Categorical Data

Abstract

We discuss nonparametric Bayesian methods that are suitable for inference with binary, ordinal and general categorical data. Modeling for such data becomes particularly interesting in the presence of covariates, when non-and semiparametric Bayesian models can generalize the link function in a generalized linear model setup, the regression on covariates or both. An important application arises in inference for diagnostic screening and related inference for ROC (receiver-operator characteristic) curves. We include some discussion of a rapidly growing literature on non-parametric Bayesian inference for ROC curves. Categorical Responses Without Covariates Binomial ResponsesWe start with an example to illustrate a number of key issues of a BNP approach for binary outcomes.Example 9 (Baseball Data) Albright (1993) describes a dataset involving the complete sequence of hits and outs for a number of players from both American and National Baseball Leagues over the [1987][1988][1989][1990] seasons. The data are available from: http://www.kelley.iu.edu/albright/Free_downloads.htm. Albright assumes the operational definition of a success to mean a player moving through the bases. We stick to that definition, and therefore, a success consists of either a hit, walk or sacrifice. From this large dataset, we consider now the total number of successes for the subset of n D 129 players from both leagues who were at bat at least on 500 occasions during the 1987 season. Denote by y i , i D 1; : : : ; n, the number of successes for the ith player. The simplest possible model for these data would assume just a single success probability, common to all players, that is, y 1 ; : : : ; y n j  ind Bin.`i; Â/;  Be.a; b/, where`i is the total number of at-bats for player i and .a; b/ are fixed hyperparameters, e.g.,