JDS Journal of Data Science 1683-86021680-743X1680-743X School of Statistics, Renmin University of China JDS1061 10.6339/22-JDS1061 Statistical Data Science Maximum Likelihood Estimation for Shape-restricted Single-index Hazard Models QinJingjingqin@niaid.nih.gov1∗ SunYifei2 YuanAo3 HuangChiung-Yu4 Biostatistics Research Branch, National Institute of Allergy and Infectious Diseases, Maryland, U.S.A. Department of Biostatistics, Mailman School of Public Health, Columbia University, New York, U.S.A. Department of Biostatistics, Bioinformatics & Biomathematics, Georgetown University, Washington D.C., U.S.A. Department of Epidemiology & Biostatistics, University of California San Francisco, California, U.S.A. Corresponding author. Email: jingqin@niaid.nih.gov. 20234112022214681695 Supplementary Material

The Supplementary Material includes the proof of Theorem 2, additonal simulation results, and the R code to implement the proposed method.

262022572022 2023 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.2023 Open access article under the CC BY license.

Single-index models are becoming increasingly popular in many scientific applications as they offer the advantages of flexibility in regression modeling as well as interpretable covariate effects. In the context of survival analysis, the single-index hazards models are natural extensions of the Cox proportional hazards models. In this paper, we propose a novel estimation procedure for single-index hazard models under a monotone constraint of the index. We apply the profile likelihood method to obtain the semiparametric maximum likelihood estimator, where the novelty of the estimation procedure lies in estimating the unknown monotone link function by embedding the problem in isotonic regression with exponentially distributed random variables. The consistency of the proposed semiparametric maximum likelihood estimator is established under suitable regularity conditions. Numerical simulations are conducted to examine the finite-sample performance of the proposed method. An analysis of breast cancer data is presented for illustration.

 isotonic regression pool-adjacent-violators algorithm profile likelihood semiparametric estimation References Ambler G, Benner A (2015). mfp: Multivariable fractional polynomials. R package version 1.5.2. Ayer M, Brunk HD, Ewing GM, Reid WT, Silverman E (1955). An empirical distribution function for sampling with incomplete information. The Annals of Mathematical Statistics, 26: 641–647. Balabdaoui F, Groeneboom P, Hendrickx K (2019). Score estimation in the monotone single-index model. Scandinavian Journal of Statistics, 46(2): 517–544. Chen K, Guo S, Sun L, Wang JL (2010). Global partial likelihood for nonparametric proportional hazards models. Journal of the American Statistical Association, 105(490): 750–760. Chen S, Zhou L (2007). Local partial likelihood estimation in proportional hazards regression. The Annals of Statistics, 35(2): 888–916. Chung Y, Ivanova A, Hudgens MG, Fine JP (2018). Partial likelihood estimation of isotonic proportional hazards models. Biometrika, 105: 133–148. Cox DR (1975). Partial likelihood. Biometrika, 62(2): 269–276. Du Y, Varadhan R (2020). SQUAREM: An R package for off-the-shelf acceleration of EM, MM and other EM-like monotone algorithms. Journal of Statistical Software, 92(7): 1–41. Fan J, Gijbels I, King M, et al. (1997). Local likelihood and local partial likelihood in hazard regression. The Annals of Statistics, 25(4): 1661–1690. Foster JC, Taylor JMG, Nan B (2013). Variable selection in monotone single-index models via the adaptive lasso. Statistics in Medicine, 32(22): 3944–3954. Gentleman R, Crowley J (1991). Local full likelihood estimation for the proportional hazards model. Biometrics, 47: 1283–1296. Groeneboom P, Hendrickx K (2018). Current status linear regression. The Annals of Statistics, 46: 1415–1444. Groeneboom P, Hendrickx K (2019). Estimation in monotone single-index models. Statistica Neerlandica, 73: 78–99. Härdle W, Hall P, Ichimura H (1993). Optimal smoothing in single-index models. The Annals of Statistics, 21: 157–178. Härdle W, Stoker TM (1989). Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association, 84(408): 986–995. Huang J, Wellner J (1997). Interval censored survival data: A review of recent progress. In: Proceedings of the First Seattle Symposium in Biostatistics: Survival Analysis. 123–169. Huang JZ, Liu L (2006). Polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form. Biometrics, 62(3): 793–802. Ichimura H (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics, 58(1–2): 71–120. Jongbloed G (1998). The iterative convex minorant algorithm for nonparametric estimation. Journal of Computational and Graphical Statistics, 7(3): 310–321. Kosorok MR (2008). Introduction to empirical processes and semiparametric inference Springer Series in Statistics. Springer, New York. Murphy S, van der Vaart AW, Wellner J (1999). Current status regression. Mathematical Methods of Statistics, 8(3): 407–425. Nelder JA, Mead R (1965). A simplex method for function minimization. The Computer Journal, 7(4): 308–313. Powell JL, Stock JH, Stoker TM (1989). Semiparametric estimation of index coefficients. Econometrica, 57: 1403–1430. Prentice RL, Self SG (1983). Asymptotic distribution theory for cox-type regression models with general relative risk form. The Annals of Statistics, 11: 804–813. R Core Team (2020). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. Robertson T, Wright FT, Dykstra R (1988). Order Restricted Statistical Inference. John Wiley & Sons, Chichester. Schumacher M, Bastert G, Bojar H, Huebner K, Olschewski M, Sauerbrei W, et al. (1994). Randomized 2 × 2 trial evaluating hormonal treatment and the duration of chemotherapy in node-positive breast cancer patients. German breast cancer study group. Journal of Clinical Oncology, 12(10): 2086–2093. Tibshirani R, Hastie T (1987). Local likelihood estimation. Journal of the American Statistical Association, 82(398): 559–567. van der Vaart A, Wellner JA (2000). Preservation theorems for Glivenko-Cantelli and uniform Glivenko-Cantelli classes. 115–133. Wang W, Wang JL, Wang Q (2009). Proportional hazards regression with unknown link function. IMS Lecture Notes-Monograph Series, 57: 47–66. Wang W, Yan J (2021). Shape-restricted regression splines with R package splines2. Journal of Data Science, 19(3): 498–517.