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ANALYSIS OF GROWTH CURVES AND STRATEGIES FOR ALTERING THEIR SHAPE 1 ,2

H. A. Fitzhugh, Jr. 3

U. S. Department of Agriculture% Clay Center, Nebraska 68933

SUMMARY

Techniques for fitting longitudinal size-age data are reviewed with emphasis on the biological interpretation of parameters, goodness of fit, comparisons and computational difficulties. Particular attention is paid to the special cases and general form of Richards function, a nonlinear, four-parameter model; to the four- parameter, deterministic growth models derived from the "quadric law of damped exponential growth"; and to a simple two component model especially suited to fitting irregular size-age data. Genetic flexibility in the shape of the growth curve and various methods of altering this shape are discussed. Selection indices and expected changes in shape of the growth curve are described for various selection criteria, including growth and maturing rates, body weight and degree of maturity. (Key Words: Growth Curves, Body Size, Degree of Maturity, Maturing Rate, Mature Weight, Biological Models.)

I N T R O D U C T I O N

Growth curves reflect the lifetime interrela- tionships between an individual's inherent im- pulse to grow and mature in all body parts and the environment in which these impulses are expressed. This environment is framed by the individual's level of productivity, the quantity and quality of food consumed and the effort required to locate, consume and digest this

Adapted from a paper presented at a symposium on Growth Curve Analyses, American Society of Animal Science, July 29, 1974, University of Mary- land, College Park.

2 Sincere appreciation is expressed to Dr. St. C. S. Taylor, ARC Animal Breeding Research Organization, Edinburgh, for his constructive commments and valu- able advice.

3 Current address: Winrock International Livestock Research and Training Center, Morrilton, Arkansas 72110.

* U.S. Meat Animal Research Center, Agricultural Research Service.

food. Knowledge of growth curves is important to all animal scientists, regardless of specializa- tion, who are concerned with the effects of their research and recommendations on lifetime production efficiency.

Development of the theory and techniques for fitting growth curves may be traced both through time and across scientific disciplines. In particular, the theory and methodology of fitting growth curves owes much to the mathe- matician, the demographer and the economist. A good starting point in a review for animal scientists is Brody's comprehensive amalgam of mathematical and biological facts and theory, h is tory and prescience--Bioenergetics and Growth. Since publication of Bioenergetics and Growth in 1945, much creative research has refined and extended theory, and the availabil- ity of high speed computers has facilitated use of more sophisticated algorithms, but it remains a usefully humbling experience to find the essence of one's "original" new idea tucked away in a footnote in Bioenergetics and GrovJtb.

The term, growth curve, usually evokes the image of a sigmoid curve depicting a lifetime sequence of measures of size, often body weight. More general terminology would be size-age curves (e.g., weight-age, hip height-age, heart girth-age), as a reminder that distinction should be made among measures of size.

Techniques for fitting and analyzing growth curves will be described followed by a discussion of the expected consequences of several procedures for changing the shape of growth curves.

TYPES OF SIZE-AGE DATA

One or more measures of size may be observed for one or more individuals at one or more times, ages or stages of development. Observation of a single measure of size, such as height, yields univariate data; observation of multiple measures of size (weight, height, width, etc.) yields multivariate data. Cock

1036 JOURNAL OF ANIMAL SCIENCE, Vol. 42, No. 4

GROWTH CURVE ANALYSIS AND ALTERATION 1037

(1966), following Tanner (1951), categorized both univariate and multivariate size-age data into three basic types-static, cross-sectional and longitudinal-and two mixed types-mixed cross-sectional and mixed longitudinal. Appro- priate application, interpretation and analytical technique vary for each type.

Static. Size, whether one or several measures, is observed one time only for a group of individuals, all at the same age or stage of development. Considerable static data have been collected and analyzed (e.g., to estimate heritability of weaning weight or body composition of choice steers); however, such data provide little information on patterns of growth and development. When all individuals are measured at the same stage of development, whether defined by age, weight, body composition, parity, etc., multivariate analysis of static data can provide useful information about shape and form (Wright, 1932;Tanner and Burt, 1954; Oxnard, t969).

Cross-Sectional. As with static data, each individual is measured only once; however, the same measurements are made at other ages (stages) on other individuals sampled from the same population. Cross-sectional data yield information on the characteristics of the mean growth curve for the sampled population, but the quality of this information depends on the degree to which each subset of individuals is truly representative of the population of infer- ence. Cross-sectional data are a necessary consequence of experiments requiring sacrifice of animals to be measured (e.g., McMeekan, 1940; Lofgreen, 1964; Butterfield and Berg, 1966). Seebeck (1968) discussed experimental design and analyses for cross-sectional data.

Longitudinal. A complete set of measurements is available for every individual at every age (stage). Longitudinal data include all information available in static and cross-sectional data, plus information on individual variation in growth. Even when longitudinal data are parti- tioned into subsets for cross-sectional or static analyses (e.g., Brown et al., 1973), confidence in the interpretation is increased since the same individuals contribute to all levels of the analysis. The emphasis in this review will be on analysis of univariate longitudinal data. An excellent survey of the literature on both univariate and multivariate longitudinal data analyses has been presented by Kowalski and Guire (1974).

Mixed Types. Mixed cross-sectional data

result when one or more traits are measured once per individual but age (or stage) at measurement is unknown. Variation in the sampled population for size and shape at a fixed age (static data) and variations resulting from growth and development over time (cross-sectional data) are confounded to an unknown degree. Such data include size measurements on captured samples of wild animal populations or on commercial beef carcasses of unknown background. Unless an alternative reference base, (e.g., dentition pattern or degree of bone ossification) is available, mixed cross- sectional data are of less value than either static or cross-sectional data because of possible misinterpretation of size-age relationships.

Mixed longitudinal data result when all measurements are not available for every individual at every age, yielding a mixture of longitudinal data for some individuals (or traits) and cross-sectional data for others (Fitzhugh et al., 1967). In many cases, it is best to discard incomplete records from the analysis; however, Tanner (1951) has suggested alternative procedures for analyzing mixed longitudinal data.

Potential yield of information from these five types of data decreases in the order: longi tudinal , mixed-longitudinal, cross-sectional, static and mixed cross-sectional. Failure to utilize the most appropriate and efficient methods of analysis for each type of data will sharply reduce information yield.

ANALYSIS OF LONGITUDINAL DATA

An information rich continuum of size-age points is a challenging analytical problem. First, the objectives for the analysis must be clearly established. Primary objectives for fitting growth curves are descriptive and predictive:

1. Descriptive--information contained in the sequence of size-age points is consolidated into a relatively few parameters.

2. Predictive-growth curve parameters are utilized-either separately or in concer t - to predict growth rates, feed requirements, responses to selection and other items of interest,

Special characteristics of the da ta set and objectives of analysis determine the method of choice for fitting the growth curve. Primary bases for comparing methods of fitting growth curves include:

1. Biological interpretability of parameters generally depends on understanding the inter- relationships of genetics and environment

1038 FITZHUGH

which yielded a particular pattern. Care should be taken not to force a biological interpretation or to casually apply interpretations found appropriate for one set of data to another. Biological interpretability includes ability to rank correctly individuals or populations for important biological characteristics, as might be required in selection programs for growth rate, maturing rate or mature size.

2. Goodness of fit to actual data refers to minimizing the deviations of actual data points from corresponding points on the fitted curve. In this respect, the best fit to n size-age points is an (n-l) polynomial, but an (n-l) polynomial has not consolidated information nor are the (n-l) estimated parameters likely to be biologically interpretable.

3. Computational difficulty varies with choice of function and the characteristics of a specific data set. Most functions are sensitive to the frequency and regularity of data on both the size and age scales. Algorithms involving iteration are sensitive to choice of starting values and may not even converge to a solution. Mathematically correct, but biologically infeasi- ble, estimates of parameters may be computed.

Cboice o f Appropriate Model. Each component of an organism-whether cell, tissue, organ or whole body-fol lows an inherent growth pattern influenced by the environment in which pattern is expressed. Thus, at any given time of observation, the growth curve for a trait, such as body weight, will represent the composite of growth curves for all the components contrib- uting to the trait. A fitted size-age relationship suitable for one component may be inappropri- ate for another component or composite trait. For example, if weight, w, varies with the cube of linear size, 1, then a model for describing linear growth (1 = F (t)) would be transformed to describe weight growth, w~l 3 = (F (0) 3. However, such "rule of thumb" generalizations about the relationships between different measures of size can be misleading. Allometric analyses of relative growth rates for different measures of size (Brody, 1945; Cock, 1966) have shown that such relationships vary for different traits, for different genotypes and for different phases of growth.

If important events, such as puberty, lactation or feed shortages do not similarly affect all components of a size trait, (e.g., lean, fat and bone are components of body weight) observation of the composite, but not the components, may obscure the impact of the event; just as

individual adolescent growth spurts are ob- scured when averaged in the population mean curve (Tanner, 1974). The effect of an event, such as puberty, on size is more obvious if each individual's size is related, not to his chronological age, but to time elapsed pre and post puberty.

A common characteristic of growth models to be discussed is that they each utilize (directly or indirectly) two biologically relevant parameters. The first parameter establishes the position of the individual (or group) in the general size space at a given reference age, usually maturity. In other words, the size parameter establishes whether the individual is large, medium or small. Correlations among different measures of size for the same individual at the same or different ages tend to be quite high (Brinks et al., 1964; Taylor and Craig, 1965; Brown et al., 1973) so the size parameter may be taken as generally indicative of body size, an exceedingly important trait given the reported relationships of size to a diverse range of traits, including litter size, dystocia, growth rate, milk production, maintenance requirements, optimal slaughter weights and economic efficiency (Joandet and Cart- wright, 1969; Eisen, 1972; Baker et al., 1973; Taylor, 1971, 1973; Long et al., 1975).

The second parameter is concerned with growth rate relative to body size. When the size parameter refers to mature size, this "rate" parameter defines average maturing time, which Taylor and Young (1966), Blaxter (1968) and others have related to intrinsic efficiency of growth. Brody (1945) used maturing time as the basis for converting from chronological to physiological age for species comparisons. Tay- lor (1965) combined the properties of Brody's physiological age and Kleiber's (1961) metabolic turnover time into metabolic age, which is proportional to the .27 power of mature weight across species. Subsequently, Taylor (1968b) and Taylor and Fitzhugh (1971) found that, within species, time taken to mature was proportional to the .3 power of mature weight.

In addition to the size and rate parameters, a third parameter is often used to partition the growth curve into two stages, which Brody (1945) called "self accelerating" and "self in- hibiting" stages during which growth rate veloc- ity is increasing and decreasing, respectively. Transition between these two stages establishes the point of inflection of the sigmoid growth curve, so this third parameter will be referred to

G R O W T H C U R V E A N A L Y S I S A N D A L T E R A T I O N 1039

as the inflection parameter. Since the growth curve is essentially linear during the interval when this transition theoretically occurs (e.g., for cattle, the age interval 6 to 18 months), the estimated point of inflection may be influenced less by the animals's phenotype and more by the properties of the specific equation chosen to fit the data.

Regression of Size on Age. Equations or functions which regress size on age are obvious choices for fitting size-age data. However, the presumption of a causal relationship betwen age and size can be misleading. Age per se does not cause size to increase and then plateau but provides opportunity for the individual's inherent potential for growth and maturation to interact with particular environments. Cumula- tive nutrient consumption is an alternative independent variable in place of age (cumulative time). This reference base would dampen fluctuations in size, especially weight, which result from temporary changes in quantity or quality of feed due to seasonal changes or from the changes in requirements associated with level of productivity. A plot of weight against cumulative ad libitum food consumption generally follows an exponential curve (Brody, 1945; Taylor and Young, 1964; Parks, 1~970).

Since lifetime feed consumption is rarely observed, age is generally the best available "independent" variable. Methods used to describe the regression relationship between size and age include average daily gain, simple and multiple regression and orthogonal polynomials (Rao, 1958, 1965; Leech and Healy, 1959; Grizzle and Allen, 1969). These techniques have generally been limited to describing fairly short intervals of growth, and estimated parameters often do not have obvious biological meaning. Models which are nonlinear in their parameters have been extensively used to fit lifetime size-age relationships. The special cases of a four parameter function (Richards, 1959; Nelder, 1961, 1962; Grosenbaugh, 1965) have been especially popular. The general indeterminate form, which will be referred to as Richards function, represents an infinite number of determinate special cases, including the monomolecular (Brody, 1945), logistic, Gompertz (Laird, 1966) and Bertalanffy (Bertatanffy, 1957).

Ricbards Function. The general and four special cases of Richards function (table 1)will be compared for biological interpretability of both estimated and derived parameters, good-

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ness of fit to actual data and computat ional difficulty. Primary references for the discussion which follows are Brody (1945), Richards (1959), Eisen et al. (1969), Fletcher (1974) and Brown et al. (1976).

Biological Interpretation. The biological interpretations rationalized for parameters from Richards function and its special cases have been a primary reason for their popularity. For example, a reparameterized version (Bertalanf- fy, 1957) of the Bertalanffy equation was directly argued from metabolic laws. Generally, however, biological interpretat ions have evolved after choice of a mathematical model on empirical grounds, so it is not surprising that they do not consistently hold for all models and all data sets.

Richards function (table 1) describes change in size y, with change in age, t:

Yt=A(lYbe'kt) M

the upper sign applies when M~>I; the lower sign for M<0. Here, M equals ( l -m) "I , used by Richards (1959). Biological interpretat ions applied to parameters est imated or derived from Richards function (or special cases) include:

A--asymptot ic value for size as t--> oo; generally interpreted as average size at maturity independent of short-term fluctuations in size due to extraneous environmental effects of climate and food supply.

u t - t h e proport ion of mature size attained at age t; u t = y t /A = (lYbe-kt) M. Degree of maturi ty for y is presumably correlated to other measures of maturi ty, both statistical and biological.

b - a scaling parameter (constant of integra- tion) which is established by the initial values of Yo and t. This parameter adjusts for the situation where Yo ~ o and/or t o o; for example, when only postnatal observations of size are available and t o is taken as birth. Brody (1945) employed the scaling parameter t* to adjust the age scale (t-t*), so that b=l .

y i - s i z e at t l , the age at which growth rate is a maximum; (yi , t l ) are the coordinates of the point of inflection (POI) when growth rate changes from an increasing to a decreasing function of age. For the monomolecular curve which has no POI, growth rate is a maximum at the coordinates (Yo ,to)-

k - a function of the ratio of maximum growth rate to mature size, commonly referred to as maturing index. The specific function of this ratio varies for special cases of Richards function and is largely determined by the value of M. Since k depends on dy/dt , A and (yl , t l ) , it serves both as a measure of growth rate and of rate of change in growth rate. Taylor (1965) referred to k -1 as the maturing interval, a standard time scale for measuring changes in degree of maturity.

M--the inflection parameter for Richards function, which establishes degree of maturi ty at point of inflection, u I =

~ M ] " is a variable in the general \ !

indeterminate form but is the defining constant for each special case of Richards function (table 1). Here, u I is undefined when O~M<l .

The first derivative of Richards function measures instantaneous absolute growth rate.

-kt -kt -1 dy/dt=+Mky be (l*--be )

= Mky(u ' l /M-1)

where the upper signs apply for M ~ I ; the lower when M<O. Formulas for instantaneous absolute (dy/dt) and relative ( y ' l d y / d t ) growth rates and absolute maturing rates ( A ' l d y / d t ) are given in table 1. Formulas for traits of interest may be derived:

1. Weighted average lifetime rates

Absolute Growth Rate = .SAMk 2M-1

.5Mk Absolute Maturing Rate =

2M-1

Mk Relative Growth Rate =

M-1

2. Instantaneous rates at POI, (yi , t l )

Absolute Growth Rate = Mk M-1 YI

Absolute Maturing Rate = k ( - - ~ ) M-1


Mk Relative Growth Rate . . . .

M-1

Fitzhugh and Taylor (1971) showed that relative rates of growth and maturing are equiva- lent; i.e., u- ldu/dt = y ' ldy /d t . For Richards function average lifetime relative growth rate is the same as relative growth rate at the POI.

Goodness o f Fit. The usual tests for goodness of fit involving the residual variance are not appropriate to longitudinal data because of correlated errors among repeated observations over time. This problem is compounded for measures of size, especially weight, which fluc- tuate sharply because of cyclic variation due to compensatory growth, climatic effects on food supply and stresses of productivity. Moreover, the choice of the model may introduce systematic errors of estimation. Suppose the true point of inflection occurs at ui=20% , but the logistic model (ui=50%) is fitted to the data using least-squares procedures. The fitted curve is likely to underestimate (or overestimate) actual size prior to the fitted POI and overestimate (or underestimate) size afterwards.

Elston and Grizzle (1962), Rao (1965), Grizzle and Allen (1969), among others have developed multivariate tests which take into account the correlations among the repeated observations. These methods were illustrated for models which were linear in their parameters. Allen (1967) described similar multivariate methods appropriate for nonlinear models, assuming that observations on different animals at the same age are uncorrelated. This assumption is not likely to hold for contemporaries unless sources of correlations between contemporary observations are fitted simultaneously.

Eisen et al. (1969), Timon and Eisen (1969) and Brown et al. (197~) have compared pooled residual variances for various cases of Richards functions on the assumption that sources of correlated errors similarly affected residual variances for the different cases; probably a reasonable assumption, unless systematic errors were introduced by the models themselves. Residual variances were generally smallest for models with a variable POI; however, differences were small.

Comparison of models for deviation of predicted from actual data at key reference points, such as POI, was suggested by Eisen et al. (1969). Rutledge et aL (1972) compared the deviation between estimated asymptote and "mature" weight. Similarly, the average devia-

tion over limited segments of the growth curve would provide information on systematic trends of bias of growth models.

Eisen et aL (1969) also suggested that estimated genetic variation would be greater for parameters from the function which best fit the underlying inherent growth pattern than for parameters from the poorer fitting functions. Again, however, correlated errors may affect estimates of genetic variation for parameters from different models.

Researchers have tended to rely on visual comparison of the fitted curve to the actual data and on inspection of parameter estimates for both biological and mathematical reason- ableness. Based on "least-eyeball" techniques, Brown et aL (1976) concluded that the four parameter Richards function "best fit" cattle growth data. However, the simpler Brody function provided almost as good a fit subsequent to 6 months of age. Estimation bias prior to 6 months, presumably due to the lack of a point of inflection, was not consistent in sign across individuals.

Computat ional Dif f icul ty . Parameters for these growth models are generally estimated using generalized least-squares, iterative procedures (Nelder, 1961; Marquardt, 1963 ; Fabens, 1965; Causton, 1969). A data base of more than a few individuals dictates use of efficient algorithms and high speed computers. Poorly conditioned matrices may increase the number of required iterations or even prevent conver- gence to a reasonable solution. Nonorthogon- ality of the fitted parameters may lead to estimates which are mathematically feasible but biologically impossible. Rutledge et aL (1972) and Brown et al. (1976) mentioned particular difficulties fitting the general form of Richards function, primarily because of the close correla- tion between k and M.

As previously mentioned, sharp fluctuation in weight-age relationships are quite common; however, the various cases of Richards function generally presume a monotonic increase in size from origin to asymptote. The fitted curve obviously smooths the irregularities of the actual data (Brown et al., 1972), but problems of interpretation occur when the irregularities substantially affect the values of the estimated parameters. Some type of adjustment for sources of major irregularities, such as weaning, parturition and lactation, seems proper. Among the alternatives used for lifetime cow weight- age data (J. E. Brown and H. A. Fitzhugh,

1042 FITZHUGH

blished results) were adjustment of data to and simultaneous with the fitting of

irowth model. Prior adjustment introduces tdditional errors of estimating the adjust- :factors. Moreover, adjustment for the cow at losses at first parturit ion and lactation :s overestimation of the asymptote, A, t on comparison to actual data. Simultane- tdjustment of data while fi t t ing nonlinear .'Is was computationaUy difficult. Other aatives were to skip observations for t ime Cals in which weight f luctuated sharply or dy use weights which equaled or exceeded

weights, but both procedures introduced lement of subjective data selection. ~rrelations Among Parameter Estimates for trds Function and Special Cases. Two

of correlations are of interest; those ~g parameters with similar biological inter- tion, but estimated from different models, those among parameters within the same ;1. Several authors (Taylor, 1968a; Eisen et 969; Timon and Eisen, 1969; Rutledge et 1972; Brown et al., 1976) have fi t ted �9 eat models to the same data and cora- l correlations among parameter estimates

different models. Their general conclu- were that parameter estimates with similar gical interpretat ions were usually positive- ~rrelated, especially the asymptotes, but

correlations were often smaller than :ted. Similar parameters in different mod-

not necessarily measure the same biologi- bhenomena. Parameter estimates may be :entially affected by the characteristics of a ta set from which they are measured. For pie, Brown (1970) found that A in several :Is was underest imated to varying degrees ever weights were not available at full rity; thus, the model of choice might ad on the importance of a good estimate and/or the age at which the last weights

observed. aother difficulty exists for interpretat ions ~rameters within models. Biological inter- tions are often based on algebraic manipu-

of the model; however, correlations Lg parameter estimates may belie these ~retations. For example, ti ,age of point of :tion, is equal to k -1 In Mb in Richards ~1. One would anticipate the correlations e e n t I and k, b and M to be negative, ire and positive, respectively; however, ' n e t al. (1976) reported values of .44, and .30. Dependencies among k, b and M

obviously had major effect on estimates of t I. Quadric Law of Damped Exponential

Growth. A four-parameter, deterministic, nonlinear growth model with mutually independent parameters based on the "quadric law of damped exponential growth" has been described by Fletcher (1974). This model provides independent parametric control over the size, rate and inflection parameters mentioned previously. Specifically for this model, these three parameters refer to the upper asymptote, absolute growth rate at the point of inflection and the size and age coordinates of the point of inflection. Unlike the special cases of Richards function, which are also special determinate cases of Fletcher 's model, the coordinates for the point of inflection are not f ixed by the choice of the model; and unlike Richards general function, these coordinates for Fletch- er's model are not determined by the particular data set. Instead the researcher retains control over this critical variate and can use, for example, age, weight and growth rate at puberty.

Independence of the parameters greatly fa- cilitates clear biological interpretat ion and definition. Problems noted in the previous section which are a consequence of interdependencies among the parameters are avoided with Fletch- er's method.

Fletcher 's approach depends on a priori knowledge of size, rate and inflection parameters. In the absence of such knowledge, he suggested use of Richards function as an appropriate explanatory model. Unfortunately, Rich- ards function suffers the disadvantages of any equation-bound method because it imposes artificial mathematical constraints on the biological variation inherent in the growth curve. Equation-free techniques, (e.g., Taylor and Fitzhugh, 1971) for estimating size, rate, inflection and other growth curve parameters, offer a more comprehensive approach to investigating the phenotypic variation for characteristics of the growth curve.

Equation-Free Analysis o f Growth Curves. Many equations used to fit growth curves are sensitive to irregularity of spacing of the size- age points. Most make the strong assumption that growth is a monotonic increasing function of time. Frequently, neither of the conditions hold, especially for weight data collected under field conditions. Even when animals are rou- tinely weighed at regular time intervals, these intervals are usually irregular with regard to


physiological and production status. Suppose a group of cows are weighed monthly throughout their lifetime, some individuals will reach puberty at 12 months, others will not; some will be pregnant, others will be lactating and others will be barren at 25 months of age; and so on. Stresses of climate, productivity and diseases will also cause weight to decline sharply from time to time. Equations which assume similar physiological status at the same age and monotonic relationships between size and age, when fitted to growth data, smooth much of the observed variation in the size-age space. This smoothing effect may obscure important, interesting genetic and environmental phenomena. As an alternative, Fitzhugh and Taylor (1971) suggested a simple two component, equation- free, model, well suited for irregular data, which also preserves the phenotypic variation in the original observations. One component in- volves proportionality to mature size; the other measures deviations from this proportionality resulting from differences in maturing rate.

This simple model is

yt=ut A

where Yt is size at age t expressed as a function of degree of maturity, u, and mature size A. Definition and estimation accuracy of A are especially critical. The logarithmic form, In y =

In u + In A, has certain computational advantages. Minimum requirements for using this model are a measure of mature size and one (but preferably more) measures of immature size. Applications and advantages of this model for analyzing growth curves in comparison with nonlinear equations are discussed by Taylor and Fitzhugh (1971).

Among the useful growth statistics which may be derived or approximated from the model are average absolute growth rate, AGR = (Y2-yl)/(t2-tl ); average absolute maturing rate, AMR = (u2-ul)/(t~-tt) = A -1 (Y2-Yl)/(t2-tl); average relative growth rate, RGR -- In y2 - l n

Yl)/(t2-h). RGR may be approximated by AGR relative to average size, i.e., AGR/�89 (Yl +Y2). Taylor and Fitzhugh (1971) used age at a given degree of maturing (t u) as a measure of time taken to mature. They showed that tu and the "k" parameter from nonlinear models (e.g., y=A(1-be-kt)) are alternatives for charac- terizing individuals for being early or late maturing. However, t u does not share the disadvantage of k; namely, k being a constant over the entire growth period and being subject

to restrictions imposed by the choice of a specific model. Use of tu (or In tu) allows more flexibility than k and is less likely to obscure interesting sources of variation in maturing patterns.

This model has been used in two published analyses of cattle data (Fitzhugh and Taylor, 1971; Smith e t aL, 1976). Some r~sults from these analyses illustrate both applications of the model and information about cattle growth.

1. Degree of maturity (or t u) for weight was slightly tess heritable than body weight, w, at the same ages prior to maturity; with genetic coefficients of variation for ut declining to zero at maturity.

2. Degrees of maturity for weight at different ages were positively correlated. Although individuals most mature at a given age tend to be more mature at other ages, genetic correlations indicated genetic flexibility in the shape of the maturity-age curve.

3. Genetic correlations between weight (w) and degree of maturity for weight (u) at similar age were generally positive, indicating that genetically heavier individuals tended to be more mature during that phase of growth. However, these correlations tended to decline to negative value as differences between ages of observing u and w increased. This result is of particular importance when performance of individuals (or breeds) which differ markedly in growth or maturing rate are compared on constant weight or age basis (say, comparison of rate of gain for Charolais vs Jersey from 300 to 500 kg).

4. Individuals growing to the heaviest mature weight tended to be less mature at a given age (ut) or older at a given degree of maturity (tu).

5. Crossbred cattle were both heavier and more mature for weight at a given chronological age but slightly lighter and less mature for weight at puberty than were contemporary straightbreds (table 2). Crossbreds being sexu- ally mature at a lower degree of maturity for weight suggests that heterosis for maturation rate of "reproductive ability", whatever its nature, exceeded heterosis for maturation rate of weight. Such a result is in agreement with the generally high heterosis for components of reproduction compared to heterosis for body weight and gains. AMR per day from birth to puberty was .121% for crossbreds and .118% for straightbreds confirming that although crossbreds were less mature for weight at

1044 FITZHUGH

TABLE 2. HETEROSIS FOR BODY WEIGHT AND DEGREE OF MATURITY a

Weight, kg Degree maturity, % St bred b St bred b

Age mean % heterosis mean % heterosis

Birth 33 5.4 6.6 3.1 200-days 193 5.4 38.8 2.6 399-days 263 7.3 52.8 4,6 550-days 364 6.4 73.0 3,7 3.3 years 428 5.2 85.7 2.5 Puberty 255 -.4 51.0 -2.7 Maturity 499 2.5 . . .

aDerived from statistics from Smith et al. (1976).

bstraightbreds were Angus, Hereford and Shorthorn females; contemporary crossbreds were the six first crosses, including reciprocals.

puberty there was heterosis (2.7%) for maturing rate to puberty.

ALTERING SHAPE OF THE GROWTH CURVE

Comparisons between species (Brody, 1945; Taylor, 1965) and within species (Taylor, 1968b; Eisen et al., 1969; Taylor and Fitzhugh, 1971; Brown et al., 1972) confirm the general negative genetic relationship between mature size and earliness of maturing. Selection for size or closely correlated traits, such as absolute growth rate, will increase time taken to mature. It is this negative relationship between the size and earliness of maturing which largely deter- mines shape of the curve (Taylor and Fitzhugh, 1971~. Among the reasons for altering the shape of the growth curve are:

1. Resolve genetic antagonism between desired rapid, efficient early growth of slaughter progeny and desired small size and lower maintenance costs of parental stocks (Gregory, 1965; Cartwright, 1970; Dickerson e t al., 1974).

2. Improve intrinsic efficiency through increased maturation rate (Taylor and Young, 1966; Blaxter, 1968).

3. Reduce dystocia by decreasing birth weight of progeny relative to dam size (Mon- teiro, 1969).

4. Lower age at first breeding by decreasing time to sexual maturity or decrease carcass fatness at preferred market weights by increasing time to chemical maturity.

The suggested effects of altering shape of growth curve on sexual and chemical maturity (item 4, above) presume that earliness of

maturity for weight (or other measures of size) are genetically correlated with earliness of maturity for other traits, such as hormonal levels and body composition. Such genetic correlations may be inferred from the usual homeostatic tendencies of growth and development; thus, physiological measures of maturity (e.g., age at puberty) are likely highly correlated with statistical measures of maturity (e.g. tu). Obviously, these inferred genetic relationships need experimental validation.

Genetic change in the shape of the growth curve wilt be limited by the degree of genetic flexibility in the shape of the curve. Genetic flexibility depends on degree of independence among the size, rate and inflection parameters. Taylor and Craig (1965) reported maximum genetic flexibility in the shape of growth curves for 12 linear measures of cattle size when cattle were approximately 12 months of age. Taylor and Fitzhugh (1971) found for Hereford females that 78% of the additive genetic variation in average time taken to mature was independent of mature weight. Brown e t al. (1972) reported for Hereford weight data fitted by Brody's equation that only 10% of the genetic variation in k was independent of A; however, for contemporary Angus, 92% of the genetic variation in k was independent o f A. For mouse weight data fitted by the logistic equation, Eisen et al. (1969) found for their control line that 78% of the genetic variation in k was independent of A; and Timon and Eisen (1969) reported 88% independent genetic variation in k. Results from experiments with mice (Mc- Carthy and Doolittle, personal c o m m u n i c a t i o n ; Wilson, 1973) and chickens (Merritt, 1974)

G R O W T H C U R V E A N A L Y S I S A N D A L T E R A T I O N 1045

have indicated selection to be an effective means of altering shape of the growth curve.

Size or Growth Rate. Results of long term selection experiments with mice indicate that selection for size or growth rate have little effect on shape of the growth curve (Timon and Eisen, 1969; Wilson, 1973; Bakker, 1974). Expected responses to selections for body weight in cattle (table 3) indicate a slightly greater increase in weight and degree of maturity at age of selection but also substantial increases in weight at all other ages, including maturity. Similarly, selection for AGR (table 4) would be expected to increase weight at all ages.

Small changes or "bumps" in the shape of the curve, as indicated by increased u at age of selection are a consequence of genetic correlations less than unity among measures of size at different ages. Selection may be for effects of transitory genotype-environment interactions affecting shape of curve over short segments. For example, beef cattle are commonly weaned at 6 to 8 months of age. Genotypes least affected by the sudden stress of weaning (e.g., calves from low milking cows which have already been effectively weaned so that their rumens are developed) may grow more rapidly after weaning regardless of their genotype for size. Selection of such individuals might alter shape of population mean curve segment imme- diately postweaning but have little effect on the entire curve (Dickinson, 1960). Similarly, selection for low milk production could indirectly increase maturing rate for weight of primipar- OUS COWS.

When weight is the selected measure of size, the type of tissue-lean, fat or bone-being increased must be considered. Early maturing and early fattening are often used as synonyms for market livestock. The negative genetic relationship between mature size and time to mature suggests that larger, faster growing individuals would also be less mature and leaner at a given age or weight prior to maturity. However, in a number of experiments reviewed by Eisen (1976), selection for weight or growth rate actually increased fatness. If the effect of selection is primarily to increase appetite but not rate or efficiency of metabolism, individuals consuming nutrients in excess of immedi- ate growth or maintenance requirements will store the excess as fat.

Direct selection for size at maturity (table 3) should decrease u at all ages. This expectation

reflects the part-whole relationship between u and A, but it also follows from the negative genetic relationship between mature size and maturing rate.

Increases in u depend on increase in early size and/or decrease in mature size. Expected responses (table 3) for early ages (UB and u6) result more from increase in weight at birth and 6 months (WB,W 6) than from decrease in mature weight (WA). Maternal influences, such as prenatal uterine environment and milk production, on wB and w 6 tend to be positively correlated with w A, thereby limiting selection pressure for decreased w A.

Selection for AGR, AMR or RGR (table 4) apparently would have similar effect within the interval of observation. However, selection for AMR or RGR would have relatively little effect on mature size, indicated by average weight between 18 months and maturity or for birth weight (Fitzhugh and Taylor, 1971). The favored genotypes for AGR, RGR and AMR in the 6 to 12 months interval were apparently quite different from favored genotypes for same measures of growth in other intervals. For these data, the 6 to 12 months interval included the stresses of weaning and wintering under fairly harsh feed and climatic conditions. Since these particular stresses would tend to be similar over years, selection for growth in this interval would favor genotypes which inter- acted best with these particular stress conditions but not the relatively better conditions in preweaning (B-6 mo) and postwinter (12 to 18 month) intervals.

Ratios and Conformation Scores. Individuals which mature early for one component of size (say, height or length) would be expected to mature early for other components; similarly, individuals which are large at maturity for one component will also tend to be large for other components (Taylor, 1962; Taylor and Craig, 1965). Selection for the ratio of a late maturing trait (weight) to an early maturing trait (height), both measured at the same age, should increase earliness of maturing and, perhaps, decrease mature size for both components. The extent of the changes will depend on the stage of development at time observation and on genetic variation and covariation for the growth curve parameters of both traits.

Conformation scores are a subjective evalua- tion of the relative proportionality among components of body size at the age of observation. Expected responses to selection for 18

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month score (table 3) were generally similar to the expected responses to selection for Wxs. This result is not surprising given the extent to which practit ioners of the art of animal breeding have long relied on visual appraisal to select early or late maturing genotypes. However, knowledge of genetic and phenotypic relationships among measures of size and maturi ty for one or several traits implemented in a selection index should be a more efficacious method of setting the shape of the growth curve.

Selection Indices. Expected responses, stan- dardized index coefficients, squared correlations between index and net merit and standard ~" deviations for indices for a number of different selection criteria are given in table 5. Restricted indices (e.g., those selecting for increased ~. 12-month weight while holding constant weight :~ at birth and maturi ty, wl 2.wB,w A) were computed according to procedures suggested by u~ Cunningham et al. (1970).

Unrestricted selection for wl 2 places positive selection pressure on weights at all other e~ ages and so would be expected to increase birth weight and mature weight, along with related traits such as dystocia and maintenance costs. Use of the index in selection for w~ 2 with or ~: without w A had essentially the same effect. ~.

Selection for u12 with WA in index in- = creased u, primarily by decreasing w A. Without a:

w A in index, u increased primarily by increas- o ing w; in fact, w 8 and wl s served as indicators wA. Indices for u would apparently change the shape of the curve more than the others z ~ considered.

Z Approximate ly 75% of the increase in w~ 2 O

with unrestricted selection was retained when selection was for Wl 2-WA, without increase in .~ w A (or WB). Failure to include w A in these particular restricted indices sharply reduced the proport ion of the variation in the selection .~ criteria accounted for by the index (R2). Unfortunately, including w A would greatly ~" lengthen the time before effective selection could be made and show progress. A means of "predict ing" mature size early in life is needed. Possibilities include mature size of relatives, such as sire or dam, or use of measures of size which are mature at earlier ages than is weight.

In most indices, w6 received small but negative emphasis. Apparent ly phenotypicaUy heavy calves at 6 months reflected the higher milk yield of their dams. Thus, we was more of a measure of environmental effect of maternal abili ty than the direct genetic growth effect.

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These expec ted responses to index select ion, genetic statistics repor ted by Brown e t al.

(1972) and a few exper imenta l results with labora tory animals indicate that the shape of the growth curve may be changed by selection. The appropr ia te change and the required selec- t ion criteria will vary. In cattle, for example, the appropr ia te change for lines used as dams in terminal crosses or those lines used in ro ta t ional crosses would be rapid early growth and reduced mature size (Cartwright e t al., 1975). Lines used as sires in terminal crosses would be selected for m a x i m u m postnatal growth re- ducing or holding constant birth size to reduce calving diff iculty. A potent ia l danger of decreasing birth weight relative to mature weight (decreasing UB) is that calves born relatively immature may have poore r survivabili ty (Mon- teiro, 1969).

A l t e r n a t i v e s to S e l e c t i o n . As men t ioned previously, cumulat ive nut r ien t consumpt ion , rath- er than cumulat ive t ime, has pr imary environmental e f fec t on both size and shape parameters of the growth curve (Taylor and Young, 1968). Nutr i t ional con t ro l o f weight matura- t ion rate in dairy cat t le (Swanson, 1967) or mature weight of swine (Elsley, 1970) is c o m m o n l y practiced. Maturat ion rate of beef cows is def ini te ly inf luenced by the stresses o f product iv i ty and cyclic variat ion in quan- t i ty and qual i ty of nutr ients (F i tzhugh et al., 1967).

Nonaddi t ive genetic effects on the shape of the curve do occur (Laird and Howard, 1967; Smith e t al., 1976). Heterot ic increases in matruring rate are l ikely due to superior "f i t - ness", especially under stressful envi ronmenta l condit ions, and super ior ef f ic iency of metabolism; perhaps, via mi tochondr ia l complementa - t ion (Wagner, 1969) and hybrid enzymes (Schwartz and Laughner, 1969).

The economic advantages of increasing early growth rate o f slaughter produce wi thou t increasing maintenance costs o f parental lines are obvious. These advantages may be gained and even accentua ted through judicious matching of complemen ta ry sire and dam lines. Results f rom compu te r s imulat ion o f integrated beef p roduc t ion systems indicated that mat ing large sire lines to small dam lines would increase return to investment by 20% over systems utilizing sires and dams of the same size geno type under the p roduc t ion condi t ions specified in the mode l (Fi tzhugh et al., 1975).

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analysis of growth curves and strategies for … · for example, if weight, w, varies with the cube...

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