On fitting generalized non-linear models with varying coefficients

This research involves varying-coefficients in an ecological regression model within a likelihood framework using the exponential family of distributions. Ecological regression is the term used when aggregate data are available, but inference for subgroups or individuals is desired. The ecological regression model considered here is non-linear and is fit to renal failure data for Texas to provide an estimate of disease prevalence by ethnicity and economic status using information available at the county level, not the subject level. An algorithm is proposed for fitting the varying-coefficients in such a non-linear regression model when the parameters are simultaneously unknown, but linear when the parameters are considered one-at-a-time. The approach is one of backfitting the estimates of the least favorable subproblems that arise in the context of profile likelihoods when the parameters are considered one-at-a-time. Backfitting is combined with the iteratively reweighted least squares formulation for fitting generalized linear models, providing an alternative to linearization techniques or a full-scale profile likelihood approach. Regression diagnostics for detecting outliers and influential points are briefly considered. The results of a small computer simulation study are reported.