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Full Description
An incomparably useful examination of statistical methods for comparison
The nature of doing science, be it natural or social, inevitably calls for comparison. Statistical methods are at the heart of such comparison, for they not only help us gain understanding of the world around us but often define how our research is to be carried out. The need to compare between groups is best exemplified by experiments, which have clearly defined statistical methods. However, true experiments are not always possible. What complicates the matter more is a great deal of diversity in factors that are not independent of the outcome.
Statistical Group Comparison brings together a broad range of statistical methods for comparison developed over recent years. The book covers a wide spectrum of topics from the simplest comparison of two means or rates to more recently developed statistics including double generalized linear models and Bayesian as well as hierarchical methods. Coverage includes:
* Testing parameter equality in linear regression and other generalized linear models (GLMs), in order of increasing complexity
* Likelihood ratio, Wald, and Lagrange multiplier statistics examined where applicable
* Group comparisons involving latent variables in structural equation modeling
* Models of comparison for categorical latent variables
Examples are drawn from the social, political, economic, and biomedical sciences; many can be implemented using widely available software. Because of the range and the generality of the statistical methods covered, researchers across many disciplines-beyond the social, political, economic, and biomedical sciences-will find the book a convenient reference for many a research situation where comparisons may come naturally.
Contents
Preface xiii
1. Introduction
1.1 Rationale for Statistical Comparison 1
1.2 Comparative Research in the Social Sciences 3
1.3 Focus of the Book 5
1.4 Outline of the Book 5
1.4.1 Chapter 2-Statistical Foundation for Comparison 6
1.4.2 Chapter 3-Comparison in Linear Regression 6
1.4.3 Chapter 4-Nonparametric Comparison 6
1.4.4 Chapter 5-Comparing Rates 6
1.4.5 Chapter 6-Comparison in Generalized Linear Models 6
1.4.6 Chapter 7-Additional Topics of Comparison in Generalized Linear Models 7
1.4.7 Chapter 8-Comparison in Structural Equation Modeling 7
1.4.8 Chapter 9-Comparison with Categorical Latent Variables 8
1.4.9 Chapter 10-Comparison in Multilevel Analysis 8
1.4.10 Summary 8
2. Statistical Foundation for Comparison 9
2.1 A System for Statistical Comparison 9
2.2 Test Statistics 10
2.2.1 The x2 Test 10
2.2.2 The t-Test 12
2.2.3 The F-test 13
2.2.4 The Likelihood Ratio Test 14
2.2.5 The Wald Test 15
2.2.6 The Lagrange Multiplier Test 17
2.2.7 A Summary Comparison of LRT WT and LMT 18
2.3 What to Compare? 19
2.3.1 Comparing Distributions 19
2.3.2 Comparing Data Structures 19
2.3.3 Comparing Model Structures 19
2.3.4 Comparing Model Parameters 20
3. Comparison in Linear Models
3.1 Introduction 21
3.2 An Example 22
3.3 Some Preliminary Considerations 23
3.4 The Linear Model 23
3.5 Comparing Two Means 24
3.6 Anova 25
3.7 Multiple Comparison Methods 27
3.7.1 Least Significance Difference Test 27
3.7.2 Tukey's Model 27
3.7.3 Scheffk's Method 28
3.7.4 Bonferroni's Method 29
3.8 Ancova 30
3.9 Multiple Linear Regression 30
3.10 Regression Decomposition 33
3.10.1 Rationale 33
3.10.2 Algebraic Presentation 34
3.10.3 Interpretation 34
3.10.4 Extension to Multiple Regression 35
3.11 Which Linear Method to Use? 36
4. Nonparametric Comparison
4.1 Nonparametic Tests 37
4.1.1 Kolmogorov-Smirnov Two-Sample Test 38
4.1.2 Mann-Whitney U-Test 38
4.2 Resampling Methods 39
4.2.1 Permutation Methods 39
4.2.2 Bootstrapping Methods 42
4.3 Relative Distribution Methods 44
5. Comparison of Rates
5.1 The Data 51
5.2 Standardization 52
5.2.1 Direct Standardization 52
5.2.2 Indirect Standardization 54
5.2.3 Model-Based Standardization 55
5.3 Decomposition 58
5.3.1 Arithmetic Decomposition 58
5.3.2 Model-Based Decomposition 59
6. Comparison in Generalized Linear Models
6.1 Introduction 62
6.1.1 The Exponential Family of Distributions 63
6.1.2 The Link Function 64
6.1.3 Maximum Likelihood Estimation 66
6.2 Comparing Generalized Linear Models 68
6.2.1 The Null Hypothesis 68
6.2.2 Comparisons Using Likelihood Ratio Tests 68
6.2.3 The Chow Test as a Special Case 70
6.3 A Logit Model Example 71
6.3.1 The Data 71
6.3.2 The Model Comparison 72
6.4 A Hazard Rate Model Example 74
6.4.1 The Model 74
6.4.2 The Data 77
6.4.3 The Model Comparison 78
6.A Data Used in Section 6.4 79
7. Additional Topics of Comparison in Generalized Linear Models 81
7.1 Introduction 81
7.2 GLM for Matched Case-Control Studies 81
7.2.1 The 1 : 1 Matched Study 83
7.2.2 The 1 : m Design 84
7.2.3 The n : m Design 85
7.3 Dispersion Heterogeneity 87
7.3.1 The Data 92
7.3.2 Group Comparison with Heterogeneous Dispersion 93
7.4 Bayesian Generalized Linear Models 96
7.4.1 Bayesian Inference 96
7.4.2 An Example 104
7.A The Data for the n : m Design 108
8. Comparison in Structural Equation Modeling
8.1 Introduction 11 1
8.2 Statistical Background 112
8.2.1 Notation and Specification 112
8.2.2 Identification 115
8.2.3 Estimation 116
8.2.4 Modification 117
8.2.5 Interpretation 118
8.3 Mean and Covariance Structures 119
8.4 Group Comparison in SEM 121
8.4.1 Equality of Multivariate Distributions 122
8.4.2 Equality of Covariance Matrices 123
8.4.3 Equality of Model Forms 124
8.4.4 Equality of Model Parameters 125
8.5 An Example 127
8.5.1 Comparing Correlation Matrices 127
8.5.2 Comparing Covariance Structures and Multivariate Distributions 130
8.5.3 Comparing Mean and Covariance Structures 132
8.A Examples of Computer Program Listings 134
8.A.1 An EQS Program for Comparing Correlation Matrices 134
8.A.2 A LISREL Program for Comparing Correlation Matrices 136
8.A.3 An EQS Program for Comparing Covariance Structures 136
8.A.4 A LISREL Program for Comparing Covariance Structures 138
8.A.5 An EQS Program for Comparing Mean and Covariance Structures 139
8.A.6 A LISREL Program for Comparing Mean and Covariance Structures 140
9 Comparison with Categorical Latent Variables142
9.1 Introduction 142
9.2 Latent Class Models 143
9.3 Latent Trait Models 147
9.4 Latent Variable Models for Continuous Indicators 148
9.5 Causal Models with Categorical Latent Variables 149
9.6 Comparison with Categorical Latent Variables 151
9.6.1 Comparing Sampling Distributions 152
9.6.2 Comparing Types and Patterns of Association between Variables 152
9.6.3 Comparing Conditional Structure and Response Probabilities 153
9.6.4 Comparing Latent Distributions and Conditional Probabilities 153
9.7 Examples 154
9.7.1 Comparison in Latent Class Analysis 154
9.7.2 Comparison in a Path Model with Categorical Latent Variables 158
9.A Software for Categorical Latent Variables 161
9.A.1 MLLSA 161
9.A.2 LAT 161
9.A.3 PANMARK 161
9.A.4 LCAG 162
9.A.5 LEM 162
9.A.6 Latent GOLD 162
9.A.7 Mplus 163
9.A.8 LATCLASS TWOMISS and POLYMISS 163
9.A.9 1ca.S and lcreg.sas 163
9.B Computer Program Listings for the Examples 163
9.B.1 A LEM Program for Comparing Sampling Zeros 163
9.B.2 A LEM Program for Comparing LCMs Assuming Two-way Interaction 164
9.B.3 A LEM Program for Comparing LCMs Assuming Linear-by-Linear Association 164
9.B.4 A LEM Program for Comparing LCMs Assuming Column-Effect RC-I1 Association 165
9.B.5 A LEM Program for Comparing LCMs Assuming Complete Homogeneity 165
9.B.6 A LEM Program for Comparing LCMs Assuming Complete Heterogeneity 166
9.B.7 A LEM Program for Comparing LCMs Assuming Partial Heterogeneity 166
9.B.8 A LEM Program for Comparing LCMs Assuming Partial Homogeneity 167
9.B.9 Data for Example 2 167
9.B.10 A LEM Program for Example 2 with Heterogeneous Groups 169
9.B.11 A LEM Program for Example 2 with Homogeneous Measurement Parameters 169
9.B.12 A LEM Program for Example 2 with Homogeneous Structural Parameters 170
9.B.13 A LEM Program for Example 2 with Homogeneous Model Parameters 171
10. Comparison in Multilevel Analysis 172
10.1 Introduction 172
10.2 An Introduction to Multilevel Analysis 173
10.2.1 The Multilevel Setting 173
10.2.2 An Introductory Bibliography 174
10.2.3 Fixed versus Random Effects 175
10.3 The Basics of the Linear Multilevel Model 176
10.3.1 The Basic Data Structure 176
10.3.2 Random Intercept Models 177
10.3.3 Random Coefficients Models 179
10.3.4 An Example 180
10.3.5 ANOVA with Random Effects 181
10.3.6 ANCOVA with Random Effects 181
10.3.7 Random Coefficient Models without Cross-Level Effects 183
10.3.8 Random Coefficient Models with Cross-Level Effects 183
10.3.9 Assumptions of the Linear Multilevel Model 186
10.4 The Basics of the Generalized Linear Multilevel Model 186
10.4.1 A Random Coefficient Logit Model with Cross- Level Effects 187
10.4.2 A Random Coefficient Probit Model with Cross- Level Effects 190
10.5 Group as an External Variable in Multilevel Analysis 190
10.6 The Relation between Multilevel Analysis and Group Comparison 191
10.6.1 Bridging Fixed and Random Effects Models 191
10.6.2 An Example 192
10.7 Multiple Membership Models 193
10.8 Summary 194
10.A Software for Multilevel Analysis 195
10.A.1 Special-Purpose Software 195
10.A.2 General-Purpose Software 195
10.A.3 Software for Other Special Purposes 196
10.B SAS Program Listings for GLMM Examples 197
10.B.1 Syntax for Producing the Logit Results in Table 10.11 197
10.B.2 Syntax for Producing the Probit Results in Table 10.11 197
10.B.3 Syntax for the Random Effects GLMM in Table 10.12 198
References 199
Index 207
