Assessing continuous bivariate effects among different groups through nonparametric regression models. 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.

keywords: Breast cancer, Bootstrap, Computer-aided diagnosis, Generalized additive models, Kernel smoothing, Interactions