Improved likelihood ratio tests for complete contingency tables

SUMMARY Lawley (1956) describes how asymptotic likelihood ratio tests can in general be improved by multiplying the -2 log A test statistic by a multiplier chosen so that the null distribution of the modified statistic is better approximated by its asymptotic x2 distribution. This paper applies this technique to asymptotic likelihood ratio tests of hypotheses concerning complete contingency tables. Improved tests are derived for hypotheses with closed form maximum likelihood estimators. An asymptotic likelihood ratio test of a composite null hypothesis against a composite alternative can in general be obtained by comparing -2 log A with a x2 distribution, where A is the ratio of maximized likelihoods under the two hypotheses. Lawley (1956) showed that such tests can be improved by multiplying -2 log A by a scale factor chosen so that the null distribution of the resulting statistic has the same moments as X2 ignoring quantities of order n-2, where n is the size of the sample. When the statistic - 2 log A can be expressed as an explicit function of the observations the multiplier is most easily found by calculating directly its expectation as far as terms of order n-1. These improved likelihood ratio tests have been most used in the area of multivariate analysis. In recent years considerable progress has been made in developing methods for analyzing multidimensional contingency tables using log linear models. A comprehensive review is given by Plackett (1 974). In these models the cell frequencies may be regarded as independent Poisson variables whose log expectations are linear in a hierarchical set of main effect and interaction parameters. The -2 log A statistic for testing the goodness of fit of a given model takes the form S = 2EX log (X/u), where the X are the cell frequencies and the ,u are the maximum likelihood estimators of their expectations. In practice X log (X/,u) can be replaced by zero when X = 0. Following Nelder & Wedderburn (1972), S will be termed the deviance of the model. To test a given model with deviance S1 against an alternative with additional estimated parameters and deviance S2, the likelihood ratio test statistic iS S1-S2