Categorical Data Analysis by Example

GRAHAM J. G. UPTON is formerly Professor of Applied Statistics, Department of Mathematical Sciences, University of Essex. Dr. Upton is author of The Analysis of Cross-tabulated Data (1978) and joint author of Spatial Data Analysis by Example (2 volumes, 1995), both published by Wiley. He is the lead author of The Oxford Dictionary of Statistics (OUP, 2014). His books have been translated into Japanese, Russian, and Welsh.

"Concise introduction to dealing with categorical data (with supporting R code) which will help the general data scientist." (Raspberry Pi March 2017)

• Preface xiAcknowledgments xiii1 Introduction 11.1 What are categorical data? 11.2 A typical data set 21.3 Visualisation and crosstabulation 31.4 Samples, populations, and random variation 41.5 Proportion, probability and conditional probability 51.6 Probability distributions 61.6.1 The binomial distribution 61.6.2 The multinomial distribution 71.6.3 The Poisson distribution 71.6.4 The normal distribution 71.6.5 The chisquared (X2) distribution 81.7 *The likelihood 92 Estimation and inference for categorical data 112.1 Goodness of fit 112.1.1 Pearson’s X2 goodness-of-fit statistic 112.1.2 * The link between X2 and the Poisson and ­I2 distributions 122.1.3 The likelihood-ratio goodness-of-fit statistic, G2 132.1.4 * Why the G2 and X2 statistics usually have similar values 142.2 Hypothesis tests for a binomial proportion (large sample) 142.2.1 The normal score test 142.2.2 * Link to Pearson’s X2 goodness-of-fit test 152.2.3 G2 for a binomial proportion 152.3 Hypothesis tests for a binomial proportion (small sample) 162.3.1 One-tailed hypothesis test 162.3.2 Two-tailed hypothesis tests 172.4 Interval estimates for a binomial proportion 182.4.1 Laplace’s method 182.4.2 Wilson’s method 182.4.3 The Agresti-Coull method 192.4.4 Small samples and exact calculations 193 The 2 X 2 contingency table 233.1 Introduction 233.2 Fisher’s exact test (for independence) 243.2.1 * Derivation of the exact test formula 263.3 Testing independence with large cell frequencies 273.3.1 Using Pearson’s goodness-of-fit test 273.3.2 The Yates correction 283.4 The 2 X 2 table in a medical context 293.5 Measuring lack of independence (comparing proportions) 313.5.1 Difference of proportions 313.5.2 Relative risk 323.5.3 Odds-ratio 334 The I x J contingency table 374.1 Notation 374.2 Independence in the I X J contingency table 384.2.1 Estimation and degrees of freedom 384.2.2 Odds-ratios and independence 394.2.3 Goodness-of-fit and lack of fit of the independence model 394.3 Partitioning 424.3.1 * Additivity of G2 424.3.2 Rules for partitioning 444.4 Graphical displays 444.4.1 Mosaic plots 454.4.2 Cobweb diagrams 454.5 Testing independence with ordinal variables 465 The exponential family 515.1 Introduction 515.2 The exponential family 525.2.1 The exponential dispersion family 535.3 Components of a general linear model 535.4 Estimation 546 A model taxonomy 576.1 Underlying questions 576.1.1 Which variables are of interest? 576.1.2 What categories should be used? 586.1.3 What is the type of each variable? 586.1.4 What is the nature of each variable? 586.2 Identifying the type of model 587 The 2 X J contingency table 617.1 A problem with X2 (and G2) 617.2 Using the logit 627.2.1 Estimation of the logit 637.2.2 The null model 647.3 Individual data and grouped data 647.4 Precision, confidence intervals, and prediction intervals 697.4.1 Prediction intervals 707.5 Logistic regression with a categorical explanatory variable 707.5.1 Parameter estimates with categorical variables (J > 2) 737.5.2 The dummy variable representation of a categorical variable 748 Logistic regression with several explanatory variables 778.1 Degrees of freedom when there are no interactions 778.2 Getting a feel for the data 798.3 Models with two variable interactions 818.3.1 Link to the testing of independence between two variables 839 Model selection and diagnostics 859.1 Introduction 859.1.1 Ockham’s razor 869.2 Notation for interactions and for models 879.3 Stepwise methods for model selection using G2 899.3.1 Forward selection 899.3.2 Backward elimination 919.3.3 Complete stepwise 939.4 AIC and related measures 939.5 The problem caused by rare combinations of events 959.5.1 Tackling the problem 969.6 Simplicity versus accuracy 989.7 DFBETAS 10010 Multinomial logistic regression 10310.1 A single continuous explanatory variable 10310.2 Nominal categorical explanatory variables 10610.3 Models for an ordinal response variable 10810.3.1 Cumulative logits 10810.3.2 Proportional odds models 10910.3.3 Adjacent-category logit models 11410.3.4 Continuation-ratio logit models 11511 Log-linear models for I X J tables 11911.1 The saturated model 11911.1.1 Cornered constraints 12011.1.2 Centered constraints 12211.2 The independence model for an I X J table 12512 Log-linear models for I X J X K tables 12912.1 Mutual independence: A=B=C 13112.2 The model AB=C 13112.3 Conditional independence and independence 13312.4 The model AB=AC 13412.5 The models AB=AC=BC and ABC 13512.6 Simpson’s paradox 13512.7 Connection between log-linear models and logistic regression 13713 Implications and uses of Birch’s result 14113.1 Birch’s result 14113.2 Iterative scaling 14213.3 The hierarchy constraint 14313.4 Inclusion of the all-factor interaction 14413.5 Mostellerising 14514 Model selection for log-linear models 14914.1 Three variables 15014.2 More than three variables 15315 Incomplete tables, dummy variables, and outliers 15715.1 Incomplete tables 15715.1.1 Degrees of freedom 15815.2 Quasi-independence 15915.3 Dummy variables 15915.4 Detection of outliers 16016 Panel data and repeated measures 16516.1 The mover-stayer model 16616.2 The loyalty model 16816.3 Symmetry 16916.4 Quasi-symmetry 17016.5 The loyalty-distance model 172A R code for Cobweb function 175Index 179Author Index 183Index of Examples 185