On the use of stochastic ordering to test for trend with clustered binary data

We introduce the use of stochastic ordering for defining treatment-related trend in clustered exchangeable binary data for both when cluster sizes are fixed and when they vary randomly. In the latter case, there is a well-documented tendency for such data to be sparse, a problem we address by making an assumption of interpretability or, equivalently, marginal compatibility. Our procedures are based on a representation of the joint distribution of binary exchangeable random variables by a saturated model, and may hence be considered nonparametric. The definition of trend by stochastic ordering is proposed to ensure a flexibility that allows for various forms of monotone increases in response to the cluster as a whole to be included in the evaluation of the trend. We obtain maximum likelihood estimates of probability functions under stochastic ordering using mixture-likelihood-based algorithms. Since the data are sparse, we avoid the use of asymptotic results and obtain p-values of the likelihood ratio procedures by permutation resampling. We demonstrate how the proposed framework can be used in risk assessment, and provide comparisons with existing procedures.