We survey effect measures for models for ordinal categorical data that can be simpler to interpret than the model parameters. For describing the effect of an explanatory variable while adjusting for other explanatory variables, we present probability-based measures, including a measure of relative size and partial effect measures based on instantaneous rates of change. We also discuss summary measures of predictive power that are analogs of R-squared and multiple correlation for quantitative response variables. We illustrate the measures for an example and provide R code for implementing them. KEYWORDScumulative link models, cumulative logits, marginal effects, multiple correlation, proportional odds, R-squared, stochastic ordering | INTRODUCTIONPopular models for ordinal categorical response variables, such as models that apply link functions to cumulative probabilities, are generalized linear models that employ nonlinear link functions. As a consequence of the nonlinearity, model parameters are not as simple to interpret as slopes and correlations for ordinary linear regression. The model effect parameters relate to measures, such as odds ratios and probits, that may not be easily understood or can even be misinterpreted by non-quantitatively oriented methodologists, see, for example, Schwartz et al. (1999).This article surveys simpler ways to interpret the effects of an explanatory variable and to summarize the model's predictive power. In Section 2, we present alternative summaries of the effect of an explanatory variable while adjusting for other explanatory variables in the model. These include simple comparisons of the probability of extreme-response outcomes at extreme ---

We deal with contingency table data that are used to examine the relationships between a set of categorical variables or factors. We assume that such relationships can be adequately described by the cond`itional independence structure that is imposed by an undirected graphical model. If the contingency table is large, a desirable simplified interpretation can be achieved by combining some categories, or levels, of the factors. We introduce conditions under which such an operation does not alter the Markov properties of the graph. Implementation of these conditions leads to Bayesian model uncertainty procedures based on reversible jump Markov chain Monte Carlo methods. The methodology is illustrated on a 2�3�4 and up to a 4�5�5�2�2 contingency table. Copyright 2005 Royal Statistical Society.

In this paper we compare two alternative MCMC samplers for the Bayesian analysis of discrete graphical models; we present both a hierarchical and a nonhierarchical version of them. We rst consider the MC 3 algorithm by Madigan and York (1995) for which we propose an extension that allows for a hierarchical prior on the cell counts. We then describe a novel methodology based on a reversible jump sampler. As a prior distribution we assign, for each given graph, a hyper-Dirichlet distribution on the matrix of cell probabilities. Two applications to real data are presented.

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractWe propose a conjugate and conditional conjugate Bayesian analysis of models of marginal independence with a bi-directed graph representation. We work with Markov equivalent directed acyclic graphs (DAGs) obtained using the same vertex set with the addition of some latent vertices when required. The DAG equivalent model is characterised by a minimal set of marginal and conditional probability parameters. This allows us to use compatible prior distributions based on products of Dirichlet distributions. For models with DAG representation on the same vertex set, the posterior distribution and the marginal likelihood is analytically available, while for the remaining ones a data augmentation scheme introducing additional latent variables is required. For the latter, we estimate the marginal likelihood using Chib's (1995) estimator. Additional implementation details including identifiability of such models is discussed. For all models, we also provide methodology for the computation of the posterior distributions of the marginal log-linear parameters based on a simple transformation of the simulated values of the probability parameters. We illustrate our method using a popular 4-way dataset.

Summary This paper extends the ordinary quasi‐symmetry (QS) model for square contingency tables with commensurable classification variables. The proposed generalised QS model is defined in terms of odds ratios that apply to ordinal variables. In particular, we present QS models based on global, cumulative and continuation odds ratios and discuss their properties. Finally, the conditional generalised QS model is introduced for local and global odds ratios. These models are illustrated through the analysis of two data sets.