Active-set algorithms based statistical inference for shape-restricted generalized additive Cox regression models
This work addresses the need for flexible yet interpretable covariate modeling in survival analysis, offering an incremental improvement by extending shape-restricted inference to Cox regression with automated knot selection.
The authors tackled the problem of incorporating shape restrictions (e.g., monotonicity, convexity) into Cox regression models for survival analysis, developing an active-set algorithm that automatically eliminates unnecessary knots in spline approximations and produces accurate linear covariate effect estimates comparable to maximum partial likelihood estimates when forms are known.
Recently the shape-restricted inference has gained popularity in statistical and econometric literature in order to relax the linear or quadratic covariate effect in regression analyses. The typical shape-restricted covariate effect includes monotonic increasing, decreasing, convexity or concavity. In this paper, we introduce the shape-restricted inference to the celebrated Cox regression model (SR-Cox), in which the covariate response is modeled as shape-restricted additive functions. The SR-Cox regression approximates the shape-restricted functions using a spline basis expansion with data driven choice of knots. The underlying minimization of negative log-likelihood function is formulated as a convex optimization problem, which is solved with an active-set optimization algorithm. The highlight of this algorithm is that it eliminates the superfluous knots automatically. When covariate effects include combinations of convex or concave terms with unknown forms and linear terms, the most interesting finding is that SR-Cox produces accurate linear covariate effect estimates which are comparable to the maximum partial likelihood estimates if indeed the forms are known. We conclude that concave or convex SR-Cox models could significantly improve nonlinear covariate response recovery and model goodness of fit.