Partly linear single-index cure models with a nonparametric incidence link function

Abstract

In cancer studies, it is commonplace that a fraction of patients participating in the study are cured, such that not all of them will experience a recurrence, or death due to cancer. Also, it is plausible that some covariates, such as the treatment assigned to the patients or demographic characteristics, could affect both the patients’ survival rates and cure/incidence rates. A common approach to accommodate these features in survival analysis is to consider a mixture cure survival model with the incidence rate modeled by a logistic regression model and latency part modeled by the Cox proportional hazards model. These modeling assumptions, though typical, restrict the structure of covariate effects on both the incidence and latency components. As a plausible recourse to attain flexibility, we study a class of semiparametric mixture cure models in this article, which incorporates two single-index functions for modeling the two regression components. A hybrid nonparametric maximum likelihood estimation method is proposed, where the cumulative baseline hazard function for uncured subjects is estimated nonparametrically, and the two single-index functions are estimated via Bernstein polynomials. Parameter estimation is carried out via a curated expectation-maximization algorithm. We also conducted a large-scale simulation study to assess the finite-sample performance of the estimator. The proposed methodology is illustrated via application to two cancer datasets.

Keywords

Bernstein polynomial expectation-maximization algorithm mixture cure models sieve estimation survival analysis

Get full access to this article

View all access options for this article.

References

Peng

and Dear

. A nonparametric mixture model for cure rate estimation. Biometrics 2000; 56: 237–243.

Lam

, Fong

and Tang

. Estimating the proportion of cured patients in a censored sample. Stat Med 2005; 24: 1865–1879.

Chen

, Ibrahim

and Sinha

. A new Bayesian model for survival data with a surviving fraction. J Am Stat Assoc 1999; 94: 909–919.

Ibrahim

, Chen

and Sinha

. Bayesian semiparametric models for survival data with a cure fraction. Biometrics 2001; 57: 383–388.

Boag

. Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc Ser B (Methodological) 1949; 11: 15–53.

Berkson

and Gage

. Survival curve for cancer patients following treatment. J Am Stat Assoc 1952; 47: 501–515.

Farewell

. A model for a binary variable with time-censored observations. Biometrika 1977; 64: 43–46.

Ghitany

, Maller

and Zhou

. Exponential mixture models with long-term survivors and covariates. J Multivar Anal 1994; 49: 218–241.

Amico

, Van Keilegom

and Legrand

. The single-index/Cox mixture cure model. Biometrics 2019; 75: 452–462.

10.

Musta

and Yuen

. Single-index mixture cure model under monotonicity constraints. arXiv preprint arXiv:221109464 2022.

11.

Farewell

. The use of mixture models for the analysis of survival data with long-term survivors. Biometrics 1982; 38: 1041–1046.

12.

Cox

. Regression models and life-tables. J R Stat Soc Ser B (Statistical Methodology) 1972; 34: 187–220.

13.

Kuk

AYC

and Chen

. A mixture model combining logistic regression with proportional hazards regression. Biometrika 1992; 79: 531–541.

14.

and Taylor

. Estimation in a Cox proportional hazards cure model. Biometrics 2000; 56: 227–236.

15.

Cox

. Partial likelihood. Biometrika 1975; 62: 269–276.

16.

. Maximum likelihood estimation in the proportional hazards cure model. Ann Inst Stat Math 2008; 60: 545–574.

17.

Wang

, Du

and Liang

. Two-component mixture cure rate model with spline estimated nonparametric components. Biometrics 2012; 68: 726–735.

18.

Amico

and Van Keilegom

. Cure models in survival analysis. Annu Rev Stat Appl 2018; 5: 311–342.

19.

Friedman

and Stuetzle

. Projection pursuit regression. J Am Stat Assoc 1981; 76: 817–823.

20.

Härdle

and Stoker

. Investigating smooth multiple regression by the method of average derivatives. J Am Stat Assoc 1989; 84: 986–995.

21.

Ichimura

. Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J Econom 1993; 58: 71–120.

22.

Carroll

, Fan

, Gijbels

et al. Generalized partially linear single-index models. J Am Stat Assoc 1997; 92: 477–489.

23.

and Ruppert

. Penalized spline estimation for partially linear single-index models. J Am Stat Assoc 2002; 97: 1042–1054.

24.

Wang

. Proportional hazards regression models with unknown link function and time-dependent covariates. Stat Sin 2004; 14: 885–905.

25.

Huang

and Liu

. Polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form. Biometrics 2006; 62: 793–802.

26.

, Chen

, Singh

et al. A class of partially linear single-index survival models. Canad J Stat 2006; 34: 97–112.

27.

Sun

, Kopciuk

and Lu

. Polynomial spline estimation of partially linear single-index proportional hazards regression models. Comput Stat Data Anal 2008; 53: 176–188.

28.

Shang

, Liu

, Zeleniuch-Jacquotte

et al. Partially linear single index Cox regression model in nested case-control studies. Comput Stat Data Anal 2013; 67: 199–212.

29.

Tibshirani

and Hastie

. Local likelihood estimation. J Am Stat Assoc 1987; 82: 559–567.

30.

Dabrowska

. Non-parametric regression with censored survival time data. Scand J Stat 1987; 14: 181–197.

31.

Hastie

and Tibshirani

. Generalized additive models for medical research. Stat Methods Med Res 1995; 4: 187–196.

32.

Xue

, Lam

and Li

. Sieve maximum likelihood estimator for semiparametric regression models with current status data. J Am Stat Assoc 2004; 99: 346–356.

33.

Lee

, Wong

, Lam

et al. Analysis of clustered interval-censored data using a class of semiparametric partly linear frailty transformation models. Biometrics 2022; 78: 165–178.

34.

Lorentz

. Bernstein polynomials. New York: Chelesa Publishing Co., 1986.

35.

Zhou

, Hu

and Sun

. A sieve semiparametric maximum likelihood approach for regression analysis of bivariate interval-censored failure time data. J Am Stat Assoc 2017; 112: 664–672.

36.

, Zhao

, Zhu

et al. Variable selection for high-dimensional partly linear additive Cox model with application to Alzheimer’s disease. Stat Med 2020; 39: 3120–3134.

37.

Lam

, Lee

, Wong

et al. Marginal analysis of current status data with informative cluster size using a class of semiparametric transformation cure models. Stat Med 2021; 40: 2400–2412.

38.

Horowitz

. Semiparametric and Nonparametric Methods in Econometrics. New York: Springer, 2009.

39.

International Neural Network Society (INNS) tINNCCS, Battiti R and Masulli F. BFGS optimization for faster and automated supervised learning. In International Neural Network Conference: July 9–13, 1990 Palais Des Congres—Paris—France. Springer, pp.757–760.

40.

Jennrich

and Sampson

. Newton-Raphson and related algorithms for maximum likelihood variance component estimation. Technometrics 1976; 18: 11–17.

41.

Taylor

. Semi-parametric estimation in failure time mixture models. Biometrics 1995; 51: 899–907.

42.

Murphy

and Van der Vaart

. On profile likelihood. J Am Stat Assoc 2000; 95: 449–465.

43.

Zeng

, Gao

and Lin

. Maximum likelihood estimation for semiparametric regression models with multivariate interval-censored data. Biometrika 2017; 104: 505–525.

44.

Farquhar

, Divaris

, Mazul

et al. Poor oral health affects survival in head and neck cancer. Oral Oncol 2017; 73: 111–117.

45.

Lenze

, Farquhar

, Sheth

et al. Socioeconomic status drives racial disparities in HPV-negative head and neck cancer outcomes. Laryngoscope 2021; 131: 1301–1309.

46.

Foekens

, Peters

, Look

et al. The urokinase system of plasminogen activation and prognosis in 2780 breast cancer patients. Cancer Res 2000; 60: 636–643.

47.

Cook

and Li

. Dimension reduction for conditional mean in regression. Ann Stat 2002; 30: 455–474.

48.

Yin

and Cook

. Dimension reduction for the conditional kth moment in regression. J R Stat Soc Ser B (Statistical Methodology) 2002; 64: 159–175.

49.

Chen

, Hall

and Müller

. Single and multiple index functional regression models with nonparametric link. Ann Stat 2011; 39: 1720–1747.

50.

Zhong

, Mueller

and Wang

. Deep learning for the partially linear Cox model. Ann Stat 2022; 50: 1348–1375.

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.00 MB

0.17 MB