marginal models for dependent clustered and longitudinal categorical data and Mixed Model SAS Code
Wicher Bergsma

recommend to download for one of them to save money since SAS code is not important this submission cannot be altered due to limitation of the uploading system
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Marginal Models for CategoricalData . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Historical and Comparable Approaches . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Coefficients for the Comparison ofMarginalDistributions . . . . . . . . 9
1.3.1 MeasuringDifferences in Location . . . . . . . . . . . . . . . . . . . . . 9
1.3.2 MeasuringDifferences in Dispersion . . . . . . . . . . . . . . . . . . . . 13
1.3.3 MeasuringAssociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.4 MeasuringAgreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Loglinear Marginal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Ordinary LoglinearModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 Basic Concepts and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.2 ModelingAssociationAmong ThreeVariables. . . . . . . . . . . . 30
2.2 Applications of LoglinearMarginalModels . . . . . . . . . . . . . . . . . . . . 34
2.2.1 Research Questions and Designs Requiring Marginal Models 34
2.2.2 ComparingOne Variable Distributions . . . . . . . . . . . . . . . . . . 36
2.2.3 More ComplexDesigns and Research Questions . . . . . . . . . . 42
2.3 Maximum Likelihood Inference for Loglinear Marginal Models . . . 51
2.3.1 Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3.2 Specifying Loglinear Marginal Models by Constraining
the Cell Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.3 Simultaneous Modeling of Joint and Marginal
Distributions: Redundancy, Incompatibility and Other Issues 61
2.3.4 ***Maximum Likelihood Estimates of Constrained Cell
Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.3.5 ***A Numerical Algorithm for ML Estimation . . . . . . . . . . . 67
2.3.6 ***Efficient Computation of ML Estimates
for Simultaneous Joint andMarginalModels . . . . . . . . . . . . . 70
2.3.7 ***Large Sample Distribution of ML estimates . . . . . . . . . . . 71
2.3.8 Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
ix
x Contents
3 NonloglinearMarginal Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.1 Comparing Item Characteristics for Different Measurement Levels . 75
3.1.1 Interval Level ofMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.1.2 Ordinal Level ofMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.1.3 Nominal Level ofMeasurement . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 ComparingAssociations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.1 Generalized exp-log Specification of Nonloglinear
MarginalModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3.2 Compatibility and Redundancy of Restrictions . . . . . . . . . . . . 93
3.3.3 Homogeneous Specification of Coefficients . . . . . . . . . . . . . . 93
3.3.4 ***Algorithm for Maximum Likelihood Estimation . . . . . . . 94
3.3.5 ***Asymptotic Distribution of ML Estimates . . . . . . . . . . . . 95
4 Marginal Analysis of Longitudinal Data . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.1 TrendData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.1.1 Comparing Net Changes in More Than One Characteristic . . 99
4.1.2 Simultaneous Tests for Restrictions on Association and Net
Change:Modeling Joint andMarginal Tables . . . . . . . . . . . . . 102
4.2 Panel Data: Investigating Net Changes in One Characteristic . . . . . . 104
4.2.1 Overall Net Changes; Cumulative Proportions; Growth
Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.2.2 Subgroup Comparisons of Net Changes . . . . . . . . . . . . . . . . . 115
4.2.3 Changes in Associations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.3 Gross Changes in One Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3.1 Comparing Turnover Tables for Different Periods . . . . . . . . . 120
4.3.2 Comparing SummaryMeasures of Gross Change . . . . . . . . . 126
4.3.3 Extensions; Net Plus Gross Changes; Multiway Turnover
Tables; Subgroup Comparisons. . . . . . . . . . . . . . . . . . . . . . . . . 129
4.4 Net and Gross Changes in Two Related Characteristics . . . . . . . . . . . 130
4.4.1 Net Changes in Two Characteristics . . . . . . . . . . . . . . . . . . . . . 131
4.4.2 Changes in Association Between Two Changing
Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.4.3 Gross Changes in Two Characteristics . . . . . . . . . . . . . . . . . . . 140
4.4.4 Combining Hypotheses about Net and Gross Changes . . . . . 147
4.5 Minimally Specified Models for Comparing Tables
with OverlappingMarginals;Detection of ProblematicModels . . . . 148
5 Causal Analyses: Structural Equation Models
and (Quasi-)Experimental Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.1 SEMs - Structural EquationModels . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.1.1 SEMs for CategoricalData . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.1.2 An Example:Women’s Role . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.3 MarginalModeling and Categorical SEM. . . . . . . . . . . . . . . . 165
5.2 Analysis of (Quasi-)ExperimentalData . . . . . . . . . . . . . . . . . . . . . . . . 172
Contents xi
5.2.1 The One-group Pretest-Posttest Design . . . . . . . . . . . . . . . . . . 173
5.2.2 The NonequivalentControl GroupDesign . . . . . . . . . . . . . . . 175
5.2.3 A Truly ExperimentalDesign . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6 Marginal Modeling with Latent Variables . . . . . . . . . . . . . . . . . . . . . . . . 191
6.1 Latent ClassModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2 LatentMarginal Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.3 Loglinear and Nonloglinear Latent Class Models: Equal Reliabilities 198
6.3.1 Restrictions on Conditional Response Probabilities . . . . . . . . 199
6.3.2 Restrictions on Odds Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.3.3 Restrictions on PercentageDifferences . . . . . . . . . . . . . . . . . . 204
6.3.4 Restrictions on Agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
6.4 Marginal causal analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
6.4.1 SEMs with latentmarginal homogeneity . . . . . . . . . . . . . . . . . 207
6.4.2 Latent Variable SEMs for Clustered Data . . . . . . . . . . . . . . . . 209
6.5 Estimation of Marginal Models with Latent Variables
Using the EMAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
6.5.1 Basic EMAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.5.2 ***General EM for MarginalModels . . . . . . . . . . . . . . . . . . . 215
6.5.3 ***Marginal Restrictions in Combination with a Loglinear
Model for the Complete Table . . . . . . . . . . . . . . . . . . . . . . . . . 217
6.5.4 ***Speeding up of the EM Algorithm for Separable Models 218
6.5.5 ***Asymptotic Distribution of ML Estimates . . . . . . . . . . . . 219
7 Conclusions, Extensions, and Applications . . . . . . . . . . . . . . . . . . . . . . . 223
7.1 MarginalModels for Continuous Variables . . . . . . . . . . . . . . . . . . . . . 224
7.1.1 Changes inMeans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.1.2 Changes in Correlation and Regression Coefficients . . . . . . . 226
7.2 Alternative Procedures andModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
7.2.1 Alternative Estimation Procedures:WLS and GEE . . . . . . . . 228
7.2.2 Modeling Dependent Observations: Marginal, Random
Effects, and Fixed EffectsModels . . . . . . . . . . . . . . . . . . . . . . 230
7.3 Specific Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
7.3.1 Multiple Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
7.3.2 CategoricalDyadic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
7.3.3 Mokken Scale Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.4 Problems and FutureDevelopments . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
7.5 Software,Generalized exp-log Routines, andWebsite . . . . . . . . . . . . 244
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263