The analysis of stratified 2 x 2 contingency tables

We consider the problem of testing for independence against the consistent superiority of one treatment over another when the response variable is binary and is compared across two treatments in each of several strata. Specifically, we consider the randomized clinical trial setting. A number of issues arise in this context. First, should tables be combined if there are small or zero margins? Second, should one assume a common odds ratio across strata? Third, if the odds ratios differ across strata, then how does the standard test (based on a common odds ratio) perform? Fourth, are there other analyzes that are more appropriate for handling a situation in which the odds ratios may differ across strata? In addressing these issues we find that the frequently used Cochran–Mantel–Haenszel test may have a poor power profile, despite being optimal when the odds ratios are common. We develop novel tests that are analogous to the Smirnov, modified Smirnov, convex hull, and adaptive tests that have been proposed for ordered categorical data. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exploring contingency tables with correspondence analysis

An algorithm for correspondence analysis is described and implementedin SAS/IML (SAS Institute, 1985a). The technique is shown, throughthe analysis of several biological examples, to supplement thelog-linear models approach to the analysis of contingency tables,both in the model identification and model interpretation stagesof analysis. A simple two-way contingency table of tumor datais analyzed using correspondence analysis. This example emphasisesthe relationships between the parameters of the log-linear modelfor the table and the graphical correspondence analysis results.The technqiue is also applied to a three-way table of surveydata concerning ulcer patients to demonstrate applications ofsimple correspondence analysis to higher dimensional tableswith fixed margins. Finally, the diets and foraging behaviorsof birds of the Hubbard Brook Forest are each analyzed and thena simultaneous display of the two separate but related tablesis constructed to highlight relationships between the tables. Received on August 29, 1988; accepted on April 25, 1989     

Incorporating prior information into the analysis of contingency tables

Contingency tables are often analyzed using log-linear models and in some situations prior information on the value of parameters in the log-linear model is available. In this article we describe a prior-posterior procedure that incorporates prior information directly into the analysis through a multivariate normal prior for the log-linear parameters. The mode and curvature of the posterior density are proposed as summary statistics.     

Quasi-independent triangular contingency tables.

The model of quasi-independence in a triangular contingency table is parametrised in a way which is both easy to interpret probabilistically, and easy to analyse.     

Bayesian analysis of matrix normal graphical models

We present Bayesian analyses of matrix-variate normal data with conditional independencies induced by graphical model structuring of the characterizing covariance matrix parameters. This framework of matrix normal graphical models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of graphical model uncertainty and model structure search. Extensions to matrix-variate time series embed matrix normal graphs in dynamic models. Examples highlight questions of graphical model uncertainty, search and comparison in matrix data contexts. These models may be applied in a number of areas of multivariate analysis, time series and also spatial modelling.     

Combining contingency tables with missing dimensions

We propose a methodology for estimating the cell probabilities in a multiway contingency table by combining partial information from a number of studies when not all of the variables are recorded in all studies. We jointly model the full set of categorical variables recorded in at least one of the studies, and we treat the variables that are not reported as missing dimensions of the study-specific contingency table. For example, we might be interested in combining several cohort studies in which the incidence in the exposed and nonexposed groups is not reported for all risk factors in all studies while the overall numbers of cases and cohort size is always available. To account for study-to-study variability, we adopt a Bayesian hierarchical model. At the first stage of the model, the observation stage, data are modeled by a multinomial distribution with fixed total number of observations. At the second stage, we use the logistic normal (LN) distribution to model variability in the study-specific cells' probabilities. Using this model and data augmentation techniques, we reconstruct the contingency table for each study regardless of which dimensions are missing, and we estimate population parameters of interest. Our hierarchical procedure borrows strength from all the studies and accounts for correlations among the cells' probabilities. The main difficulty in combining studies recording different variables is in maintaining a consistent interpretation of parameters across studies. The approach proposed here overcomes this difficulty and at the same time addresses the uncertainty arising from the missing dimensions. We apply our modeling strategy to analyze data on air pollution and mortality from 1987 to 1994 for six U.S. cities by combining six cross-classifications of low, medium, and high levels of mortality counts, particulate matter, ozone, and carbon monoxide with the complication that four of the six cities do not report all the air pollution variables. Our goals are to investigate the association between air pollution and mortality by reconstructing the tables with missing dimensions, to determine the most harmful pollutant combinations, and to make predictions about these key issues for a city other than the six sampled. We find that, for high levels of ozone and carbon monoxide, the number of cases with a high number of deaths increases as the levels of particulate matter, PM10, increases and that the most harmful combinations corresponds to high levels of PM10, confirming prior findings that levels of PM10 higher than the NAAQS standard are harmful.     

2 k contingency tables in ecology

Suppose observations are made on the presence or absence of k different species in N sampling units. Denote by s the number of species per sampling unit and let m₂ and Var(s) respectively be the observed variance of s and its expectation under the null hypothesis that all k species are independent. It is shown that the difference m₂—Var(s) is directly interpretable as a measure of the overall association among the species. Examples using field data are given and it is shown how the proposed measure of association may be used to judge: (i) whether some chosen group of species contributes markedly to the total amount of interdependence within a community; and (ii) whether some chosen group of species may be disregarded without affecting the result when ecological data are to be classified or ordinated.