ADVANCE METHODS OF STATISTICAL ANALYSIS USED IN ANIMAL BREEDING

Dr. Barada Shankar MohantyM-5428

Biostatistics ,Division Of LES & IT

CON

TEN

TSTypes of Data

Nature of Data

Models Used

Types of Model (Linear , Nonlinear )

Different types of data analysis

Bayesian Method of Analysis

References

Gathering or obtaining the desired information under study. Primary Source

Data is collected by researcher himself can be collected by using experiments, surveys, questionnaires,

interviews, and observations. Secondary Source

comes from resources that have already been published. Data collected, compiled or written by other researchers eg. books,

journals, newspapers Any reference must be acknowledged

DATA

Variables : Any measurable characteristics or quantity which can assume a range of numerical values within certain limits, i.e. income, Height, age, weight ,price etc.

• A discrete variable may assume only a countable number of values: intermediate values are not meaningful.

• Mastitis, DiseaseDiscrete

• A continuous variable may assume any real value within some range. Takes fractional or integral values.

• Milk yield ,fat yield,Body wt etcContinuous

• variable is controlled by the experimenter.Dependent

• variable is measured.Independent

A mathematical model is an equation or a set of equations which represents the behavior of a system.• Linear Model : Unit increase in independent variable cause a proportionate increase

dependent variable.• Y= β0+ β1X + β2X2 + e A linear model will exactly spell out which effects are affecting which observation and

the different effects (such as breed and feeding regime) are estimated simultaneously and during this process they are corrected for each other.

Linear models are the most common type of statistical models used in animal breeding to predict breeding values based on phenotypic observations.

•Non-linear Model : If one or all the parameter of a model are not appear • linearly, the model is known as nonlinear model.

• Y= a Xb e(-cx) + e

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MODEL

The model usually consists of factors. • Discrete factors or class variables such as sex, year, herd• Continuous factors or covariables such as age

Model Contains 3 components : Predictor-Dependent Variable Predictant- Independent Variable Error term

PredictorModelPredictant

Error

Types of Analysis :

(A)Univariate Analysis : we group the individuals on the basis of single performance.

When we use one variable to describe a person, place, or thing.

(B)Bivariate Analysis : When we use two variable .

(C)Multivariate Analysis : When we use more than two variables.

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Univariate Analysis :

1. Linear Regression Model2. Least square model

a. Random effect Model- Heritability,Repeatabilty estimation. b. Fixed effect Model-c. Mixed effect Model- BLUP

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Linear Regression Model :

Where Yi = Dependent VariableXi = Independent Variable β0 & β1 remains fixed ,we can’t found them exactly.e= Error term

The principle of estimation of regression coefficient is based on Least Square Analysis.

Yi = β0+ β1Xi + e

Random

Effects Fixe

d

Effec

t

Rand

om E

ffect

s

Effects which have levels that are considered to be drawn from an infinite large population of levels.

Animal effects are often random. In repeated experiments there maybe other animals drawn from

the population. e,g. Sire ,Dam effect

Fixe

d Eff

ects

Effects for which the defined classes comprise all the possible levels of interest, e.g. Herd , Season ,Year effect. Effects can be considered as fixed when the number of levels are relatively small and is confined to this number after repeated sampling.

Predictors- Used for Estimation of Random effect.

Estimators- Used for Estimation of Fixed effect.

Principles of Least Square Analysis :The Square of difference between observed and estimated/ predicted value of dependent variable must be least or Zero.

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= b0+ b1Xi + e then

0

Random Effect Model :

Where , Yij = jth observation of ith Sire μ= General mean effectSi = Effect of ith sire eij = Error term

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Yij = μ+ Si + eij

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Fixed Effect :

Where , Yij = jth observation of ith Herd μ= General mean effectFi = Effect of ith herd eij = Error term

Yij = μ+ Fi + eij

To achieve this ‘mixed models’ are used in which fixed effects and breeding values (indicated as ‘random effects’) will be estimated jointly.

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Mixed Effect Models

Where , Yijk = kth observation in ith farm of jth Sireμ= General mean effectSj = Effect of jth sire in ith farm Fi = ith farm effect eijk = Error term

Yijk=μ+ Fi+Sj + eijk

Interaction effect :

Yijk=μ+ Si+Fj + (SF)ij + eijk

BLUP estimation of breeding values is based on a mixed model, which is a linear basis of Best Linear Unbiased Prediction.

BLUP

Best in the sense that they have minimum mean squared error within the class of linear unbiased estimators; and predictors to distinguish them from estimators of fixed effects.

BLUP estimates of the realized values of the random variables u are linear in the sense that they are linear functions of the data, y;

The Best prediction is that which minimises the prediction error. Unbiased in the sense that the average value of the estimate is

equal to the average value of the quantity being estimated;

(G.K Robinson,1991)

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Maximizes the correlation between true and estimated value of effects by minimizing the error variance.

The factors for which estimates are required linear functions of the observations.

Estimates of fixed effects and estimable functions are such that E(T) = .

Best -

Linear

Unbiased

3 different kinds of BLUP, (Henderson, 1973 )

Henderson model-1

Henderson model-2

Henderson model-3

Only Random Effect

Random + Fixed + Interaction Effect

Both Random & Fixed Effect

The general linear model equation in matrix form is

Y = Xβ + Zu + eWhere ,

Y is an n × 1 vector of n observed records

X is a known incidence matrix of order n × p, which relates the records in y to the fixed effects in b

β is a p × 1 vector of p levels of fixed effects (to be estimated )

Z is a known incidence matrix of order n × q, which relates the records in y to the random effects in u

u is a q × 1 vector of q levels of random effects such as individual genetic value(to be estimated )

e is an n × 1 vector of random, residual terms

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Expectations and Variance Covariance (VCV) Matrices In general the expectation of y is

> which is also known as the 1st moment. The 2nd moments describe the variance covariance structure of y:

G is a dispersion matrix for random effects other than errors , R is the dispersion matrix of error terms, for which both are general square

matrices assumed to be non-singular and positive definite, with elements that are assumed known.

We usually write V = ZGZT + R

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Estimating fixed Effects & Predicting Random Effects :-

For a mixed model, y, X, and Z , , u, R, and G are generally unknown Two complementary estimation issues Estimation of and u Estimation of fixed effects, BLUE = Best Linear Unbiased Estimator

Prediction of random effects BLUP = Best Linear Unbiased Predictor

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= (XTX) -1 XTy

= GZT ( y-X )

• The BLUP eliminates the non genetic biases in estimating Breeding Value.• It also removes the genetic biases taking in to account the effects of non-

random mating , genetic merit of Dams and selection.

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Advantages:– Handles unbalanced designs– Uses information for all relatives measured to improve estimatesBLUP can be used to estimate a variety of genetic values– GCA, SCA, line values (i.e., genotypic values ofpure lines)– One can also use BLUP to estimate environmental effects, G x E

REML = Restricted Maximum Likelihood. Standard ML variance estimation assumes fixed factors are known without error. REML is an approach that produces unbiased estimators for these special

cases and produces less biased estimates than ML in general.

Depending on whom you ask, REML stands for Residual Maximun Likelihood or Restricted Maximum Likelihood

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REML Variance Component Estimation

variance components by REML are estimated based on residuals calculated after fitting by ordinary least squares from fixed effects part of the model.

It Maximizes a marginal likelihood function. So it is also called Residual Maximun Likelihood or Marginal Maximun

Likelihood.

For linear mixed effects models, the REML estimators of variance components produce the same estimates as the unbiased ANOVA-based estimators formed by taking appropriate linear combinations of mean squares when the latter are positive and data are balanced.

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ESTIMATION

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Before getting onto iterative algorithms, it is helpful to review the differencebetween the log-likelihood function l used to calculate maximum likelihood estimates, and that () used for REML:

The term log ()makes the adjustment for degrees of freedom used in estimating treatment effects, so that REML estimates of variance components are less biased than ML estimates. The other major differences are: is not a function of the fixed effects The constant in is a function of the fixed design matrix X

ML vrs REML

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ML vrs REML ML estimates are biased because no account is taken of degree of

freedom in estimating the variance components. REML takes care of bias in estimates as well as avoids –ve estimates of

component variance . (Searle et al, 1992 )

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Genetic EvaluationREML and BLUP applied to multi trait mixed models have become the standard method for genetic evaluation in all terrestrial animal species.The main benefits of using this methodology include:(1) Increasing accuracy of selection;(2) Managing accumulation of inbreeding;(3) estimating genetic trend without a control;(4) the possibility of conducting large scale genetic evaluation across populations.

N.H. Nguyen and R.W. Ponzoni Vol. 29 No. 3 & 4 Jul-Dec 2006

Bivariate Analysis :

•Tests of statistical significance.•Chi-square.

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Multivariate analysis

Multivariate analysis consists of a collection of methods that can be used when several measurements are made on each individual or object in one or more samples.

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Types :1. Multivariate Regression2.Discriminant Analysis3. Principal Component Analysis4. Genetic Divergence Analysis5. Canonical Variate Analysis

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Yi =β0+ β1X1+β2X2+…………… βnXn+ ei

Yi=β0+ + ei

When the number of population is more than one and each animal in the population has multiple characters.

1. Multivariate Regression :

2. Discriminant Analysis :

• Purpose : To find out the Discriminant Function (D) that increases the differences among populations by minimising the variances within the population and maximising the mean differences between the populations with respect of characters

• D=λ1 d1 + λ2 d2+ λ3 d3

• = ∑ λi di

Where , di = ith mean difference of the populations in relation to the character, λi = weighting coefficient attached to the difference.

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A Linear combinations of independent characters are involved to maximise the variance accounted for in the original set of characters.• Z1=a1 x1 + a2 x2+ a3 x3+ a4 x4

• Z2=b1 x1 + b2 x2+ b3 x3+ b4 x4

• Z3=c1 x1 + c2 x2+ c3 x3+ c4 x4

• Z4=d1 x1 + d2 x2+ d3 x3+ d4 x4

ai , bi , ci , di are relative weighting factor attached to each character.∑ai

2= ∑bi 2= ∑ci

2 =∑di

2=1 It is mainly confined to single population

Principal Component shows highest variance - 1st Principal Component Principal Component shows next highest variance – 2nd Principal Component .

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3. Principal Component (Z) Analysis :

First time this analysis In India was reported by Dhara and Chakravarty (1996) in large animals for predicting the breeding value of milk production on the basis of selected number of principal components.

4. Genetic Divergence Analysis / Genetic distance analysis / D2

analysis : (Given by P C Mahalanobis,1928 )by summing the squares of deviation of the same

transformed or untransformed traits between the two genotypes in various combinations.

i= No. of traits varrying from 1-p, J,k= genotypes. j k

Follows chi-square dist. at p degrees of freedom More critical is the trait the more no. of distinctly different group develop.

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D2=

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Few Notes on GD : To keep variation within population between different animals,thus many

groups can be formed within population. Explain about extent of variability and range of variability within

population. It explains the evolutionary divergence. Progeny testing programme needs GD for distiguishing unrelated sires.

Used in single population Traits of the Animals can be divided into two sets and the relationship

between two sets Is to be evaluated. 2 set of characters – Y set- response character

X set- Predictor characterY Set is maximally correlated with X set.• M= p1 y1 + p2 y2

• N= q1 x1 + q2 x2+ q3 x3

The Canonical coefficients (p1 , p2 and q1 , q2 and q3) in such a way so that the correlation between two sets of characters or Canonical Variate (M and N) become maximum and that correlation is called Canonical correlation.

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5. Canonical Variate Analysis

In India First used on dairy buffaloes by Thomas and Chakravarty ( 1999 )

Fundamentals of a Bayesian Analysis :

Formulate a probability model for the data. Decide on a prior distribution, which quantifies the uncertainty in the values

of the unknown model parameters before the data are observed. Summarize important features of the posterior distribution, or calculate

quantities of interest based on the posterior distribution. These quantities constitute statistical outputs, such as point estimates and

intervals.

BAYESIAN METHODS OF ANALYSIS

Bayesian inference: A form of inference which regards parameters as being random variables possessed of prior distributions reflecting the accumulated state of knowledge.

Bayes estimation: The estimation of population parameters by the use of methods of inverse probability.

( A.L. Pretorius and A.J. van der Merwe (2000)

This theorem is based on Conditional probability.Conditional probability : The probability of event B occurring when it is known that some event A has only occurred and it is noted by P().

when A & B are dependent event.

P()=

or

If B1 , B2….. BK are mutually disjoint events with probability P(BK)≠ 0 (i=1,2,….K) than for any arbitrary event ‘A’ which is a subset of such that P(A)≠ 0 then we have

𝑷 (𝑩𝒊

𝑨 )=𝑷 (𝑩𝒊 ) .𝑷 ( 𝑨𝑩𝒊 )

∑𝒊=𝟏

𝑲

𝑷 (𝑩𝒊 ) .𝑷 ( 𝑨𝑩𝒊 )

1. The probability i.e. P(B1), P(B2),……. P(BK) are called “A Priori Probability” as they exist before we get any information of the experiment itself.

2. The probability i.e. , i=1,2,3…..k are called “Likelihood” because they indicate how how likely the event under consideration is to occur for given each and “A Priori Probability”.

3. The probability i.e. are called “Posterior Probability” because they are determined after the result of experiment are known.

P(Bi)= “A Priori Probability”=“Likelihood” =“Posterior Probability”

𝑷 (𝑩𝒊

𝑨 )=𝑷 (𝑩𝒊 ) .𝑷 ( 𝑨𝑩𝒊 )

∑𝒊=𝟏

𝑲

𝑷 (𝑩𝒊 ) .𝑷 ( 𝑨𝑩𝒊 )

Few Important Notes :

The Notation of “priori” and “posterior” in Bayes’ theorem are relative to a given sample outcome. That is, if a posterior distribution has been determined from a particular sample, this Posterior distribution would be considered the prior distribution relative to a new sample.

Priori Posterior-1 Posterior-2

priori

priori

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References :

Robin Thompson and Esa Mantysaari , Prospects for Statistical Methods in Animal Breeding ,Jour. Ind. Soc. Ag. Statistics 57 (Special Volume), 2004: 15-25

P Narain, Statistics And Its Application To Agriculture And Genetics , IARI,New Delhi.

A.K.Chakravarty, Multivariate Analysis In Animal Breeding, NDRI,Karnal. Verbyla, A. P. (1990) A conditional derivation of residual maximum

likelihood. Australian Journal of Statistics, 32, 227-230. Henderson CR (1975) Best linear unbiased estimation and prediction

under a selection model. Biometrics 31:423–447. Henderson CR (1976) A simple method for the inverse of a numerator

relationship matrix used in prediction of breeding values. Biometrics 32:69–83

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2003) Bayesian Data Analysis, 2nd ed. London, Chapman & Hall.Carlin, B. P., and Louis, T. A. (2000) Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed. Boca Raton, Chapman & Hall. Duchateau, L., Janssen, P., & Rowlands,G.L., 1998. Linear Mixed Models. An

introduction with applications in veterinary research. ILRI, Nairobi, Kenya, 159-170.

05/01/2023 baradashankar.mohanty@gmail.com