Tests of Independence, Treatment Heterogeneity Dose-Related Trend with Exchangeable Binary Data

Existing methods of testing for treatment effects for clustered binary data include the beta-binomial, quasilikelihood GEE procedures. All of these methods revolve around the mean response and the second-order correlation. However, these two parameters alone do not fully determine the effect of treatment. This article develops nonparametric likelihood ratio procedures to test for independence, heterogeneity dose-related trend in dose-response studies involving exchangeable binary data. The hypotheses of independence, heterogeneity trend are expressed in terms of joint probabilities of similar responses among cluster mates. Constrained maximum likelihood estimates of these probabilities are computed and used to construct test statistics. Unlike the test statistics for independence and heterogeneity, the asymptotic distribution of the likelihood ratio test for trend is not exactly a chi-square. However, an upper bound of its p value is obtained by using a chi-squared distribution. A set of clustered binary data from the Shell Toxicology Laboratory, on the developmental effect of a chemical agent on banded Dutch rabbits, is used to illustrate the various test procedures. The same dataset is used to compare the proposed trend test with some existing trend tests, such as those based on a beta-binomial model, generalized estimating equations survey sampling methods.