A New Index of Fit Based on Mixture Methods for the Analysis of Contingency Tables

SUMMARY 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 = (1 – π)π 1 + ππ 2, with π the mixing proportion (0 ≤ π ≤ 1) and π 1 and π 2 the tables of probabilities for each latent class or component. In the unstructured approach recommended here, the mixture model applies H to π 1 but does not impose any restrictions on π 2. A contingency table P can generally be represented as such a two-point mixture for an interval of π-values. We define our index of lack of fit, π*, to be the smallest such π, i.e. π* 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 π 1 and π 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.