Spline-based models for predictiveness curves and surfaces

A biomarker is defined to be a biological characteristic that is objectively measured and evaluated as an indicator of normal biologic processes, pathogenic processes, or pharmacologic responses to a therapeutic intervention. The use of biomarkers in cancer has been advocated for a variety of purposes, which include use as surrogate endpoints, early detection of disease, proxies for environmental exposure and risk prediction. We deal with the latter issue in this paper. Several authors have proposed use of the predictiveness curve for assessing the capacity of a biomarker for risk prediction. For most situations, it is reasonable to assume monotonicity of the biomarker effects on disease risk. In this article, we propose the use of flexible modelling of the predictiveness curve and its bivariate analogue, the predictiveness surface, through the use of spline algorithms that incorporate the appropriate monotonicity constraints. Estimation proceeds through use of a two-step algorithm that represents the "smooth, then monotonize" approach. Subsampling procedures are used for inference. The methods are illustrated to data from a melanoma study.