# Log-Linear Modeling: Concepts, Interpretation, and Application

## Books An easily accessible introduction to log-linear modeling for non-statisticians

Highlighting advances that have lent to the topic's distinct, coherent methodology over the past decade, Log-Linear Modeling: Concepts, Interpretation, and Application provides an essential, introductory treatment of the subject, featuring many new and advanced log-linear methods, models, and applications.

The book begins with basic coverage of categorical data, and goes on to describe the basics of hierarchical log-linear models as well as decomposing effects in cross-classifications and goodness-of-fit tests. Additional topics include:

• The generalized linear model (GLM) along with popular methods of coding such as effect coding and dummy coding
• Parameter interpretation and how to ensure that the parameters reflect the hypotheses being studied
• Symmetry, rater agreement, homogeneity of association, logistic regression, and reduced designs models

Throughout the book, real-world data illustrate the application of models and understanding of the related results. In addition, each chapter utilizes R, SYSTAT®, and §¤EM software, providing readers with an understanding of these programs in the context of hierarchical log-linear modeling.

Log-Linear Modeling is an excellent book for courses on categorical data analysis at the upper-undergraduate and graduate levels. It also serves as an excellent reference for applied researchers in virtually any area of study, from medicine and statistics to the social sciences, who analyze empirical data in their everyday work.

Preface xiii

Acknowledgments xvii

1 Basics of Hierarchical Log-linear Models 1

1.1 Scaling: Which Variables Are Considered Categorical? 2

1.2 Crossing Two or More Variables 4

1.3 Goodman’s Three Elementary Views 8

1.4 Assumptions Made for Log-linear modeling 9

2 Effects in a Table 13

2.1 The Null Model 13

2.2 The Row Effects-Only Model 15

2.3 The Column Effects-Only Model 15

2.4 The Row-and Column-Effects-Model 16

2.5 Log-Linear Models 18

3 Goodness-of-Fit 23

3.1 Goodness of Fit I: Overall Fit Statistics 23

3.2 Goodness-of-Fit II: R2 Equivalents and Information Criteria 30

3.3 Goodness-of-Fit III: Null Hypotheses Concerning Parameters 35

3.4 Goodness-of-fit IV: Residual Analysis 36

3.5 The Relationship Between Pearson’s X2 and Log-linear

Modeling 52

4 Hierarchical Log-linear Models and Odds Ratio Analysis 55

4.1 The Hierarchy of Log-linear Models 55

4.2 Comparing Hierarchically Related Models 57

4.3 Odds Ratios and Log-linear-Models 63

4.4 Odds Ratios in Tables Larger than 2 x 2 65

4.5 Testing Null Hypotheses in Odds Ratio Analysis 70

4.6 Characteristics of the Odds Ratio 72

4.7 Application of the Odds Ratio 75

4.8 The Four Steps to Take When Log-linear-Modeling 81

4.9 Collapsibility 86

5 Computations I: Basic Log-linear Modeling 97

5.1 Log-linear Modeling in R 97

5.2 Log- linear Modeling in SYSTAT 102

5.3 Log-linear Modeling in lEM 106

6 The Design Matrix Approach 111

6.1 The Generalized Linear Model (GLM) 111

6.2 Design Matrices: Coding 115

7 Parameter Interpretation and Significance Tests 129

7.1 Parameter Interpretation Based on Design Matrices 130

7.2 The Two Sources of Parameter Correlation: Dependency of Vectors and Data Characteristics 139

7.3 Can Main Effects Be Interpreted? 143

7.4 Interpretation of Higher Order Interactions 150

8 Computations II: Design Matrices and Poisson GLM 157

8.1 GLM-based Log-linear-Modeling in R 157

8.2 Design Matrices in SYSTAT 164

8.3 Log-linear-Modeling with Design Matrices in lEM 170

9 Nonhierarchical and Nonstandard Log-linear Models 181

9.1 Defining Nonhierarchical and Nonstandard Log-linear-Models 182

9.2 Virtues of Nonhierarchical and Nonstandard Log-linear-Models 182

9.3 Scenarios for Nonstandard Log-linear-Models 184

9.4 Nonstandard Scenarios: Summary and Discussion 240

9.5 Schuster’s Approach to Parameter Interpretation 242

10 Computations III: Nonstandard Models 251

10.1 Non-Hierarchical and Nonstandard Models in R 251

10.2 Estimating Non-Hierarchical and Nonstandard Models with SYSTAT 256

10.3 Estimating Non-Hierarchical and Nonstandard Models with lEM 265

11 Sampling Schemes and Chisquare Decomposition 273

11.1 Sampling Schemes 273

11.2 Chi-Square Decomposition 276

12 Symmetry Models 289

12.1 Axial Symmetry 289

12.2 Point-symmetry 294

12.3 Point-axial Symmetry 295

12.4 Symmetry in Higher-Dimensional Cross-Classifications 296

12.5 Quasi-Symmetry 298

12.6 Extensions and Other Symmetry Models 301

12.7 Marginal Homogeneity: Symmetry in the Marginals 305

13 Log-linear Models of Rater Agreement 309

13.1 Measures of Rater Agreement in Contingency Tables 309

13.2 The Equal Weight Agreement Model 313

13.3 The Differential Weight Agreement Model 315

13.4 Agreement in Ordinal Variables 316

13.5 Extensions of Rater Agreement Models 319

14 Homogeneity of Associations 327

14.1 The Mantel-Haenszel and Breslow-Day Tests 327

14.2 Log-linear-Models to Test Homogeneity of Associations 330

14.3 Extensions and Generalizations 335

15 Logistic Regression and Logit Models 339

15.1 Logistic Regression 339

15.2 Log-linear Representation of Logistic Regression Models 344

15.3 Overdispersion in Logistic Regression 347

15.4 Logistic Regression Versus Log-linear Modeling Modules 349

15.5 Logit Models and Discriminant Analysis 351

15.6 Path Models 357

16 Reduced Designs 363

16.1 Fundamental Principles for Factorial Design 364

16.2 The Resolution Level of a Design 365

16.3 Sample Fractional Factorial Designs 368

17 Computations IV: Additional Models 379

17.1 Additional Log-linear-Models in R 379

17.2 Additional Log-linear-Models in SYSTAT 388

17.3 Additional Log-linear-Models in lEM 404

References 417

## Books & Journals

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### Journals

#### Stat #### Scandinavian Journal of Statistics View all