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Statistical methods are introduced for latent structure analysis of a set of two or more multidimensional contingency tables. Three basic classes of models are considered: (a) models that assume complete homogeneity across tables, (b) models that allow partial homogeneity across tables, and (c) models that allow complete heterogeneity. Methods are presented for testing whether these models are congruent with the data in the tables and for assessing the significance of differences among the tables in the estimated parameters. To illustrate the wide applicability of these models and methods, we present analyses of two quite different sets of data.

A framework based on mixture methods is proposed for evaluating goodness of fit in the analysis of contingency tables. For a given model H applied to a contingency table P, we consider the two-point mixture P = (I -1I")H t + 1I"H 2 , with 11" the mixing proportion (0~11"~1) and HI and H 2 the tables of probabilities for each latent class or component. In the unstructured approach recommended here, the mixture model applies H to HI but does not impose any restrictions on H 2 • A contingency table P can generally be represented as such a two-point mixture for an interval of 1I"-values. We define our index of lack of fit, 11"*, to be the smallest such 11", i.e. 11"* is the fraction of the population that cannot be described by model H. This approach can be contrasted with the structured approach that applies model H to both HI and H 2 and leads to conventional latent class models when H is the hypothesis of independence. The case where H is the hypothesis of row-column independence and P is a two-way contingency table is covered in detail, but the procedure is quite general.

Goodman recently presented a class of models for the analysis of association between two discrete, ordinal variables. The association was measured in terms of the odds ratios in 2 x 2 subtables formed from adjacent rows and adjacent columns of the cross-classification, and models were devised that allowed the odds ratios to depend on an overall effect, on row effects, on column effects, and on other effects. This article presents some generalizations of this approach appropriate for multiway crossclassifications, including (a) models for the analysis of conditional association, (b) models for the analysis of partial association, and (c) models for the analysis of symmetric association. Three cross-classifications are analyzed with these models and methods, and rather simple interpretations of the association in each are provided.

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