A penalized approach to the bivariate logistic regression model for the association between ordinal responses

摘要

Bivariate ordered logistic models (BOLMs) are appealing to jointly model themarginal distribution of two ordered responses and their association, given aset of covariates. When the number of categories of the responses increases,the number of global odds ratios (or their re-parametrizations) to be estimatedalso increases and estimating the association structure becomes crucial forthis type of data. In fact, such data could be too "rich" to be fully modelledwith an ordinary BOLM while, sometimes, the well-known Dale's model could betoo parsimonious to provide a good fit. In addition, when the cross-tabulationof the responses contains some zeros, for a number of model configurations,including the bivariate version of the partial proportional odds model (PPOM),estimation of a BOLM by the Fisher-scoring algorithm may either fail orestimate a too "irregular" association structure. In this work, we propose touse a nonparametric approach for the maximum likelihood estimation of a BOLM.We apply penalties to the differences between adjacent row and column effects.As a result, estimation is less demanding than an ordinary BOLM, permitting thefit of PPOMs and/or the smoothing of the marginal and association parameters bypolynomial curves and surfaces, with scores chosen by the data. Model selectionis based on the penalized log-likelihood ratio, whose limiting distribution hasbeen studied through simulations, and AIC. Our proposal is compared to theGoodman's model and the Dale's model, in terms of goodness-of-fit andparsimony, on a literature data set. Finally, an application on an originaldata set of liver disease patients is proposed.