The Statistical Analysis of Categorical Data

The aim of this book is to give an up to date account of the most commonly uses statisti­ cal models for categorical data. The emphasis is on the connection between theory and applications to real data sets. The book only covers models for categorical data. Various models for mixed continuous and categorical data are thus excluded. The book is written as a textbook, although many methods and results are quite recent. This should imply, that the book can be used for a graduate course in categorical data analysis. With this aim in mind chapters 3 to 12 are concluded with a set of exer­ cises. In many cases, the data sets are those data sets, which were not included in the examples of the book, although they at one point in time were regarded as potential can­ didates for an example. A certain amount of general knowledge of statistical theory is necessary to fully benefit from the book. A summary of the basic statistical concepts deemed necessary pre­ requisites is given in chapter 2. The mathematical level is only moderately high, but the account in chapter 3 of basic properties of exponential families and the parametric multinomial distribution is made as mathematical precise as possible without going into mathematical details and leaving out most proofs.

1. Categorical Data.- 2. Preliminaries.- 2.1 Statistical models.- 2.2 Estimation.- 2.3 Testing statistical hypotheses.- 2.4 Checking the model.- 3. Statistical Inference.- 3.1 Log-linear models.- 3.2 The one-dimensional case.- 3.3 The multi-dimensional case.- 3.4 Testing composite hypotheses.- 3.5 The parametric multinomial distribution.- 3.6 Generalized linear models.- 3.7 Solution of likelihood equations.- 3.8 Exercises.- 4. Two-way Contingency Tables.- 4.1 Three models.- 4.2 The 2×2 table.- 4.3 The log-linear parameterization.- 4.4 The hypothesis of no interaction.- 4.5 Residual analysis.- 4.6 Exercises.- 5. Three-way Contingency Tables.- 5.1 The log-linear parameterization.- 5.2 Hypothesis in a three-way table.- 5.3 Hypothesis testing.- 5.4 Decomposition of the test statistic.- 5.5 Detection of model departures.- 5.6 Exercises.- 6. Multi-dimension Contingency Tables.- 6.1 The log-linear model.- 6.2 Interpretation of log-linear models.- 6.3 Search for a model.- 6.4 Diagnostics for model departures.- 6.5 Exercises.- 7. Incomplete Tables, Separability and Collapsibility.- 7.1 Incomplete tables.- 7.2 Two-way tables and quasi-independence.- 7.3 Higher order tables. Separability.- 7.4 Collapsibility.- 7.5 Exercises.- 8. The Logit Model.- 8.1 The logit-model with binary explanatory variables.- 8.2 The logit model with polytomous explanatory variables.- 8.3 Exercises.- 9. Logistic Regression Analysis.- 9.1 The logistic regression model.- 9.2 Regression diagnostics.- 9.3 Predictions.- 9.4 Polytomous response variables.- 9.5 Exercises.- 10. Models for the Interactions.- 10.1 Introduction.- 10.2 Symmetry models.- 10.3 Marginal homogeneity.- 10.4 Models for mobility tables.- 10.5 Association models.- 10.6 RC-association models.- 10.7 Log-linear association models.- 10.8 Exercises.- 11. Correspondence Analysis.- 11.1 Correspondence analysis for two-way tables.- 11.2 Correspondence analysis for multi-way tables.- 11.3 Comparison of models.- 11.4 Exercises.- 12. Latent Structure Analysis.- 12.1 Latent structure models.- 12.2 Latent class models.- 12.3 Continuous latent structure models.- 12.4 The EM-algorithm.- 12.5 Estimation in the latent class model.- 12.6 Estimation in the continuous latent structure model.- 12.7 Testing the goodness of fit.- 12.8 Diagnostics.- 12.9 Score models with varying discriminating powers.- 12.10 Comparison of latent structure models.- 12.11 Estimation of the latent variable.- 12.12 Exercises.- 13. Computer Programs.- References.- Author Index.- Examples with Data.