Applied categorical data analysis

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Goodness Of Fit Statistics For Discrete Multivariate Data

Summary

This article surveys various strategies for modeling ordered categorical (ordinal) response variables when the data have some type of clustering, extending a similar survey for binary data by Pendergast, Gange, Newton, Lindstrom, Palta & Fisher (1996). An important special case is when repeated measurement occurs at various occasions for each subject, such as in longitudinal studies. A much greater variety of models and fitting methods are available than when a similar survey for repeated ordinal response data was prepared a decade ago (Agresti, 1989). The primary emphasis of the review is on two classes of models, marginal models for which effects are averaged over all clusters at particular levels of predictors, and cluster-specific models for which effects apply at the cluster level. We present the two types of models in the ordinal context, review the literature for each, and discuss connections between them. Then, we summarize some alternative modeling approaches and ways of estimating parameters, including a Bayesian approach. We also discuss applications and areas likely to be popular for future research, such as ways of handling missing data and ways of modeling agreement and evaluating the accuracy of diagnostic tests. Finally, we review the current availability of software for using the methods discussed in this article.

Resume

Cet article passe en revue diverses strategies pour la modelisation de variables de reponse qualitatives et ordinales, quand les donees presentent des phenomenes de grappes. II prolonge une etude similaire realisee pour les variables dichtomiques par Pendergast et al (1996). Les mesures repetees a plusieurs reprises pour chaque individu, par exemple dans les etudes longitudinales, constituent un cas particulier important. II existe a present une variete beaucoup plus grande de modeles et de methodes d'ajustement, que lors de la relisation d'une etude similarire pour les variables ordinales et repetees il ya une dizaine d'annees (Agresti, 1989). II est mis particulierement l'accent sur deux Deux types de modeles, les modeles, marginaux pour lesquels on prend la moyenne des effets sur toutes les grappes pour des niveaux donnes des variables explicatives, et les modeles, par grappes pour lesquels les effets sont pris en compte au niveau de chaque grappe.Nous presentons les deux types de modeles, dans le cas de variables ordinales, brosons un panorama de la litterature pour chacun des deux types, et discutons les relations entre eux. Puis nous reurmons d'autres approches pour la modelisation et l'estimation de parametres, notamment une approche bayesienne, Nous discutons aussi de applications et des domaines qui devraien susciter de nouvelles rechlles recherches, par exemple les methodes de trailtement des donnees manquantes ou d'eualuation de l'exactitude de tests. Enfin, nous considerons la disponibilite actuelle de logiciesl pour mettre en oeuvre les methodes discutees dans cet article.

Modeling clustered ordered categorical data: A survey

The Analysis of Cross-Tabulated Data.

Three kinds of decompositions for the conditional symmetry model in a square contingency table

Pour l'analyse des tableaux carres, Caussinus (1965) a propose le modele de quasi-symetrie et montre qu'un tableau est symetrique si et seulement s'il satisfait a la fois quasi-symetrie et egalite des distributions marginales. Bishop, Fienberg et Holland (1975, p. 307) ont note qu'un theoreme semblable valait pour les tableaux a trois dimensions, tandis que Bhapkar et Darroch l'ont donne pour des tableaux de dimension quelconque. Le but de cet article est (1) de passer en revue les questions de symetrie, les modeles eux-memes, leur decomposition et les mesures d'ecart au modele pour divers concepts de symetrie et asymetrie, (2) de montrer que, pour les tableaux multiples, la statistique du rapport de vraisemblance pour tester la symetrie est asymptotiquement equivalente a la somme des statistiques analogues testant respectivement la quasi-symetrie d'un certain ordre et l'egalite des marges pour l'ordre correspondant.

The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables

For the analysis of square contingency tables with ordered categories, Goodman (1979, Biometrika 66, 413-418) considered the diagonals-parameter symmetry (DPS) model, which has a multiplicative form for cell probabilities. This paper proposes another DPS model which has a similar multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (>i) or above. Special cases of the proposed model include conditional symmetry and symmetry. The relationship with a stochastic ordering for marginal distributions is also described. Moreover, the relationships between the proposed model and the (generalized) palindromnic symmetry models are described. An example is given.

Diagonals-parameter symmetry model for cumulative probabilities in square contingency tables with ordered categories

For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

For the analysis of square contingency tables with nominal categories, Tomizawa and coworkers have considered measures that represent the degree of departure from symmetry. This paper proposes a measure that represents the degree of asymmetry for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed using the Cressie–Read power-divergence or Patil–Taillie diversity index, defined for the cumulative probabilities that an observation falls in row (column) category i or below and column (row) category j(> i) or above. The measure depends on the order of listing the categories. It should be useful for comparing the degree of asymmetry in several tables with ordered categories. The relationship between the measure and the normal distribution is shown.

Sadao Tomizawa

Papers

Three kinds of decompositions for the conditional symmetry model in a square contingency table

The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables

Diagonals-parameter symmetry model for cumulative probabilities in square contingency tables with ordered categories

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

Theory & Methods: Measure of Asymmetry for Square Contingency Tables Having Ordered Categories