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

Abstract

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.

Mixture Models of Missing Data

“…This is the idea underlying the mixture index of fit proposed by Rudas et al, (1994). Let (1 ) p) denote the relative size of the fraction where the model fits and p that of the fraction where the model does not fit.…”

Rater agreement and the generalized Rudas-Clogg-Lindsay index of fit

“…Following Rudas et al [1], for a given model H applied to a contingency table P the twocomponent mixture…”

“…Whereas in the case of very large sample sizes, as argued in Rudas et al (Reference [1], p. 623), the concept of statistical signi"cance may become meaningless or even misleading, in the case of (very) small sample sizes, the sparseness of the data may raise doubts that the asymptotic properties of the conventional chi-squared goodness-of-"t statistics hold. Rudas et al [1] demonstrated the usefulness of their "t index for large samples; that the RCL mixture approach remains applicable to small samples resulting in empty cells will be shown here. As a consequence, the RCL index of "t may be recommended as a supplement to conventional goodness-of-"t tests in the presence of moderate sample sizes, and as their alternative in extreme situations of sample sizes.…”

“…For evaluating goodness of "t in the analysis of contingency tables, Rudas et al [1] presented a new index of "t related to the mixture approach used in latent class analysis. This index of "t does not result in chi-square distributed test statistics as conventional goodness-of-"t tests do, but it gives a directly interpretable quantitative measure of lack of "t: the proportion of cases incompatible with the model H to be tested.…”

“…and H @ . The second 4;4 contingency table with empty cells will be analysed by the same method to study the e!ect of sampling zeros, mentioned as a possible problem by Rudas et al (Reference [1], p. 636). A further analysis of the 4;4 table assuming the two-class latent class model will serve to support the argument that the RCL index of "t is not only applicable to simple standard models.…”