Mixed modelling is very useful, and easier than you think! Mixed modelling is now well established as a powerful approach to statistical data analysis. It is based on the recognition of random-effect terms in statistical models, leading to inferences and estimates that have much wider applicability and are more realistic than those otherwise obtained. Introduction to Mixed Modelling leads the reader into mixed modelling as a natural extension of two more familiar methods, regression analysis and analysis of variance. It provides practical guidance combined with a clear explanation of the underlying concepts. Like the first edition, this new edition shows diverse applications of mixed models, provides guidance on the identification of random-effect terms, and explains how to obtain and interpret best linear unbiased predictors (BLUPs). It also introduces several important new topics, including the following: * Use of the software SAS, in addition to GenStat and R. * Metaanalysis and the multiple testing problem. * The Bayesian interpretation of mixed models. Including numerous practical exercises with solutions, this book provides an ideal introduction to mixed modelling for final year undergraduate students, postgraduate students and professional researchers. It will appeal to readers from a wide range of scientific disciplines including statistics, biology, bioinformatics, medicine, agriculture, engineering, economics, archaeology and geography. Praise for the first edition: "One of the main strengths of the text is the bridge it provides between traditional analysis of variance and regression models and the more recently developed class of mixed models.Each chapter is well-motivated by at least one carefully chosen example.demonstrating the broad applicability of mixed models in many different disciplines.most readers will likely learn something new, and those previously unfamiliar with mixed models will obtain a solid foundation on this topic."--Kerrie Nelson University of South Carolina, in American Statistician, 2007

InhaltsangabePreface xi 1 The need for more than one random-effect term when fitting a regression line 1 1.1 A data set with several observations of variable Y at each value of variable X 1 1.2 Simple regression analysis: Use of the software GenStat to perform the analysis 2 1.3 Regression analysis on the group means 9 1.4 A regression model with a term for the groups 10 1.5 Construction of the appropriate F test for the significance of the explanatory variable when groups are present 13 1.6 The decision to specify a model term as random: A mixed model 14 1.7 Comparison of the tests in a mixed model with a test of lack of fit 16 1.8 The use of REsidual Maximum Likelihood (REML) to fit the mixed model 17 1.9 Equivalence of the different analyses when the number of observations per group is constant 21 1.10 Testing the assumptions of the analyses: Inspection of the residual values 26 1.11 Use of the software R to perform the analyses 28 1.12 Use of the software SAS to perform the analyses 33 1.13 Fitting a mixed model using GenStat's Graphical User Interface (GUI) 40 1.14 Summary 46 1.15 Exercises 47 References 51 2 The need for more than one random-effect term in a designed experiment 52 2.1 The split plot design: A design with more than one random-effect term 52 2.2 The analysis of variance of the split plot design: A random-effect term for the main plots 54 2.3 Consequences of failure to recognize the main plots when analysing the split plot design 62 2.4 The use of mixed modelling to analyse the split plot design 64 2.5 A more conservative alternative to the F and Wald statistics 66 2.6 Justification for regarding block effects as random 67 2.7 Testing the assumptions of the analyses: Inspection of the residual values 68 2.8 Use of R to perform the analyses 71 2.9 Use of SAS to perform the analyses 77 2.10 Summary 81 2.11 Exercises 82 References 86 3 Estimation of the variances of random-effect terms 87 3.1 The need to estimate variance components 87 3.2 A hierarchical random-effects model for a three-stage assay process 87 3.3 The relationship between variance components and stratum mean squares 91 3.4 Estimation of the variance components in the hierarchical random-effects model 93 3.5 Design of an optimum strategy for future sampling 95 3.6 Use of R to analyse the hierarchical three-stage assay process 98 3.7 Use of SAS to analyse the hierarchical three-stage assay process 100 3.8 Genetic variation: A crop field trial with an unbalanced design 102 3.9 Production of a balanced experimental design by 'padding' with missing values 106 3.10 Specification of a treatment term as a random-effect term: The use of mixed-model analysis to analyse an unbalanced data set 110 3.11 Comparison of a variance component estimate with its standard error 112 3.12 An alternative significance test for variance components 113 3.13 Comparison among significance tests for variance components 116 3.14 Inspection of the residual values 117 3.15 Heritability: The prediction of genetic advance under selection 117 3.16 Use of R to analyse the unbalanced field trial 122 3.17 Use of SAS to analyse the unbalanced field trial 125 3.18 Estimation of variance components in the regression analysis on grouped data 128 3.19 Estimation of variance components for block effects in the split-plot experimental design 130 3.20 Summary 132 3.21 Exercises 133 References 136 4 Interval estimates for fixed-effect terms in mixed models 137 4.1 The concept of an interval estimate 137 4.2 Standard errors for regression coefficients in a mixed-model analysis 138 4.3 Standard errors for differences between treatment means in the split-plot design 142 4.4 A significance test for the difference between treatment means 144 4.5 The least significant difference (LSD) between treatment means 147 4.6 Standard errors f

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