Non-parametric estimation of the odds ratios for continuous exposures using generalized additive models with an unknown link function

The generalized additive, model (GAM) is a powerful and widely used tool that allows researchers to fit, non-parametrically, the effect of continuous predictors on a transformation of the mean response variable. Such a transformation is given by a so-called link function, and in GAMs this link function is assumed to be known. Nevertheless, if an incorrect choice is made for the link, the resulting GAM is misspecified and the results obtained may be misleading. In this paper, we propose a modified version of the local scoring algorithm that allows for the non-parametric estimation of the link function, by using local linear kernel smoothers. To better understand the effect that each covariate produces on the outcome, results are expressed in terms of the non-parametric odds ratio (OR) curves. Bootstrap techniques were used to correct the bias in the OR estimation and to construct point-wise confidence intervals. A simulation study was carried out to assess the behaviour of the resulting estimates. The proposed methodology was illustrated using data from the AIDS Register of Galicia (NW Spain), with a view to assessing the effect of the CD4 lymphocyte count on the probability of being AIDS-diagnosed via Tuberculosis (TB). This application shows how the link's flexibility makes it possible to obtain OR curve estimates that are less sensitive to the presence of outliers and unusual values that are often present in the extremes of the covariate distributions.