Partial Exchangeability for Contingency Tables

The Robustness of a Hierarchical Model for Multinomials and Contingency Tables

Biomtlrika (1991), 78, 1, pp. 229-32 Printed in Great Britain Reduced rank models for contingency tables BY JAN DE LEEUW Departments of Psychology and Mathematics, University of California, Los Angeles, California 90024-1563, U.S.A. AND PETER G. M. VAN DER HEIJDEN Department of Empirical and Theoretical Sociology, University of Utrecht, 3508 TC Utrecht, The Netherlands SUMMARY Some key words: Canonical analysis; Correspondence analysis; Latent class analysis; Reduced rank models. 1. INTRODUCTION In recent years much attention has been given to models for two-way contingency tables that can be formulated in terms of reduced rank of a matrix with probabilities. A well-known reduced rank model is the independence model, where the rank is one. For rank higher than one distinct classes of reduced rank models are possible. Each has the independence model as the special case for rank one. A first class of such models is closely related to what is known under names as canonical analysis or correspondence analysis. Recently much attention has been given to the maximum likelihood estimation of versions of these models by Goodman (1985, 1986, 1987) and Gilula & Haberman (1986, 1988). A second class of models that can be formulated in terms of reduced rank is latent class analysis, LCA, for two-way tables. Latent class analysis was proposed by Lazersfeld (1950a, b). See Clogg (1981) for a more recent review. In this paper we relate these classes of models to each other. The relation has been discussed earlier by Gilula (1979, 1983, 1984), Gilula & Haberman (1986), Goodman (1987), and van der Heijden, Mooijaart & de Leeuw (1989). We summarize existing results in a simple way using new proofs. Gilula (1979) provided conditions that had to hold for rank-2 correspondence analysis to imply rank-2 latent class analysis. We show here that rank-2 correspondence analysis always implies rank-2 latent class analysis. This implies that the theorem and the example given by Gilula (1979) are incorrect. 2. GENERAL REDUCED RANK MODELS The basic model studied in this paper assumes that a n n x m probability matrix II has rank p, where p = min (n,m). We call this model R p . The probability matrix II has all elements nonnegative, while the sum of the ny is equal to one. We suppose, unless indicated otherwise, that II is full, in the sense that its row sums ir l+ and its column sums v +J are all positive. Thus no row or column is equal to zero. We compare this model with the canonical model C p , in which at most p - 1 of the canonical correlations between the row and the column variables of the table are nonzero. These canonical correlations are the stationary values of the product moment correlation coefficient, seen as a function of scores for rows and scores for columns. Downloaded from biomet.oxfordjournals.org at University of California, Los Angeles on May 18, 2011 Reduced rank models for the analysis of two-way contingency tables are introduced. Two classes of reduced rank models are discerned, with well-known exponents canonical analysis and latent class analysis. The relation between these two classes is discussed. Results on the subject mentioned earlier in the literature are shown to be either redundant or inaccurate.

Generalized linear models build a unified framework containing many extensions of a linear model. Important examples include logistic regression for binary responses, Poisson regression for count data or log-linear models for contingency tables. Penalizing the negative log-likelihood with the l1-norm, still called the Lasso, is in many examples conceptually similar to the case with squared error loss in linear regression due to the convexity of the negative log-likelihood. This implies that the statistical properties as well as the computational complexity of algorithms are attractive. A noticeable difference, however, occurs with log-linear models for large contingency tables where the computation is in general much more demanding. We present in this chapter the models and estimators while computational algorithms and theory are described in more details in Chapters 4 and 6, respectively.

Categorical time series models for contingency tables

ABSTRACTThis paper proposes a new model for square contingency tables. The proposed model tests the equality of local odds ratios between the one side of the main diagonal and corresponding other side and it represents the non-symmetric structure of the square contingency table. The proposed model is compared with twenty-five models introduced for analysing the square contingency tables for both symmetric and non-symmetric structures. The results show that the proposed model provides best fit performance than other existing models for square contingency tables.

In the analysis of square contingency tables, which are two-way contingency tables in which the row and column variables consist of the same classification, statistical models regarding the symmetry of row and column variables are often used rather than the independence. This study proposes an index for measuring the degree of departure from the anti-sum-symmetry model. The proposed index is constructed using the Kullback–Leibler divergence. The anti-sum-symmetry model is useful to evaluate whether symmetric and asymmetric structures exist with respect to the anti-diagonal of the table. We derive the plug-in estimator and large-sample confidence interval for the proposed index. The usefulness of the proposed index is demonstrated by applying it to real data.

This article discusses through three examples several new methods to aid in the analysis of large contingency tables. The general goal is to give better understanding of specific contingency tables, both by comparing how various log-linear/logistic models fit and through clearer interpretations of the resulting fits. For model selection, we show how to focus on a subset of simple, good-fitting models, beginning with a plot of a goodness-of-fit statistic versus residual degrees of freedom for all of the fitted models. To assess whether a particular model is adequate, we demonstrate that certain plots of residuals can reveal interesting effects that are often otherwise hidden. For model summarization and interpretation, we plot odds-ratio factors with confidence intervals to show the effects of explanatory variables in a concise and appealing way. The first example involves the relationship of job satisfaction to demographic variables for craft employees of a large corporation. The data presented consist of a five-way contingency table with about 10,000 counts. Job satisfaction for such employees increased with age and was higher in the Southwest and West than in the Northeast. Of four race-by-sex groups, the most satisfied was nonwhite males; the least satisfied was nonwhite females. Another example gives a six-way table with about 1,200 counts concerning whether or not high-school students think they will need mathematics in their future work. Among other results, for students planning to take a job right after graduation, those from suburban schools had odds about 2.6 times those from urban schools of thinking that mathematics will be useful. Moreover, among urban students, males had odds of finding mathematics useful about 2.1 times those for females, but there was little difference between the odds for males and females among suburban students. The third example, drawn from the literature, relates knowledge about cancer to four dichotomous variables. We compare our analysis with earlier ones. Key Words: Graphical methodsLog-linear modelsModel assessmentOdds ratios

For the analysis of square contingency tables with the same row and column ordinal classifications, this article pro- poses a new model, which may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate t-distribution with equal marginal variances having any degrees of freedom. The proposed model indicates that the difference between the symmetric two probabilities raised to the power is proportional to the distance from the main diagonal of the table. The model is an extension of the symmetry model. An example is given. The simulation studies based on bivariate t-distribution are given.

For the analysis of C × C square contingency tables, we usually estimate using a statistical model an unknown probability distribution with high confidence from obtained observations. The statistical model that fits the data well and is easy to interpret is preferred. The anti-sum-symmetry (ASS) and anti-conditional sum-symmetry (ACSS) models have a structure that the ratio of the probability with which the sum of row and column levels is t, for t = 2, . . . , C, and the probability with which the sum of row and column levels is 2(C + 1) − t is always one and constant, respectively. This study proposes two kinds of models that the ratio of those changes exponentially depending on the sum of row and column levels. This study also gives the decomposition theorems of the ASS model using the proposed models. Moreover, we show that the value of the likelihood ratio chi-squared statistics for the ASS model is asymptotically equivalent to the sum of those for the decomposed models. We evaluate the advantage of the proposed models by applying they to a single data set of real-world grip strength data.

For the analysis of square contingency tables with ordered categories, this paper introduces the quasi­diagonals­parameter symmetry (QDPS) model which is an extension of the quasi­symmetry (QS) model. It is shown that the QDPS model is preferable to the QS model for unaided vision data of British women earlier analysed by Stuart (1955). Various extended models of QS are expressed in terms of the local odds ratios and the other odds ratios.

For square contingency tables with ordered categories, this paper decomposes the well-known marginal homogeneity (MH) model (Stuart 1955) into an extended MH model and the model of equality of row and column marginal means. When the MH model fits the data poorly, this decomposition may be useful for knowing the reason why the MH model fits the data poorly. Such examples are given.

This is a review paper, beginning with de Finetti's work on partial exchangeability, continuing with his approach to approximate exchangeability, and then his (surprising) approach to assigning informative priors in nonstandard situations. Recent progress on Markov chain Monte Carlo methods for drawing conclusions is supplemented by a review of work by Gerencsér and Ottolini on getting honest bounds for rates of convergence. The paper concludes with a speculative approach to combining classical asymptotics with Monte Carlo. This promises real speed‐ups and makes a nice example of how theory and computation can interact.

The -divergence association models for two-way contingency tables is a family of models that includes the association and correlation models as special cases. We present this family of models, discussing its features and demonstrating the role of -divergence in building this family. The most parsimonious member of this family, the model of -scaled uniform local association, is considered in detail. It is implemented and representative examples are commented on.

Bayesian Methods and Simulation-Based Computation for Contingency Tables

One of the most important tasks of the application of mathematical‐statistical methods consists in giving help in the search for possible relationships, and connected with this, the specification of new hypotheses.The progress of both the special diciplines of sciences and mathematical statistics itself leads to the application of more and more complex, that means multivariate, methods. In medical fields, especially in epidemiological and medicin‐sociological studies, this fact means the necessity of analysing multidimensional contingency tables.The above formulated problem is equivalent to the problem of fitting an appropriate mathematical model (for contingency tables is this a log‐linear model) to the data in a way which makes the structural relationships clear to us.In this paper it is shown that one is able to get to well‐interpretable models of independence with relatively simple means. Two stepwise test procedures are described yielding essentially the same results: a so called reduction procedure which is particularly profitable in sparsely occupied tables and a procedure which uses a combination of hypotheses of conditional pairwise independence.