Marginal Correlation from Logit- and Probit-Beta-Normal Models for Hierarchical Binary Data

In hierarchical data settings, be it of a longitudinal, spatial, multi-level, clustered, or otherwise repeated nature, often the association between repeated measurements attracts at least part of the scientific interest. Quantifying the association frequently takes the form of a correlation function, including but not limited to intraclass correlation. Vangeneugden et al. (2010) derived approximate correlation functions for longitudinal sequences of general data type, Gaussian and non-Gaussian, based on generalized linear mixed-effects models. Here, we consider the extended model family proposed by Molenberghs et al. (2010). This family flexibly accommodates data hierarchies, intra-sequence correlation, and overdispersion. The family allows for closed-form means, variance functions, and correlation function, for a variety of outcome types and link functions. Unfortunately, for binary data with logit link, closed forms cannot be obtained. This is in contrast with the probit link, for which such closed forms can be derived. It is therefore that we concentrate on the probit case. It is of interest, not only in its own right, but also as an instrument to approximate the logit case, thanks to the well-known probit-logit ‘conversion.’ Next to the general situation, some important special cases such as exchangeable clustered outcomes receive attention because they produce insightful expressions. The closed-form expressions are contrasted with the generic approximate expressions of Vangeneugden et al. (2010) and with approximations derived for the so-called logistic-beta-normal combined model. A simulation study explores performance of the method proposed. Data from a schizophrenia trial are analyzed and correlation functions derived.