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Recently the shape-restricted inference has gained popularity in statisticaland econometric literature in order to relax the linear or quadratic covariateeffect in regression analyses. The typical shape-restricted covariate effectincludes monotonic increasing, decreasing, convexity or concavity. In thispaper, we introduce the shape-restricted inference to the celebrated Coxregression model (SR-Cox), in which the covariate response is modeled asshape-restricted additive functions. The SR-Cox regression approximates theshape-restricted functions using a spline basis expansion with data drivenchoice of knots. The underlying minimization of negative log-likelihoodfunction is formulated as a convex optimization problem, which is solved withan active-set optimization algorithm. The highlight of this algorithm is thatit eliminates the superfluous knots automatically. When covariate effectsinclude combinations of convex or concave terms with unknown forms and linearterms, the most interesting finding is that SR-Cox produces accurate linearcovariate effect estimates which are comparable to the maximum partiallikelihood estimates if indeed the forms are known. We conclude that concave orconvex SR-Cox models could significantly improve nonlinear covariate responserecovery and model goodness of fit.

Active-set algorithms based statistical inference for shape-restricted generalized additive Cox regression models