Multilevel Models (eBook)

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Jichuan Wang, Wright State University, Dayton, Ohio, USA; HaiyiXie, Dartmouth Medical School, Hanover, New Hampshire, USA; James H. Fisher, Wright State University, Dayton, Ohio, USA.

Preface 6
Contents 8
1 Introduction 12
1.1 Conceptual framework of multilevel modeling 12
1.2 Hierarchically structured data 14
1.3 Variables in multilevel data 15
1.4 Analytical problems with multilevel data 17
1.5 Advantages and limitations of multilevel modeling 19
1.6 Computer software for multilevel modeling 21
2 Basics of linear multilevel models 24
2.1 Intraclass correlation coefficient (ICC) 24
2.2 Formulation of two-level multilevel models 26
2.3 Model assumptions 28
2.4 Fixed and random regression coefficients 29
2.5 Cross-level interactions 31
2.6 Measurement centering 32
2.7 Model estimation 34
2.8 Model fit, hypothesis testing, and model comparisons 38
2.8.1 Model fit 38
2.8.2 Hypothesis testing 39
2.8.3 Model comparisons 41
2.9 Explained level-1 and level-2 variances 41
2.10 Steps for building multilevel models 44
2.11 Higher-level multilevel models 48
3 Application of two-level linear multilevel models 50
3.1 Data 50
3.2 Empty model 53
3.3 Predicting between-group variation 59
3.4 Predicting within-group variation 64
3.5 Testing level-1 random 68
3.6 Across-level interactions 73
3.7 Other issues in model development 77
4 Application of multilevel modeling to longitudinal data 84
4.1 Features of longitudinal data 84
4.2 Limitations of traditional approaches for modeling longitudinal data 85
4.3 Advantages of multilevel modeling for longitudinal data 86
4.4 Formulation of growth models 86
4.5 Data and variable description 88
4.6 Linear growth models 90
4.6.1 The shape of average outcome change over time 91
4.6.2 Random intercept growth models 91
4.6.3 Random intercept-slope growth models 95
4.6.4 Intercept and slope as outcomes 97
4.6.5 Controlling for individual background variables in models 99
4.6.6 Coding time score 100
4.6.7 Residual variance/covariance structures 102
4.6.8 Time-varying covariates 106
4.7 Curvilinear growth models 109
4.7.1 Polynomial growth model 109
4.7.2 Dealing with collinearity in higher order polynomial growth model 111
4.7.3 Piecewise (linear spline) growth model 117
5 Multilevel models for discrete outcome measures 124
5.1 Introduction to generalized linear mixed models 124
5.1.1 Generalized linear models 124
5.1.2 Generalized linear mixed models 126
5.2 SAS Procedures for multilevel modeling with discrete outcomes 127
5.3 Multilevel models for binary outcomes 128
5.3.1 Logistic regression models 128
5.3.2 Probit models 129
5.3.3 Unobserved latent variables and observed binary outcome measures 130
5.3.4 Multilevel logistic regression models 130
5.3.5 Application of multilevel logistic regression models 131
5.3.6 Application of multilevel logit models to longitudinal data 147
5.4 Multilevel models for ordinal outcomes 150
5.4.1 Cumulative logit models 150
5.4.2 Multilevel cumulative logit models 152
5.5 Multilevel models for nominal outcomes 157
5.5.1 Multinomial logit models 157
5.5.2 Multilevel multinomial logit models 158
5.5.3 Application of multilevel multinomial logit models 159
5.6 Multilevel models for count outcomes 165
5.6.1 Poisson regression models 166
5.6.2 Poisson regression with over-dispersion and a negative binomial model 168
5.6.3 Multilevel Poisson and negative binomial models 169
5.6.4 Application of multilevel Poisson and negative binomial models 169
6 Other applications of multilevel modeling and related issues 186
6.1 Multilevel zero-inflated models for count data with extra zeros 186
6.1.1 Fixed-effect zero-inflated Poisson (ZIP) model 187
6.1.2 Random effect zero-inflated Poisson (RE-ZIP) models 188
6.1.3 Random effect zero-inflated negative binomial (RE-ZINB) models 189
6.1.4 Application of RE-ZIP and RE-ZINB models 189
6.2 Mixed-effect mixed-distribution models for semi-continuous outcomes 199
6.2.1 Mixed-effect mixed distribution model 200
6.2.2 Application of the mixed-effect mixed distribution model 201
6.3 Bootstrap multilevel modeling 206
6.3.1 Nonparametric residual bootstrap multilevel modeling 207
6.3.2 Parametric residual bootstrap multilevel modeling 208
6.3.3 Application of nonparametric residual bootstrap multilevel modeling 209
6.4 Group-based models for longitudinal data analysis 221
6.4.1 Introduction to group-based trajectory model 223
6.4.2 Group-based logit trajectory model 225
6.4.3 Group-based zero-inflated Poisson (ZIP) trajectory model 233
6.4.4 Group-based censored normal trajectory models 241
6.5 Missing values issue 248
6.5.1 Missing data mechanisms and their implications 249
6.5.2 Handling missing data in longitudinal data analyses 250
6.6 Statistical power and sample size for multilevel modeling 252
6.6.1 Sample size estimation for two-level designs 252
6.6.2 Sample size estimation for longitudinal data analysis 253
References 258
Index 270