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Weighted Likelihood for Semiparametric Models and Two‐phase Stratified Samples, with Application to Cox Regression

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  • NORMAN E. BRESLOW
  • JON A. WELLNER

Abstract

. We consider semiparametric models for which solution of Horvitz–Thompson or inverse probability weighted (IPW) likelihood equations with two‐phase stratified samples leads to consistent and asymptotically Gaussian estimators of both Euclidean and non‐parametric parameters. For Bernoulli (independent and identically distributed) sampling, standard theory shows that the Euclidean parameter estimator is asymptotically linear in the IPW influence function. By proving weak convergence of the IPW empirical process, and borrowing results on weighted bootstrap empirical processes, we derive a parallel asymptotic expansion for finite population stratified sampling. Several of our key results have been derived already for Cox regression with stratified case–cohort and more general survey designs. This paper is intended to help interpret this previous work and to pave the way towards a general Horvitz–Thompson approach to semiparametric inference with data from complex probability samples.

Suggested Citation

  • Norman E. Breslow & Jon A. Wellner, 2007. "Weighted Likelihood for Semiparametric Models and Two‐phase Stratified Samples, with Application to Cox Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(1), pages 86-102, March.
    • Handle: RePEc:bla:scjsta:v:34:y:2007:i:1:p:86-102
    DOI: 10.1111/j.1467-9469.2006.00523.x
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    2. Issa J. Dahabreh & Sarah E. Robertson & Lucia C. Petito & Miguel A. Hernán & Jon A. Steingrimsson, 2023. "Efficient and robust methods for causally interpretable meta‐analysis: Transporting inferences from multiple randomized trials to a target population," Biometrics, The International Biometric Society, vol. 79(2), pages 1057-1072, June.
    3. Patrice Bertail & Emilie Chautru & Stephan Clémençon, 2017. "Empirical Processes in Survey Sampling with (Conditional) Poisson Designs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 97-111, March.
    4. Yucong Lin & Jinhua Su & Yang Liu & Jue Hou & Feifei Wang, 2024. "Implicit profiling estimation for semiparametric models with bundled parameters," Statistical Papers, Springer, vol. 65(5), pages 3203-3234, July.
    5. Jieli Ding & Tsui-Shan Lu & Jianwen Cai & Haibo Zhou, 2017. "Recent progresses in outcome-dependent sampling with failure time data," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(1), pages 57-82, January.
    6. Menggang Yu & Bin Nan, 2010. "Regression Calibration in Semiparametric Accelerated Failure Time Models," Biometrics, The International Biometric Society, vol. 66(2), pages 405-414, June.
    7. Qingning Zhou & Jianwen Cai & Haibo Zhou, 2018. "Outcome†dependent sampling with interval†censored failure time data," Biometrics, The International Biometric Society, vol. 74(1), pages 58-67, March.
    8. Matthew R. Williams & Terrance D. Savitsky, 2021. "Uncertainty Estimation for Pseudo‐Bayesian Inference Under Complex Sampling," International Statistical Review, International Statistical Institute, vol. 89(1), pages 72-107, April.
    9. Soyoung Kim & Yayun Xu & Mei‐Jie Zhang & Kwang‐Woo Ahn, 2020. "Stratified proportional subdistribution hazards model with covariate‐adjusted censoring weight for case‐cohort studies," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(4), pages 1222-1242, December.
    10. Jan Feifel & Madlen Gebauer & Martin Schumacher & Jan Beyersmann, 2020. "Nested exposure case-control sampling: a sampling scheme to analyze rare time-dependent exposures," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 26(1), pages 21-44, January.
    11. Jing Zhang & Haibo Zhou & Yanyan Liu & Jianwen Cai, 2021. "Conditional screening for ultrahigh-dimensional survival data in case-cohort studies," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 27(4), pages 632-661, October.
    12. Takumi Saegusa, 2015. "Variance Estimation under Two-Phase Sampling," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1078-1091, December.
    13. Zhiguo Li & Peter Gilbert & Bin Nan, 2008. "Weighted Likelihood Method for Grouped Survival Data in Case–Cohort Studies with Application to HIV Vaccine Trials," Biometrics, The International Biometric Society, vol. 64(4), pages 1247-1255, December.
    14. Sangwook Kang & Jianwen Cai, 2009. "Marginal Hazards Regression for Retrospective Studies within Cohort with Possibly Correlated Failure Time Data," Biometrics, The International Biometric Society, vol. 65(2), pages 405-414, June.
    15. Rebecca Payne & Ming Yang & Yingye Zheng & Majken K. Jensen & Tianxi Cai, 2016. "Robust risk prediction with biomarkers under two‐phase stratified cohort design," Biometrics, The International Biometric Society, vol. 72(4), pages 1037-1045, December.
    16. Yanqing Sun & Xiyuan Qian & Qiong Shou & Peter B. Gilbert, 2017. "Analysis of two-phase sampling data with semiparametric additive hazards models," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 23(3), pages 377-399, July.
    17. Gustavo Amorim & Ran Tao & Sarah Lotspeich & Pamela A. Shaw & Thomas Lumley & Bryan E. Shepherd, 2021. "Two‐phase sampling designs for data validation in settings with covariate measurement error and continuous outcome," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 184(4), pages 1368-1389, October.
    18. Dandan Liu & Tianxi Cai & Yingye Zheng, 2012. "Evaluating the Predictive Value of Biomarkers with Stratified Case-Cohort Design," Biometrics, The International Biometric Society, vol. 68(4), pages 1219-1227, December.
    19. Jon Arni Steingrimsson & Robert L. Strawderman, 2017. "Estimation in the Semiparametric Accelerated Failure Time Model With Missing Covariates: Improving Efficiency Through Augmentation," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(519), pages 1221-1235, July.
    20. Olli Saarela & James A. Hanley, 2015. "Case-base methods for studying vaccination safety," Biometrics, The International Biometric Society, vol. 71(1), pages 42-52, March.
    21. Jing Zhang & Haibo Zhou & Yanyan Liu & Jianwen Cai, 2021. "Feature screening for case‐cohort studies with failure time outcome," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(1), pages 349-370, March.
    22. Han, Bo & Wang, Xiaoguang, 2020. "Semiparametric estimation for the non-mixture cure model in case-cohort and nested case-control studies," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).

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