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For certain subtypes of breast cancer, study findings show that their level of estrogen receptor expression is associated with their risk of cancer death, and also suggests a non-linear effect on the hazard of death. A flexible form of the proportional hazards model, λ ( t ∣ x, z  ) = λ ( t ) exp(  z    T    β  ) q ( x ), is desirable to facilitate a rich class of covariate effect on a survival outcome to provide meaningful insight, where the functional form of q ( x ) is not specified except for its shape. Prior biologic knowledge on the shape of the underlying distribution of the covariate effect in regression models can be used to enhance statistical inference. Despite recent progress, major challenges remain for the semiparametric shape-restricted inference due to lack of practical and efficient computational algorithms to accomplish non-convex optimization. We propose an alternative algorithm to maximize the full log-likelihood with two sets of parameters iteratively under monotone constraints. The first set consists of the non-parametric estimation of the monotone-restricted function q ( x ), while the second set includes estimating the baseline hazard function and other covariate coefficients. The iterative algorithm in conjunction with the pool-adjacent-violators algorithm makes the computation efficient and practical. The Jackknife resampling effectively reduces the estimator bias, when sample size is small. Simulations show that the proposed method can accurately capture the underlying shape of q ( x ), and outperforms the estimators when q ( x ) in the Cox model is mis-specified. We apply the method to model the effect of estrogen receptor on breast cancer patients' survival.

Estrogen receptor expression on breast cancer patients’ survival under shape-restricted Cox regression model