Multivariate Methods for Clustered Ordinal Data With Applications to Survival Analysis

Clustered data are the rule in many clinical specialties such as ophthalmology. Methods have been developed for the treatment of clustered continuous or binary outcome data. Less attention has been given to ordinal outcomes which occur frequently in ophthalmology. For example, grading systems of cataract and diabetic retinopathy are commonly used where a photograph is graded by comparison with a series of reference photographs of increasing severity. Some commonly used methods for the analysis of ordered categorical data include the proportional odds and continuation ratio models. It is difficult, however, to incorporate clustering effects into these models. Instead, for clusters of size two, we propose a generalization of the adjacent category model given by logPr(i + 1,j)/Pr(i,j) = ui + (j - 1) lambda + beta' x, where Pr(i,j) denotes the probability that the right (left) eye has grade i(j), x is a vector of (person or eye-specific) covariates for the right eye, u and beta are vectors of location and covariate parameters and lambda is a clustering parameter. Based on this model, and a similar model interchanging the role of i and j, we derived a closed-form expression for Pr(i,j) as a function of u, lambda and beta and use Newton-Raphson method to maximize the likelihood. An extension of the method allows for extra agreement along the diagonal and is then a generalization of the agreement plus linear-by-linear association model proposed by Agresti in the setting of no covariates. We apply these methods to a data set of 43 diabetic subjects from the Harvard Clinical Cataract Research Center, where cortical cataract grade was the outcome. We also extend this methodology to a survival setting, where both censored and uncensored outcomes are available for individual cluster members, and one wishes to take clustering into account. We apply the survival analysis model to a data set of 1807 children (two ears per child) in the greater Boston area, who were followed for the development of otitis media over the first year of life.