Modelling bivariate ordinal responses smoothly with examples from ophthalmology and genetics

A non-parametric implementation of the bivariate Dale model (BDM) is presen ted as an extension of the generalized additive model (GAM) of Hastie and T ibshirani. The original BDM is an example of a bivariate generalized linear model. In this paper smoothing is introduced on the marginal as well as on the association level. Our non-parametric procedure can be used as a diagn ostic tool for identifying parametric transformations of the covariates in the linear BDM, hence it also provides a kind of goodness-of-fit test for a bivariate generalized linear model. Cubic smoothing spline functions for t he covariates are estimated by maximizing a penalized version of the log-li kelihood. The method is applied to two studies. The first study is the clas sical Wisconsin Epidemiologic Study of Diabetic Retinopathy. The second stu dy is a twin study, where the association between the elements of twin pair s is of primary interest. The results show that smoothing on the associatio n level can give a significant improvement to the model fit. Copyright (C) 2001 John Wiley & Sons, Ltd.