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Statistical modelling of birth weight variability within litter in pigs
Inauguraldissertation
zur
Erlangung des akademischen Grades eines
Doktors der Naturwissenschaften
der
Mathematisch-Naturwissenschaftlichen Fakult at
der
Ernst-Moritz-Arndt-Universit at Greifswald
vorgelegt von
D orte Wittenburg
geboren am 16. Oktober 1978
in Rostock
Greifswald, 28. April 2008Dekan: Prof. Dr. K. Fesser
1. Gutachter: Prof. Dr. V. Liebscher
2. Gutachter: Prof. Dr. N. Reinsch
3. Gutachter: Prof. Dr. H.-P. Piepho
Tag der Promotion: 24. September 2008CONTENTS III
Contents
1 Introduction 1
2 Analysis of e ects on within-litter variance 6
2.1 Measures of birth weight variability within litter . . . . . . . . . . . . . . . . 6
2.1.1 Distribution of the trait and its transformations . . . . . . . . . . . . 6
2.1.2 Linear mixed model (LMM) . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Generalized linear mixed model (GLMM) . . . . . . . . . . . . . . . . 13
2.2 Parameter estimation and prediction . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Fixed and random e ects . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Estimation of variance components . . . . . . . . . . . . . . . . . . . 24
2.2.3 of heritability . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Model comparison and residual diagnostics . . . . . . . . . . . . . . . . . . . 34
2.3.1 Studentized residuals for the LMM . . . . . . . . . . . . . . . . . . . 35
2.3.2 Studentized residuals for the GLMM . . . . . . . . . . . . . . . . . . 37
2.3.3 Testing for normality . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.4 Testing for skewness and kurtosis . . . . . . . . . . . . . . . . . . . . 38
2.3.5 Rank correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Testing for xed e ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5 Testing for random e ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.1 Asymptotic distribution of the likelihood ratio test statistic . . . . . . 45
2.5.2 Exact distribution of the likelihood ratio test statistic . . . . . . . . . 47
2.5.3 Approximation of the distribution of the likelihood ratio test statistic
via parametric bootstrap simulations . . . . . . . . . . . . . . . . . . 49
2.5.4 Approximation of the distribution of the likelihood ratio test statistic
by an appropriate distribution . . . . . . . . . . . . . . . . . . . . . . 50
2.5.5 Goodness of t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.5.6 Power of the likelihood ratio test . . . . . . . . . . . . . . . . . . . . 59CONTENTS IV
2.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 Data analysis 65
3.1 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.1 Setup of birth weight variability . . . . . . . . . . . . . . . . . . . . . 65
3.1.2 Testing for sex e ect and estimates of variance components . . . . . . 67
3.1.3 Model comparison and residual diagnostics . . . . . . . . . . . . . . . 69
3.1.4 Testing for random boar e ect . . . . . . . . . . . . . . . . . . . . . . 72
3.2 EAS data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.1 Description of the data . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2.2 Parameter estimates and testing for sex e ect . . . . . . . . . . . . . 86
3.2.3 Total born piglets versus liveborn piglets . . . . . . . . . . . . . . . . 89
3.2.4 Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.2.5 Testing for the random boar e ect . . . . . . . . . . . . . . . . . . . . 91
3.3 BHZP data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3.1 Description of the data . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3.2 Parameter estimates and testing for sex e ect . . . . . . . . . . . . . 96
3.3.3 Model evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3.4 Testing for the random boar e ect . . . . . . . . . . . . . . . . . . . . 99
4 Discussion 105
4.1 Analysis of heterogeneous variances . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Heritability and weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.3 Paternal e ect on within-litter variance . . . . . . . . . . . . . . . . . . . . . 106
4.4 Di erence between sexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.5 Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.6 Testing a single variance component . . . . . . . . . . . . . . . . . . . . . . . 109
4.7 Testing the correlation and xing one variance component . . . . . . . . . . 110CONTENTS V
5 Summary 115
A Appendix 119
A.1 Numerator relationship matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.2 Selection on minimal trait expression . . . . . . . . . . . . . . . . . . . . . . 122
A.3 R script for data simulation and parametric bootstrap . . . . . . . . . . . . . 126
A.4 ASReml worksheet for the analysis of a simulated dataset . . . . . . . . . . . 133
Bibliography 136NOMENCLATURE VI
Nomenclature
2 2 -distribution with m degrees of freedomm
Diagonal matrix of rst derivatives
‘() Log-likelihood function
‘ () Residual log-likelihood functionR
Convergence limit
Linear predictor
Type I error
( ) Gamma function
Skewness1
Kurtosis2
Gamma distribution with parameters and ;
Non-centrality parameter
Scaling factor to achieve the scaled F -test statistic F
H Hessian matrix
K Matrix used in residual diagnostics
S Matrix used in residual
m k-th momentk
2 Expectation of S in a GLMM
Adjusted denominator degrees of freedom
Total number of levels of xed e ects1
Number of random e ects2
Dispersion parameterNOMENCLATURE VII
() Standard normal distribution function0;1
2 Power of likelihood ratio test based on the 50:50 mixture of -distributionsself
Restricted likelihood ratio test based on methodj2fself; nite ; boot; mix; aUD; gammagj
2 Vector of variance components
2 Additive genetic variance of sowa
2 Residual variancee
2 Variance of boar heterozygosityp
2 Variance of permanent environment within litterlitter
2 Variance of permanent environmentpe
Parameter space of
Parameter vector to specify null hypothesis testing problems for random e ects
Restricted parameter space of under H0 0
" Vector of residuals corresponding to a GLMM
2’ 2() Density of a normal distribution with parameters and ;
bF () Empirical distribution functionn
b Vector of di erences between response vector Y and its predictor Y
Genetic gain after one period of selection
A Numerator relationship matrix
a Scaling parameter for RLRT
AI Average information matrix
B Total number of bootstrap simulations
b Vector of xed e ects
b Fixed e ect of farm-year-seasonf
b Fixed e ect of pig linebsNOMENCLATURE VIII
b Fixed e ect of paritypa
b Fixed e ect of sex separated by line of sowsg
C Coe cient matrix referring to the mixed model equations
C Coe cient matrix referring to the generalized mixed model equations
D Residual covariance matrix in a GLMM
d Number of samples in a subset of bootstrap simulations
d Number of simulated t = 00
D Kolmogorov test statisticK
2
2D -test statistic
e Vector of residuals corresponding to a (pseudo) LMM
E(Y ) Expectation of Y
e Vector of observed errors
F F -test statistic
f() Density function
F Scaled F -test statistic
2F () Distribution function of a -distribution with m degrees of freedomm
F () Distribution function of a gamma distribution with parameters and

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