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

Abstract

Recently the shape-restricted inference has gained popularity in statistical and econometric literature to relax the linear or quadratic covariate effect in regression analyses. The typical shape-restricted covariate effect includes monotone 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 modelled 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.

Keywords:

• Shape-restricted inference
• Cox regression model
• automatic knots selection
• spline basis expansion
• optimization

AMS Subject Classifications:

• 62N02
• 62P05
• 65K99

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The algorithm of the method, the shape-constrained maximum likelihood estimator (SC-MLE) is implemented in the R package scar.

2 https://stat.ethz.ch/R-manual/R-devel/library/survival/html/pbc.html

3 In addition, the sixth variable, hepatomegaly, had been independently predictive of survival until the logarithm transformation of bilirubin was introduced. Given that the variable does not include the values of the additional 106 cases, we did not include them in our regression analysis.

4 The public website of Freddie Mac's Single Family Loan-Level dataset is http://www.freddiemac.com/research/datasets/sf_loanlevel_dataset.html

5 https://www.bls.gov/