Modelling and analysing exchangeable binary data with random cluster sizes.

Correlated binary data occur very frequently in cluster sample surveys, dependent repeated cancer screening, teratological experiments, ophthalmologic and otolaryngologic studies, and other clinical trials. The standard methods to analyse these data include the use of beta-binomial models and generalized estimating equations with third and fourth moments specified by 'working matrices'. However, in many applications it is reasonable to assume that the data from the same cluster are exchangeable. When all sampled clusters have equal sizes, Bowman and George introduced maximum likelihood estimates (MLEs) of the population parameters such as the marginal means, moments, and correlations of order two and higher. They also extended their approach to sampled clusters with unequal sizes. It seems that their extension has a gap. This paper points out the source of this gap and shows that estimates introduced by Bowman and George are not the MLEs of the parameters which are used to identify the joint distribution of correlated binary data. We show that the MLEs of the population parameters have no closed form in general and should be calculated by numerical methods. We apply our results and a generalized estimating equation procedure to a data set from a double-blind randomized clinical trial comparing two antibiotics, cefaclor and amoxicillin, used for the treatment of acute otitis media. To see the performance of the MLEs with small or moderate sample sizes, several simulation studies are also conducted.