Exploratory analysis of contingency tables by loglinear formulation and generalizations of correspondence analysis

Goodman's (1979, 1981, 1985) loglinear formulation for bi-way contingency tables is extended to tables with or without missing cells and is used for exploratory purposes. A similar formulation is done for three-way tables and generalizations of correspondence analysis are deduced. A generalized version of Goodman's algorithm, based on Newton's elementary unidimensional method is used to estimate the scores in all cases.This research was partially supported by National Science and Engineering Research Council of Canada, Grant No. A8724. The author is grateful to the reviewers and the editor for helpful comments.     

Analysis of contingency tables by ideal point discriminant analysis

Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.Presented as the Presidential Address to the Psychometric Society's Annual and European Meetings, June, 1987. Preparation of this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Chikio Hayashi of University of the Air in Japan for providing the ISM data, and to Jim Ramsay and Ivo Molenaar for their helpful comments on an earlier draft of this paper.     

Tests of symmetry in three-way contingency tables

Three-dimensional contingency tables are analyzed, with one variable (e.g., sex) as a factor, and with a natural relation between the other variables (e.g. left and right eye vision). Models of special interest, like symmetry and proportional symmetry between the related variables, and homogeneity across the factor levels, are investigated. Maximum likelihood estimators of parameters and partitions of chi-square goodness-of-fit statistics are explicitly presented; the independence of certain models is noted, and an example is discussed.     

The analysis of group contingency data

The data analyses utilized in group contingency projects are reviewed. Previous studies are cited to emphasize advantages of nonconsolidated ("individual") over consolidated analyses. Several procedures are described that enable applied researchers to incorporate nonconsolidated data analyses in group contingency studies.     

Correspondence analysis used complementary to loglinear analysis

Loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables. In this paper we will show that there are cases in which these two techniques can be used complementary to each other. More specifically, we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices, each following a specific loglinear model. Therefore, in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models. A generalization of correspondence analysis, recently proposed by Escofier, will also be discussed. With this decomposition, which includes classical correspondence analysis as a special case, it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis. In this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models.     

Exact procedures for the analysis of multidimensional contingency tables

The log-linear model for contingency tables expresses the logarithm of a cell frequency as an additive function of main effects, interactions, etc., in a way formally identical with an analysis of variance model. Exact statistical tests are developed to test hypotheses that specific effects or sets of effects are zero, yielding procedures for exploring relationships among qualitative variables which are suitable for small samples. The tests are analogous to Fisher's exact test for a 2 × 2 contingency table. Given a hypothesis, the exact probability of the obtained table is determined, conditional on fixed marginals or other functions of the cell frequencies. The sum of the probabilities of the obtained table and of all less probable ones is the exact probability to be considered in testing the null hypothesis. Procedures for obtaining exact probabilities are explained in detail, with examples given.     

The analysis of three-way contingency tables by three-mode association models

TheRC(M) association model (Goodman, 1979, 1985, 1986, 1991) is useful for analyzing the relationship between the variables of a 2-way cross-classification. The models presented here are generalizations of theRC(M) association model for 3-way tables. The family of models proposed here, 3-mode association models, use Tucker's 3-mode components model (Tucker, 1964, 1966; Kroonenberg, 1983) to represent either the three factor interaction or the combined effects of two and three factor interactions. An example from a study in developmental psychology (Kramer & Gottman, 1992) is provided to illustrate the usefulness of the proposed models.I thank Stanley Wasserman, Laurie Kramer, Ulf Böckenholt, Larwence Hubert, Jeffrey Tanaka, and five anonymous reviewers for valuable comments.     

Strategy changes in human contingency judgments as a function of contingency tables.

Judgment strategies of 169 undergraduate students on problems to judge the contingency between two binary events were identified by the method of rule-based analysis to clarify whether or not the strategies the subjects used would be affected by the concrete nature of the contingency table. Problems were constructed along two factors: total cell frequency and width of range of objective contingencies. Although the factor of total cell frequency had no effect on subjects' strategies, the number of subjects who changed strategies corresponding with problem instances increased when the objective contingencies were set closer to zero or when problems became more difficult. These results are discussed in the context of previous studies of this issue in the literature.     

Evaluation of a trial-based interview-informed synthesized contingency analysis

We evaluated the viability of an interview-informed synthesized-contingency analysis (IISCA) conducted in a trial-based format with 3 children with autism spectrum disorders who engaged in problem behavior. We compared results to those from typical trial-based and traditional functional analyses and found high degrees of correspondence. The trial-based IISCA format took the least amount of time to conduct and was associated with the lowest frequencies of problem behavior. Results are discussed in terms of merits of each of the 3 types of functional analysis arrangements and directions for future research.     

An incomplete data approach to the analysis of covariance structures

In this paper, linear structural equation models with latent variables are considered. It is shown how many common models arise from incomplete observation of a relatively simple system. Subclasses of models with conditional independence interpretations are also discussed. Using an incomplete data point of view, the relationships between the incomplete and complete data likelihoods, assuming normality, are highlighted. For computing maximum likelihood estimates, the EM algorithm and alternatives are surveyed. For the alternative algorithms, simplified expressions for computing function values and derivatives are given. Likelihood ratio tests based on complete and incomplete data are related, and an example on using their relationship to improve the fit of a model is given.This research forms part of the author's doctoral thesis and was supported by a Commonwealth Postgraduate Research Award. The author also wishes to acknowledge the support of CSIRO during the preparation of this paper and the referees' comments which led to substantial improvements.     

Multivariate normal integrals and contingency tables with ordered categories

Ak-dimensional multivariate normal distribution is made discrete by partitioning thek-dimensional Euclidean space with rectangular grids. The collection of probability integrals over the partitioned cubes is ak-dimensional contingency table with ordered categories. It is shown that loglinear model with main effects plus two-way interactions provides an accurate approximation for thek-dimensional table. The complete multivariate normal integral table is computed via the iterative proportional fitting algorithm from bivariate normal integral tables. This approach imposes no restriction on the correlation matrix. Comparisons with other numerical integration algorithms are reported. The approximation suggests association models for discretized multivariate normal distributions and contingency tables with ordered categories.The contingency-table approach occurred to me while I was collaborating with Paul Holland of the Educational Testing Service in 1985 on bivariate dependence functions. Holland maintains a belief that the continuous can learn from the discrete. This work is a reassertion of his claim.This research was sponsored by the National Science Council, Republic of China.     

A measure of effect size for r x c contingency tables

Goodman and Kruskal's tau measure of categorical association is advanced as a replacement for conventional measures of effect size for r x c contingency tables. Goodman and Kruskal's tau is an asymmetric measure of categorical association which is based entirely on the observed data and possesses a clear interpretation in terms of proportional reduction in error. Comparisons with conventional measures of effect size based on chi-squared such as Pearson's phi2, Tschuprow's T2, and Cramer's V2 demonstrate the advantages of employing tau as a measure of effect size.     

A modification of kendall's tau for measuring association in contingency tables

A coefficient of association is described for a contingency table containing data classified into two sets of ordered categories. Within each of the two sets the number of categories or the number of cases in each category need not be the same.=+1 for perfect positive association and has an expectation of 0 for chance association. In many cases also has –1 as a lower limit. The limitations of Kendall's _a and _b and Stuart's _c are discussed, as is the identity of these coefficients to' under certain conditions. Computational procedure for is given.