣADΦΣ

ADΦΣΣ

ΣA

DΦΣ

ΣΣΣ

ΣADΦΣ

ADΦΣADΦ

Σ

ΣΣ

ADΦΣΣADΦΣ

Tr

Tr

Tr

11

1

e

1

e

1

1

e

11

e

1

e

1

e

111

e

1

oooK

Koooo

ooK

Koooo

oooo

231

( ) ( ) ( ) ( ) ( )[ ]( )

( ) ( )[ ] ( ) ( )[ ] . Tr Tr

Tr

1111

11111

BCΣADΦΣADΦΣBCΣ

BCΣADΦΣADΦΣBCΣ0Σ

oooooo

oooooo

−−−−

−−−−−

−−=

−−=

Summing term gives:

( ) ( ) ( )[ ]( ) ( )[ ]

( ) ( )[ ]

( ) ( )[ ] . Tr

Tr

Tr

TrLln

E

11

11

11

11

e

2

BCΣADΦΣ

BCΣADΦΣ

ADΦΣBCΣ

ADΦΣBCΣθ

ooo

oooK

Kooo

ooo

−−

−−

−−

−−

−=

−

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂γ∂∂

−

Eq. C94

The last mixed term in IF involves the environmental parameters. It is evaluated as

follows:

( ) ( )

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′+−∂

−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡α∂

∂γ∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

−−−

Tr

21

E

Lln

ELln

E

e

111

eeee

2

ΔBΣΣΔBΣ

θθ

Eq. C95

[ ]

.

21

E Tr21

E

21 Tr

21

E

e

11

e

1

e

11

e

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

+γ∂

∂=

−−−

−−−

ΔBΣΣΔBΣ

ΔBΣΣΔBΣ

On evaluating the first term, we find:

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂ −

−−−

BΣB

ΣBΣBΣ

e

1

e

1

e

1

e

1

Tr21

Tr21

Tr21

E Eq. C96

232

( )

( ) ( )[ ]

( )[ ] ( )[ ] . Tr21

Tr21

Tr21

Tr21

111

111

1

e

11

BΣBCΣBCΣ

BΣBCΣBCΣ

BΣΣ

ΣBCΣ

−−−

−−−

−−−

−=

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−=

oo

oo

o

The second term is found to be:

[ ] [ ]

[ ] [ ] [ ]

( )

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

−γ∂

∂+

γ∂∂

−=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

γ∂∂

+γ∂

∂−=

⎟⎟⎠

⎞⎜⎜⎝

⎛′

γ∂∂

−=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=

⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂∂′−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂′−∂

−−−−

−

−−−

−

−−−−

−−−−

Tr21

Tr21

ETr21

Tr21

E

21

E 21

E

1

e

11

e

1

e

1

e

111

e

1

e

11

e

11

e

11

e

11

ΣΣΣ

BΣΣBΣB

Σ

ΣΣ

BΣΣBΣ

ΔΔBΣΣ

ΔBΣΣ

Δ

ΔBΣΣ

ΔΔBΣΣΔ

Eq. C97

( ) ( )

( ) ( ) ( )[ ]( )

( )[ ] ( )[ ] ( )[ ] . Tr21

Tr21

Tr21

Tr21

Tr21

11111

11111

111

e

11

BCBΣΣBΣBCΣBCΣ

BCBΣΣBΣBCΣBCΣ

IBCBΣΣBΣΣ

ΣBCΣ

ooo

ooo

oo

−−−−−

−−−−−

−−−−−

++−=

−−−=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−

γ∂∂

−−=

On summing terms, we find:

233

( ) ( )[ ] ( )[ ]

( )[ ] ( )[ ]( )[ ]

( )[ ] . Tr21

Tr21

Tr21

Tr21

Tr21

Tr21Lln

E

11

11

111

111

ee

2

BCBΣΣ

BCBΣΣ

BΣBCΣBCΣ

BΣBCΣBCΣθ

o

oK

Koo

oo

−−

−−

−−−

−−−

=

+

+−

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

Eq. C98

It is perhaps desirable to summarize these results by presenting the elements of the score

vector and all of the unique elements in IF in the order they were computed. For the sake

of completeness, the partial derivatives with respect to the mean and covariate effects are

also reported. For the elements in the score vector, we have:

( ) ( )( ) n, . . . 1, ,0i ; ββ

Lln 1ni

1

ii

=′′=′′

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂=

∂∂ −− ΔΣXeΔΣXβθ

( ) ( )[ ] ( ) ΔΣAΦΣΔAΦΣθ 111

g

TrLln −−− ′+−=α∂

∂oo

( ) ( )[ ] ( ) ΔΣACΦΣΔACΦΣθ 111

g

TrLln −−− ′+−=γ∂

∂oooo

( ) ( )[ ] ( ) ΔΣADΦΣΔADΦΣθ 111Tr

Lln −−− ′−=λ∂

∂oooo

( ) ( ) ΔBΣΣΔBΣθ 111

e 21

Tr21Lln −−− ′+−=

α∂∂

( ) ( ) ( )

21

Tr21Lln 111

e

ΔΣBCΣΔBCΣθ −−− ′+−=

γ∂∂

oo

For all of the unique elements in IF , we have:

234

( ) ( )( ) ( ) n , . . . j, i, ββββ

LlnE ni

1nj

i

1

jji

2

∀′′=∂∂′

′

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂∂− −− XeΣXeβXΣXβθ

( ) ( ) ⎥⎦⎤

⎢⎣⎡=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

− − 2 1

gg

2

Tr2Lln

E AΦΣθ

o

( ) ( ) ⎥⎦⎤

⎢⎣⎡=⎟

⎟⎠

⎞⎜⎜⎝

⎛

∂∂∂

− − 2 1

gg

2

Tr2γγθLln

E ACΦΣ oo

( ) ( ) ⎥⎦⎤

⎢⎣⎡=⎟

⎟⎠

⎞⎜⎜⎝

⎛

λ∂λ∂∂

− − 2 12

Tr2Lln

E ADΦΣθ

oo

( ) ( ) ⎥⎦⎤

⎢⎣⎡=⎟

⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

− − 2 1

ee

2

Tr21Lln

E BΣθ

( ) ( ) ⎥⎦⎤

⎢⎣⎡=⎟

⎟⎠

⎞⎜⎜⎝

⎛

γ∂γ∂∂

− − 2 1

ee

2

Tr21Lln

E BCΣθ

o

( ) ( ) ( )[ ]ACΦΣAΦΣθ

ooo11

gg

2

Tr2Lln

E −−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

( ) ( ) ( )[ ]ADΦΣAΦΣθ

ooo11

g

2

Tr2Lln

E −−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂λ∂∂

−

( ) ( )[ ]BΣAΦΣθ 11

ge

2

TrLln

E −−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂α∂∂

− o

( ) ( ) ( )[ ]BCΣAΦΣθ

oo11

ge

2

TrLln

E −−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

( ) ( ) ( )[ ]ADΦΣACΦΣθ

oooo11

g

2

Tr2Lln

E −−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂λ∂∂

−

235

( ) ( )[ ]BΣACΦΣθ 11

ge

2

TrLln

E −−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂α∂∂

− oo

( ) ( ) ( )[ ]BCΣACΦΣθ

ooo11

ge

2

TrLln

E −−=⎟⎟⎠

⎞⎜⎜⎝

⎛

γ∂γ∂∂

−

( ) ( )[ ]BΣADΦΣθ 11

e

2

TrLln

E −−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂α∂∂

− oo

( ) ( ) ( )[ ]BCΣADΦΣθ

ooo11

e

2

TrLln

E −−−=⎟⎟⎠

⎞⎜⎜⎝

⎛

λ∂γ∂∂

−

( ) ( )[ ]BCBΣΣθ

o11

ee

2

Tr21Lln

E −−=⎟⎟⎠

⎞⎜⎜⎝

⎛

α∂γ∂∂

−

The 15 unique elements corresponding to the variance components are arranged in the

Fisher information matrix in the following page. This formulation assumes that

(Williams and Blangero, 1999a): 1) 0Xβ = ; and 2) Σ is determined completely by the

variances in the genetic and environmental effects. Because IF is symmetrical, only the

upper triangular part is reported.

( ) [ ] ( ) ( )[ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]

( ) [ ] ( ) ( )[ ] ( )[ ] ( ) ( )[ ]

( ) [ ] ( )[ ] ( ) ( )[ ]

( )[ ] ( )[ ]

( ) [ ].

Tr21

Tr21

Tr21

TrTrTr2

TrTrTr2Tr2

TrTrTr2Tr2Tr2

2 1

112 1

11112 1

1111112 1

111111112 1

I

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−

−

=

−

−−−

−−−−−

−−−−−−−

−−−−−−−−−

BCΣ

BCBΣΣBΣ

BCΣADΦΣBΣADΦΣADΦΣ

BCΣACΦΣBΣACΦΣADΦΣACΦΣACΦΣ

BCΣAΦΣBΣAΦΣADΦΣAΦΣACΦΣAΦΣAΦΣ

F

o

o

ooooooo

ooooooooooo

oooooooooo

Eq. C99

By Equation 139, we also have for the sampling covariance matrix of the parameter estimates:

. 2

2

2

2

2

ˆ1

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

==−

e

eee

ee

egeggg

egeggggg

I

γ

γαα

λγλαλ

γγαγλγγ

γαααλαγαα

σσσσσσσσσσσσσσσ

θΣF Eq. C100

The actual elements in the Fisher information and sampling covariance matrices can be solved for numerically using GaussTM.

237

Appendix D: Geometry of the Likelihood Function

There is an elegant geometrical interpretation of θ

Σ ˆ that follows from the results

of vector calculus (Fig. D1). If IF is visualized as giving a hyperparaboloid tangent to

the hypersurface at which ( ) 0θ =ˆS or, approximately, the curvature where ( )θLln is a

maximum, then θ

Σ ˆ , being the reciprocal of the curvature, gives the radius of curvature in

the vicinity where ( )θLln is a maximum (Huzurbazar, 1949; Rao, 1960; Efron, 1975;

Edwards, 1992). In other words, θ

Σ ˆ measures the curvature under the maximum. As

such, θ

Σ ˆ also measures the precision of estimates in the parameter vector (Huzurbazar,

1949; Rao, 1960; Efron, 1975; Thompson, 1986; Edwards, 1992). To justify the

geometric interpretation, we may take the case of a simple ln-likelihood function for θ

scalar, i.e., ( )θLln , which with advanced differential geometry approaches can be

generalized to the multivariable case (see Huzurbazar, 1949; Rao, 1960; Efron, 1975;

Kass, 1989). Let there be an osculating circle, defined as the circle that best fits under the

maximum and is tangent to the point at the maximum. The osculating circle of radius r is

given by a vector-valued function in θ :

( ) )θsinθ(cosrθ jiτ += , Eq. D1

where i and j are vectors in the plane 2ℜ . Equation D1 is a parametric equation in

terms of x and y functions, namely:

( ) θcosrθx = , Eq. D2

and

( ) θsinrθy = , Eq. D3

238

( )0

θd

θLlnd=

SEr ; r2 ±=

( )θLln

θ Figure D1. Geometry of the Ln-Likelihood Function. For a simple ln-likelihood function, the ideal maximum likelihood estimate is indicated by downward concavity and tight curvature in the vicinity of the maximum. In the figure, r denotes the radius of the osculating circle of diameter 2r.

respectively. The curvature of a vector-valued function, denoted by ψ , is given by:

( ) ( ) ( ) ( )( ) ( ) [ ] 23 22

θyθx

θxθyθyθx

′+′

′′⋅′−′′⋅′=ψ , Eq. D4

where the prime notation now indicates differentiation with respect to θ (instead of

vector or matrix transpose). On differentiating accordingly and recalling that

1θcosθsin 22 =+ is a Pythagorean identity, we have:

( )( ) ( )( )( ) ( )[ ] r

1

r

r

θcosrθsinr

θcosrθcosrθsinrθsinr 3

2

23 22==

+−

−−−−=ψ , Eq. D5

239

which tells us that the curvature is equal to the reciprocal of the radius of curvature and

vice versa. To complete the justification of the geometric interpretation, it may be argued

that the Taylor expansion approximations (about the maximum likelihood estimate) of

( )θLln and ( )θτ agree at least up to their quadratic terms (Efron, 1975). Kass (1989)

reviews extensions of these concepts to more complicated likelihood functions using

advanced differential geometry.

240

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