Statistical Group Comparison

TIM FUTING LIAO, PhD, is Associate Professor of Sociology and Statistics at the University of Illinois at Urbana-Champaign. He currently teaches as Senior Lecturer in Sociology at the University of Essex, UK.

• Preface xiii1. Introduction1.1 Rationale for Statistical Comparison 11.2 Comparative Research in the Social Sciences 31.3 Focus of the Book 51.4 Outline of the Book 51.4.1 Chapter 2-Statistical Foundation for Comparison 61.4.2 Chapter 3-Comparison in Linear Regression 61.4.3 Chapter 4-Nonparametric Comparison 61.4.4 Chapter 5-Comparing Rates 61.4.5 Chapter 6-Comparison in Generalized Linear Models 61.4.6 Chapter 7-Additional Topics of Comparison in Generalized Linear Models 71.4.7 Chapter 8-Comparison in Structural Equation Modeling 71.4.8 Chapter 9-Comparison with Categorical Latent Variables 81.4.9 Chapter 10-Comparison in Multilevel Analysis 81.4.10 Summary 82. Statistical Foundation for Comparison 92.1 A System for Statistical Comparison 92.2 Test Statistics 102.2.1 The x2 Test 102.2.2 The t-Test 122.2.3 The F-test 132.2.4 The Likelihood Ratio Test 142.2.5 The Wald Test 152.2.6 The Lagrange Multiplier Test 172.2.7 A Summary Comparison of LRT WT and LMT 182.3 What to Compare? 192.3.1 Comparing Distributions 192.3.2 Comparing Data Structures 192.3.3 Comparing Model Structures 192.3.4 Comparing Model Parameters 203. Comparison in Linear Models3.1 Introduction 213.2 An Example 223.3 Some Preliminary Considerations 233.4 The Linear Model 233.5 Comparing Two Means 243.6 Anova 253.7 Multiple Comparison Methods 273.7.1 Least Significance Difference Test 273.7.2 Tukey’s Model 273.7.3 Scheffk’s Method 283.7.4 Bonferroni’s Method 293.8 Ancova 303.9 Multiple Linear Regression 303.10 Regression Decomposition 333.10.1 Rationale 333.10.2 Algebraic Presentation 343.10.3 Interpretation 343.10.4 Extension to Multiple Regression 353.11 Which Linear Method to Use? 364. Nonparametric Comparison4.1 Nonparametic Tests 374.1.1 Kolmogorov-Smirnov Two-Sample Test 384.1.2 Mann-Whitney U-Test 384.2 Resampling Methods 394.2.1 Permutation Methods 394.2.2 Bootstrapping Methods 424.3 Relative Distribution Methods 445. Comparison of Rates5.1 The Data 515.2 Standardization 525.2.1 Direct Standardization 525.2.2 Indirect Standardization 545.2.3 Model-Based Standardization 555.3 Decomposition 585.3.1 Arithmetic Decomposition 585.3.2 Model-Based Decomposition 596. Comparison in Generalized Linear Models6.1 Introduction 626.1.1 The Exponential Family of Distributions 636.1.2 The Link Function 646.1.3 Maximum Likelihood Estimation 666.2 Comparing Generalized Linear Models 686.2.1 The Null Hypothesis 686.2.2 Comparisons Using Likelihood Ratio Tests 686.2.3 The Chow Test as a Special Case 706.3 A Logit Model Example 716.3.1 The Data 716.3.2 The Model Comparison 726.4 A Hazard Rate Model Example 746.4.1 The Model 746.4.2 The Data 776.4.3 The Model Comparison 786.A Data Used in Section 6.4 797. Additional Topics of Comparison in Generalized Linear Models 817.1 Introduction 81  7.2 GLM for Matched Case-Control Studies 817.2.1 The 1 : 1 Matched Study 837.2.2 The 1 : m Design 847.2.3 The n : m Design 857.3 Dispersion Heterogeneity 877.3.1 The Data 927.3.2 Group Comparison with Heterogeneous Dispersion 937.4 Bayesian Generalized Linear Models 967.4.1 Bayesian Inference 967.4.2 An Example 1047.A The Data for the n : m Design 1088. Comparison in Structural Equation Modeling8.1 Introduction 11 18.2 Statistical Background 1128.2.1 Notation and Specification 1128.2.2 Identification 1158.2.3 Estimation 1168.2.4 Modification 1178.2.5 Interpretation 1188.3 Mean and Covariance Structures 1198.4 Group Comparison in SEM 1218.4.1 Equality of Multivariate Distributions 1228.4.2 Equality of Covariance Matrices 1238.4.3 Equality of Model Forms 1248.4.4 Equality of Model Parameters 1258.5 An Example 1278.5.1 Comparing Correlation Matrices 1278.5.2 Comparing Covariance Structures and Multivariate Distributions 1308.5.3 Comparing Mean and Covariance Structures 1328.A Examples of Computer Program Listings 1348.A.1 An EQS Program for Comparing Correlation Matrices 1348.A.2 A LISREL Program for Comparing Correlation Matrices 1368.A.3 An EQS Program for Comparing Covariance Structures 1368.A.4 A LISREL Program for Comparing Covariance Structures 1388.A.5 An EQS Program for Comparing Mean and Covariance Structures 1398.A.6 A LISREL Program for Comparing Mean and Covariance Structures 1409 Comparison with Categorical Latent Variables1429.1 Introduction 1429.2 Latent Class Models 1439.3 Latent Trait Models 1479.4 Latent Variable Models for Continuous Indicators 1489.5 Causal Models with Categorical Latent Variables 1499.6 Comparison with Categorical Latent Variables 1519.6.1 Comparing Sampling Distributions 1529.6.2 Comparing Types and Patterns of Association between Variables 1529.6.3 Comparing Conditional Structure and Response Probabilities 1539.6.4 Comparing Latent Distributions and Conditional Probabilities 1539.7 Examples 1549.7.1 Comparison in Latent Class Analysis 1549.7.2 Comparison in a Path Model with Categorical Latent Variables 1589.A Software for Categorical Latent Variables 1619.A.1 MLLSA 1619.A.2 LAT 1619.A.3 PANMARK 1619.A.4 LCAG 1629.A.5 LEM 1629.A.6 Latent GOLD 1629.A.7 Mplus 1639.A.8 LATCLASS TWOMISS and POLYMISS 1639.A.9 1ca.S and lcreg.sas 1639.B Computer Program Listings for the Examples 1639.B.1 A LEM Program for Comparing Sampling Zeros 1639.B.2 A LEM Program for Comparing LCMs Assuming Two-way Interaction 1649.B.3 A LEM Program for Comparing LCMs Assuming Linear-by-Linear Association 1649.B.4 A LEM Program for Comparing LCMs Assuming Column-Effect RC-I1 Association 1659.B.5 A LEM Program for Comparing LCMs Assuming Complete Homogeneity 1659.B.6 A LEM Program for Comparing LCMs Assuming Complete Heterogeneity 1669.B.7 A LEM Program for Comparing LCMs Assuming Partial Heterogeneity 1669.B.8 A LEM Program for Comparing LCMs Assuming Partial Homogeneity 1679.B.9 Data for Example 2 1679.B.10 A LEM Program for Example 2 with Heterogeneous Groups 1699.B.11 A LEM Program for Example 2 with Homogeneous Measurement Parameters 1699.B.12 A LEM Program for Example 2 with Homogeneous Structural Parameters 1709.B.13 A LEM Program for Example 2 with Homogeneous Model Parameters 17110. Comparison in Multilevel Analysis 17210.1 Introduction 17210.2 An Introduction to Multilevel Analysis 17310.2.1 The Multilevel Setting 17310.2.2 An Introductory Bibliography 17410.2.3 Fixed versus Random Effects 17510.3 The Basics of the Linear Multilevel Model 17610.3.1 The Basic Data Structure 17610.3.2 Random Intercept Models 17710.3.3 Random Coefficients Models 17910.3.4 An Example 18010.3.5 ANOVA with Random Effects 18110.3.6 ANCOVA with Random Effects 18110.3.7 Random Coefficient Models without Cross-Level Effects 18310.3.8 Random Coefficient Models with Cross-Level Effects 18310.3.9 Assumptions of the Linear Multilevel Model 18610.4 The Basics of the Generalized Linear Multilevel Model 18610.4.1 A Random Coefficient Logit Model with Cross- Level Effects 18710.4.2 A Random Coefficient Probit Model with Cross- Level Effects 19010.5 Group as an External Variable in Multilevel Analysis 19010.6 The Relation between Multilevel Analysis and Group Comparison 19110.6.1 Bridging Fixed and Random Effects Models 19110.6.2 An Example 19210.7 Multiple Membership Models 19310.8 Summary 19410.A Software for Multilevel Analysis 19510.A.1 Special-Purpose Software 19510.A.2 General-Purpose Software 19510.A.3 Software for Other Special Purposes 19610.B SAS Program Listings for GLMM Examples 19710.B.1 Syntax for Producing the Logit Results in Table 10.11 19710.B.2 Syntax for Producing the Probit Results in Table 10.11 19710.B.3 Syntax for the Random Effects GLMM in Table 10.12 198References 199Index 207

"...presents a range of statistical methods for comparison of data sets." (SciTech Book News, Vol. 26, No. 2, June 2002) "...excellent resource book...highly recommended for graduate students, faculty, and researchers." (Choice, Vol. 40, No. 3, November 2002)"...graduate students and applied researchers may find the book a convenient reference..." (Short Book Reviews, December 2002)"...brings together a broad range of statistical methods for comparison developed over recent years." (Mathematical Reviews, 2003b)"...covers a wide variety of statistical methodologies in which group comparisons play a key role...worth reading...to get an overview of...topics covered." (Technometrics, Vol. 45, No. 2, May 2003)