\contentsline {chapter}{\numberline {1}CONSTRUCTING A LINEAR MODEL}{1} \contentsline {section}{\numberline {1.1}Simple Regression Model}{3} \contentsline {section}{\numberline {1.2}One Way Random Model}{3} \contentsline {section}{\numberline {1.3}Two Trait Additive Genetic Model}{4} \contentsline {section}{\numberline {1.4}Two Way Mixed Model}{5} \contentsline {section}{\numberline {1.5}Equivalent Models}{6} \contentsline {section}{\numberline {1.6}Subclass Means Model}{8} \contentsline {section}{\numberline {1.7}Determining Possible Elements In The Model}{8} \contentsline {chapter}{\numberline {2}LINEAR UNBIASED ESTIMATION}{11} \contentsline {section}{\numberline {2.1}Verifying Estimability}{11} \contentsline {subsection}{\numberline {2.1.1}Second method}{12} \contentsline {subsection}{\numberline {2.1.2}Third method}{12} \contentsline {subsection}{\numberline {2.1.3}Fourth method}{13} \contentsline {chapter}{\numberline {3}BEST LINEAR UNBIASED ESTIMATION}{15} \contentsline {section}{\numberline {3.1}Mixed Model Method For BLUE}{16} \contentsline {section}{\numberline {3.2}Variance of BLUE}{18} \contentsline {section}{\numberline {3.3}Generalized Inverses and Mixed Model Equations.}{19} \contentsline {subsection}{\numberline {3.3.1}First type of g-inverse}{19} \contentsline {subsection}{\numberline {3.3.2}Second type of g-inverse}{21} \contentsline {subsection}{\numberline {3.3.3}Third type of g-inverse}{22} \contentsline {section}{\numberline {3.4}Reparameterization}{23} \contentsline {section}{\numberline {3.5}Precautions in Solving Equations}{24} \contentsline {chapter}{\numberline {4}TEST OF HYPOTHESES CONCERNING {\pbf ${\beta }$}}{25} \contentsline {section}{\numberline {4.1}Equivalent Hypotheses}{26} \contentsline {section}{\numberline {4.2}Test Criteria}{27} \contentsline {subsection}{\numberline {4.2.1}Differences between residuals}{27} \contentsline {subsection}{\numberline {4.2.2}Differences between reductions}{28} \contentsline {subsection}{\numberline {4.2.3}Method based on variances of linear functions}{29} \contentsline {subsection}{\numberline {4.2.4}Comparison of reductions under reduced models}{30} \contentsline {chapter}{\numberline {5}PREDICTION OF RANDOM VARIABLES}{33} \contentsline {section}{\numberline {5.1}Best Prediction}{33} \contentsline {section}{\numberline {5.2}Best Linear Prediction}{34} \contentsline {section}{\numberline {5.3}Best Linear Unbiased Prediction}{36} \contentsline {section}{\numberline {5.4}Alternative Derivations Of BLUP}{37} \contentsline {subsection}{\numberline {5.4.1}Translation invariance}{37} \contentsline {subsection}{\numberline {5.4.2}Selection index using functions of {\pbf y} with zero means}{38} \contentsline {section}{\numberline {5.5}Variance Of Prediction Errors}{39} \contentsline {section}{\numberline {5.6}Mixed Model Methods}{39} \contentsline {section}{\numberline {5.7}Variances from Mixed Model Equations}{40} \contentsline {section}{\numberline {5.8}Prediction Of Errors}{41} \contentsline {section}{\numberline {5.9}Prediction Of Missing {\pbf u}}{42} \contentsline {section}{\numberline {5.10}Prediction When {\pbf G} Is Singular}{43} \contentsline {section}{\numberline {5.11}Illustration Of Prediction Of Missing {\pbf u}}{48} \contentsline {section}{\numberline {5.12}A Singular Submatrix In {\pbf G}}{51} \contentsline {section}{\numberline {5.13}Prediction Of Future Records}{52} \contentsline {section}{\numberline {5.14}When Rank of MME Is Greater Than n}{53} \contentsline {section}{\numberline {5.15}Prediction When {\pbf R} Is Singular}{57} \contentsline {subsection}{\numberline {5.15.1}{\pbf X } and {\pbf Z } linearly dependent on {\pbf R }.}{57} \contentsline {subsection}{\numberline {5.15.2}{\pbf X} linearly independent of {\pbf V}, and {\pbf Z} linearly dependent on {\pbf R}}{59} \contentsline {subsection}{\numberline {5.15.3}{\pbf Z} linearly independent of {\pbf R}}{59} \contentsline {section}{\numberline {5.16}Another Example of Prediction Error Variances}{59} \contentsline {section}{\numberline {5.17}Prediction When {\pbf u } And {\pbf e } Are Correlated}{61} \contentsline {section}{\numberline {5.18}Direct Solution To {\pbf ${\beta }$} And {\pbf u }+{\pbf T} {\pbf ${\beta }$}}{64} \contentsline {section}{\numberline {5.19}Derivation Of MME By Maximizing f({\pbf y},{\pbf w})}{66} \contentsline {chapter}{\numberline {6}G AND R KNOWN TO PROPORTIONALITY}{69} \contentsline {section}{\numberline {6.1}BLUE and BLUP}{70} \contentsline {section}{\numberline {6.2}Tests of Hypotheses}{71} \contentsline {section}{\numberline {6.3}Power Of The Test Of Null Hypotheses}{71} \contentsline {chapter}{\numberline {7}KNOWN FUNCTIONS OF FIXED EFFECTS}{75} \contentsline {section}{\numberline {7.1}Tests of Estimability}{75} \contentsline {section}{\numberline {7.2}BLUE when ${\pbf {\beta }}$ Subject to ${\pbf T}'{\pbf {\beta }}$}{77} \contentsline {section}{\numberline {7.3}Sampling Variances}{79} \contentsline {section}{\numberline {7.4}Hypothesis Testing}{80} \contentsline {chapter}{\numberline {8}UNBIASED METHODS FOR G AND R UNKNOWN}{83} \contentsline {section}{\numberline {8.1}Unbiased Estimators}{83} \contentsline {section}{\numberline {8.2}Unbiased Predictors}{87} \contentsline {section}{\numberline {8.3}Substitution Of Fixed Values For {\pbf G} And {\pbf R}}{89} \contentsline {section}{\numberline {8.4}Mixed Model Equations With Estimated {\pbf G} and {\pbf R}}{89} \contentsline {section}{\numberline {8.5}Tests Of Hypotheses Concerning {\pbf $\beta$}}{90} \contentsline {chapter}{\numberline {9}BIASED ESTIMATION AND PREDICTION}{93} \contentsline {section}{\numberline {9.1}Derivation Of BLBE And BLBP}{93} \contentsline {section}{\numberline {9.2}Use Of An External Estimate Of {\pbf ${\beta }$}}{95} \contentsline {section}{\numberline {9.3}Assumed Pattern Of Values Of {\pbf ${\beta }$}}{96} \contentsline {section}{\numberline {9.4}Evaluation Of Bias}{96} \contentsline {section}{\numberline {9.5}Evaluation Of Mean Squared Errors}{97} \contentsline {section}{\numberline {9.6}Estimability In Biased Estimation}{99} \contentsline {section}{\numberline {9.7}Tests Of Hypotheses}{101} \contentsline {subsection}{\numberline {9.7.1}Var $({\pbf e}) = {\pbf I}\sigma _{e}^{2}$}{101} \contentsline {subsection}{\numberline {9.7.2}${\prm Var} \ ({\pbf e}) = {\pbf R}$}{102} \contentsline {section}{\numberline {9.8}Estimation of {\pbf P}}{102} \contentsline {section}{\numberline {9.9}Illustration}{102} \contentsline {section}{\numberline {9.10}Relationships Among Methods}{110} \contentsline {subsection}{\numberline {9.10.1}Bayesian estimation}{110} \contentsline {subsection}{\numberline {9.10.2}Minimum mean squared error estimation}{111} \contentsline {subsection}{\numberline {9.10.3}Invariance property of Bayesian estimator}{111} \contentsline {subsection}{\numberline {9.10.4}Maximum likelihood estimation}{112} \contentsline {section}{\numberline {9.11}Pattern Of Values Of {\pbf P}}{112} \contentsline {chapter}{\numberline {10}QUADRATIC ESTIMATION OF VARIANCES}{113} \contentsline {section}{\numberline {10.1}A General Model For Variances And Covariances}{113} \contentsline {section}{\numberline {10.2}Quadratic Estimators}{117} \contentsline {section}{\numberline {10.3}Variances Of Estimators}{118} \contentsline {section}{\numberline {10.4}Solutions Not In The Parameter Space}{118} \contentsline {section}{\numberline {10.5}Form Of Quadratics}{119} \contentsline {section}{\numberline {10.6}Expectations of Quadratics}{120} \contentsline {section}{\numberline {10.7}Quadratics in $\mathaccent "705E {\pbf u}$ and $\mathaccent "705E {\pbf e}$}{121} \contentsline {section}{\numberline {10.8}Henderson's Method 1}{122} \contentsline {section}{\numberline {10.9}Henderson's Method 3}{129} \contentsline {section}{\numberline {10.10}A Simple Method for General {\pbf X}{\pbf ${\beta }$}}{133} \contentsline {section}{\numberline {10.11}Henderson's Method 2}{137} \contentsline {section}{\numberline {10.12}An Unweighted Means ANOVA}{139} \contentsline {section}{\numberline {10.13}Mean Squares For Testing ${\pbf K}'{\pbf u}$}{141} \contentsline {chapter}{\numberline {11}MIVQUE OF VARIANCES AND COVARIANCES}{143} \contentsline {section}{\numberline {11.1}La Motte Result For MIVQUE}{144} \contentsline {section}{\numberline {11.2}Alternatives To La Motte Quadratics}{144} \contentsline {section}{\numberline {11.3}Quadratics Equal To La Motte's}{145} \contentsline {section}{\numberline {11.4}Computation Of Missing $\mathaccent "705E {\pbf u}$}{149} \contentsline {section}{\numberline {11.5}Quadratics In $\mathaccent "705E {\pbf e}$ With Missing Observations}{150} \contentsline {section}{\numberline {11.6}Expectations Of Quadratics In $\mathaccent "705E {\pbf u}$ And $\mathaccent "705E {\pbf e}$}{151} \contentsline {section}{\numberline {11.7}Approximate MIVQUE}{152} \contentsline {section}{\numberline {11.8}MIVQUE (0)}{154} \contentsline {section}{\numberline {11.9}MIVQUE For Singular {\pbf G}}{155} \contentsline {section}{\numberline {11.10}MIVQUE For The Case {\pbf R} = {\pbf I}${\sigma }_{e}^{2}$}{155} \contentsline {section}{\numberline {11.11}Sampling Variances}{156} \contentsline {subsection}{\numberline {11.11.1}Result when ${\sigma }_{e}^{2}$ estimated from OLS residual}{157} \contentsline {section}{\numberline {11.12}Illustrations Of Approximate MIVQUE}{158} \contentsline {subsection}{\numberline {11.12.1}MIVQUE with $\mathaccent "705E {\sigma }_{e}^{2}$ = OLS residual}{158} \contentsline {subsection}{\numberline {11.12.2}Approximate MIVQUE using a diagonal g-inverse}{161} \contentsline {subsection}{\numberline {11.12.3}Approximate MIVQUE using a block diagonal approximate g-inverse}{162} \contentsline {subsection}{\numberline {11.12.4}Approximate MIVQUE using a triangular block diagonal approximate g-inverse}{163} \contentsline {section}{\numberline {11.13}An Algorithm for {\pbf R} = ${\pbf R}_{*}{\sigma }_{e}^{2}$ and Cov $({\pbf u}_{i},{\pbf u}_{j}^{'})$ = {\pbf 0}}{164} \contentsline {section}{\numberline {11.14}Illustration \ Of \ MIVQUE \ In \ Multivariate \ Model}{165} \contentsline {section}{\numberline {11.15}Other Types Of MIVQUE}{173} \contentsline {subsection}{\numberline {11.15.1}Not translation invariant and biased}{173} \contentsline {subsection}{\numberline {11.15.2}Translation invariant and biased}{174} \contentsline {section}{\numberline {11.16}Expectations Of Quadratics In $\mathaccent "705E {\pbf {\alpha }}$}{174} \contentsline {chapter}{\numberline {12}REML AND ML ESTIMATION}{177} \contentsline {section}{\numberline {12.1}Iterative MIVQUE}{177} \contentsline {section}{\numberline {12.2}An Alternative Algorithm For REML}{178} \contentsline {section}{\numberline {12.3}ML Estimation}{179} \contentsline {section}{\numberline {12.4}Approximate REML}{180} \contentsline {section}{\numberline {12.5}A Simple Result For Expectation Of Residual Sum Of Squares}{180} \contentsline {section}{\numberline {12.6}Biased Estimation With Few Iterations}{180} \contentsline {section}{\numberline {12.7}The Problem Of Finding Permissible Estimates}{182} \contentsline {section}{\numberline {12.8}Method For Singular {\pbf G }}{184} \contentsline {chapter}{\numberline {13}EFFECTS OF SELECTION}{185} \contentsline {section}{\numberline {13.1}Introduction}{185} \contentsline {section}{\numberline {13.2}An Example of Selection}{186} \contentsline {section}{\numberline {13.3}Conditional Means And Variances}{187} \contentsline {section}{\numberline {13.4}BLUE And BLUP Under Selection Model}{189} \contentsline {section}{\numberline {13.5}Selection On Linear Functions Of {\pbf y}}{190} \contentsline {subsection}{\numberline {13.5.1}Selection with ${\pbf L}'{\pbf X} \ = \ {\pbf 0}$}{190} \contentsline {section}{\numberline {13.6}With Non-Observable Random Factors}{192} \contentsline {section}{\numberline {13.7}Selection On A Subvector Of {\pbf y}}{193} \contentsline {section}{\numberline {13.8}Selection On {\pbf u}}{195} \contentsline {section}{\numberline {13.9}Inverse Of Conditional {\pbf A} Matrix}{197} \contentsline {section}{\numberline {13.10}Minimum Variance Linear Unbiased Predictors}{199} \contentsline {chapter}{\numberline {14}RESTRICTED BEST LINEAR PREDICTION}{203} \contentsline {section}{\numberline {14.1}Restricted Selection Index}{203} \contentsline {section}{\numberline {14.2}Restricted BLUP}{204} \contentsline {section}{\numberline {14.3}Application}{205} \contentsline {chapter}{\numberline {15}SAMPLING FROM FINITE POPULATIONS}{207} \contentsline {section}{\numberline {15.1}Finite {\pbf e}}{207} \contentsline {section}{\numberline {15.2}Finite {\pbf u}}{208} \contentsline {section}{\numberline {15.3}Infinite By Finite Interactions}{209} \contentsline {section}{\numberline {15.4}Finite By Finite Interactions}{210} \contentsline {section}{\numberline {15.5}Finite, Factorial, Mixed Models}{210} \contentsline {section}{\numberline {15.6}Covariance Matrices}{211} \contentsline {section}{\numberline {15.7}Estimability and Predictability}{213} \contentsline {section}{\numberline {15.8}BLUP When Some ${\pbf u}_{i}$ Are Finite}{217} \contentsline {section}{\numberline {15.9}An Easier Computational Method}{220} \contentsline {section}{\numberline {15.10}Biased Estimation}{222} \contentsline {chapter}{\numberline {16}THE ONE WAY CLASSIFICATION}{223} \contentsline {section}{\numberline {16.1}Estimation and Tests For Fixed ${\pbf a}$}{223} \contentsline {section}{\numberline {16.2}Levels of ${\pbf a}$ Equally Spaced}{226} \contentsline {section}{\numberline {16.3}Biased Estimation of $\mu + a_{i}$}{227} \contentsline {section}{\numberline {16.4}Model with Linear Trend of Fixed Levels of {\pbf a}}{229} \contentsline {section}{\numberline {16.5}The Usual One Way Covariate Model}{230} \contentsline {section}{\numberline {16.6}Nonhomogenous Regressions}{230} \contentsline {section}{\numberline {16.7}The Usual One Way Random Model}{232} \contentsline {section}{\numberline {16.8}Finite Levels of ${\pbf a}$}{234} \contentsline {section}{\numberline {16.9}One Way Random and Related Sires}{235} \contentsline {chapter}{\numberline {17}THE TWO WAY CLASSIFICATION}{239} \contentsline {section}{\numberline {17.1}The Two Way Fixed Model}{239} \contentsline {section}{\numberline {17.2}BLUE For The Filled Subclass Case}{240} \contentsline {section}{\numberline {17.3}The Fixed, Missing Subclass Case}{245} \contentsline {section}{\numberline {17.4}A Method Based On Assumption ${\gamma }_{ij}$ = 0 If $n_{ij}$ = 0}{247} \contentsline {section}{\numberline {17.5}Biased Estimation By Ignoring {\pbf $\gamma$}}{249} \contentsline {section}{\numberline {17.6}Priors On Squares And Products Of {\pbf $\gamma$}}{250} \contentsline {section}{\numberline {17.7}Priors On Squares And Products Of {\pbf a}, {\pbf b}, And {\pbf $\gamma$}}{254} \contentsline {section}{\numberline {17.8}The Two Way Mixed Model}{258} \contentsline {chapter}{\numberline {18}THE THREE WAY CLASSIFICATION}{265} \contentsline {section}{\numberline {18.1}The Three Way Fixed Model}{265} \contentsline {section}{\numberline {18.2}The Filled Subclass Case}{266} \contentsline {section}{\numberline {18.3}Missing Subclasses In The Fixed Model}{272} \contentsline {section}{\numberline {18.4}The Three Way Mixed Model}{278} \contentsline {chapter}{\numberline {19}NESTED CLASSIFICATIONS}{281} \contentsline {section}{\numberline {19.1}Two Way Fixed Within Fixed}{281} \contentsline {section}{\numberline {19.2}Two Way Random Within Fixed}{284} \contentsline {subsection}{\numberline {19.2.1}Sires within treatments}{284} \contentsline {subsection}{\numberline {19.2.2}Sires within breeds}{287} \contentsline {section}{\numberline {19.3}Random Within Random}{287} \contentsline {chapter}{\numberline {20}ANALYSIS OF REGRESSION MODELS}{289} \contentsline {section}{\numberline {20.1}Simple Regression Model}{289} \contentsline {section}{\numberline {20.2}Multiple Regression Model}{291} \contentsline {chapter}{\numberline {21}ANALYSIS OF COVARIANCE MODELS}{295} \contentsline {section}{\numberline {21.1}Two Way Fixed Model With Two Covariates}{295} \contentsline {section}{\numberline {21.2}Two Way Fixed Model With Missing Subclasses}{298} \contentsline {section}{\numberline {21.3}Covariates All Equal At The Same Level Of A Factor}{300} \contentsline {section}{\numberline {21.4}Random Regressions}{301} \contentsline {chapter}{\numberline {22}ANIMAL MODEL, SINGLE RECORDS}{303} \contentsline {section}{\numberline {22.1}Example With Dam-Daughter Pairs}{304} \contentsline {chapter}{\numberline {23}SIRE MODEL, SINGLE RECORDS}{309} \contentsline {chapter}{\numberline {24}ANIMAL MODEL, REPEATED RECORDS}{313} \contentsline {chapter}{\numberline {25}SIRE MODEL, REPEATED RECORDS}{321} \contentsline {chapter}{\numberline {26}ANIMAL MODEL, MULTIPLE TRAITS}{325} \contentsline {section}{\numberline {26.1}No Missing Data}{325} \contentsline {section}{\numberline {26.2}Missing Data}{329} \contentsline {section}{\numberline {26.3}EM Algorithm}{334} \contentsline {chapter}{\numberline {27}SIRE MODEL, MULTIPLE TRAITS}{341} \contentsline {section}{\numberline {27.1}Only One Trait Observed On A Progeny}{341} \contentsline {section}{\numberline {27.2}Multiple Traits Recorded On A Progeny}{345} \contentsline {section}{\numberline {27.3}Relationship To Sire Model With Repeated Records On Progeny}{348} \contentsline {chapter}{\numberline {28}JOINT COW AND SIRE EVALUATION}{349} \contentsline {section}{\numberline {28.1}Block Diagonality Of Mixed Model Equations}{349} \contentsline {section}{\numberline {28.2}Single Record On Single Trait}{351} \contentsline {section}{\numberline {28.3}Simple Repeatability Model}{354} \contentsline {section}{\numberline {28.4}Multiple Traits}{356} \contentsline {section}{\numberline {28.5}Summary Of Methods}{357} \contentsline {section}{\numberline {28.6}Gametic Model To Reduce The Number Of Equations}{358} \contentsline {subsection}{\numberline {28.6.1}Single record model}{359} \contentsline {subsection}{\numberline {28.6.2}Repeated records model}{361} \contentsline {chapter}{\numberline {29}NON-ADDITIVE GENETIC MERIT}{365} \contentsline {section}{\numberline {29.1}Model for Genetic Components}{365} \contentsline {section}{\numberline {29.2}Single Record on Every Animal}{366} \contentsline {section}{\numberline {29.3}Single or No Record on Each Animal}{369} \contentsline {section}{\numberline {29.4}A Reduced Set of Equations}{372} \contentsline {section}{\numberline {29.5}Multiple or No Records}{375} \contentsline {section}{\numberline {29.6}A \ Reduced Set of Equations for Multiple Records}{377} \contentsline {chapter}{\numberline {30}LINE CROSS AND BREED CROSS ANALYSES}{381} \contentsline {section}{\numberline {30.1}Genetic Model}{381} \contentsline {section}{\numberline {30.2}Covariances Between Crosses}{382} \contentsline {section}{\numberline {30.3}Reciprocal Crosses Assumed Equal}{384} \contentsline {section}{\numberline {30.4}Reciprocal Crosses With Maternal Effects}{386} \contentsline {section}{\numberline {30.5}Single Crosses As The Maternal Parent}{387} \contentsline {section}{\numberline {30.6}Breed Crosses}{388} \contentsline {section}{\numberline {30.7}Same Breeds Used As Sires And Dams}{388} \contentsline {chapter}{\numberline {31}MATERNAL EFFECTS}{395} \contentsline {section}{\numberline {31.1}Model For Maternal Effects}{395} \contentsline {section}{\numberline {31.2}Pedigrees Used In Example}{396} \contentsline {section}{\numberline {31.3}Additive And Dominance Maternal And Direct Effects}{398} \contentsline {chapter}{\numberline {32}THREE WAY MIXED MODELS}{399} \contentsline {section}{\numberline {32.1}The Example}{399} \contentsline {section}{\numberline {32.2}Estimation And Prediction}{400} \contentsline {section}{\numberline {32.3}Tests Of Hypotheses}{401} \contentsline {section}{\numberline {32.4}REML Estimation By EM Method}{403} \contentsline {chapter}{\numberline {33}SELECTION WHEN VARIANCES ARE UNEQUAL}{405} \contentsline {section}{\numberline {33.1}Sire Evaluation With Unequal Variances}{405} \contentsline {section}{\numberline {33.2}Cow Evaluation With Unequal Variances}{409}