Abstract
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 suggest a nonlinear effect on the hazard of death. A flexible form of the proportional hazards model, , is desirable to facilitate a rich class of covariate effect on a survival outcome to provide meaningful insight, where the functional form of 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 nonconvex 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 nonparametric estimation of the monotone-restricted function , 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 and outperforms the estimators when in the Cox model is misspecified. We apply the method to model the effect of estrogen receptor on breast cancer patients’ survival.
Funding Statement
The research is partially supported by grant R01 CA193878 (JN, YS).
Acknowledgments
The authors wish to thank Dr. Naoto Ueno and his group for sharing breast cancer patient cohort data which motivated this research. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.
Citation
Jing Qin. Geng Deng. Jing Ning. Ao Yuan. Yu Shen. "Estrogen receptor expression on breast cancer patients’ survival under shape-restricted Cox regression model." Ann. Appl. Stat. 15 (3) 1291 - 1307, September 2021. https://doi.org/10.1214/21-AOAS1446
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