Assessing continuous bivariate effects among different groups through nonparametric regression models: An application to breast cancer detection

In many applications, the joint effect of two continuous covariates on the target binary response may vary across groups defined by levels of a given factor. A testing procedure that would enable this type of surface-by-factor interactions to be detected has been designed. To accomplish this goal, a logistic generalized additive model (GAM) with bivariate continuous interactions varying across groups defined by levels of a factor is considered. A local scoring algorithm based on local linear kernel smoothers was implemented to estimate the proposed logistic GAM. Bootstrap resampling techniques were used for the purpose of testing for factor-by-surface interactions. Given the high computational cost involved, binning techniques were used to speed up computation in the estimation and testing processes. The adequacy of the bootstrap-based test was assessed by means of a simulation study. If a factor-by-surface interaction is detected in the model, it is then established that the use of the odds-ratio curves is very useful in obtaining a direct interpretation of the fitted model. The benefits of using this methodology when analyzing real data are illustrated by applying the technique to the outputs produced by a computerized system dedicated to the early detection of breast cancer.