Statistical estimation in the proportional hazards model with risk set sampling. (English) Zbl 1047.62089

Chen, Kani

Ann. Stat. 32, No. 4, 1513-1532 (2004).

Summary: D.C. Thomas’ [J. R. Stat. Soc., Ser. A 140, 483–485 (1977)] partial likelihood estimator of regression parameters is widely used in the analysis of nested case-control data with Cox’s model. This paper proposes a new estimator of the regression parameters, which is consistent and asymptotically normal. Its asymptotic variance is smaller than that of Thomas’ estimator away from the null. Unlike some other existing estimators, the proposed estimator does not rely on any more data than strictly necessary for Thomas’ estimator and is easily computable from a closed form estimating equation with a unique solution. The variance estimation is obtained as minus the inverse of the derivative of the estimating function and therefore inference is easily available. A numerical example is provided in support of the theory.

MSC:

62N02	Estimation in survival analysis and censored data
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F12	Asymptotic properties of parametric estimators
62J05	Linear regression; mixed models

Keywords:

extended nested case-control data; time-restricted nested case-control data; empirical approximation; asymptotic relative efficiency

References:

[1]	Andersen, P. K. and Gill, R. D. (1982). Cox’s regression model for counting processes: A large sample study. Ann. Statist. 10 1100–1120. JSTOR: · Zbl 0526.62026 · doi:10.1214/aos/1176345976
[2]	Borgan, Ø., Goldstein, L. and Langholz, B. (1995). Methods for the analysis of sampled cohort data in the Cox proportional hazards model. Ann. Statist. 23 1749–1778. JSTOR: · Zbl 0843.62094 · doi:10.1214/aos/1176324322
[3]	Borgan, Ø. and Olsen, E. F. (1999). The efficiency of simple and counter-matched nested case-control sampling. Scand. J. Statist. 26 493–509. · Zbl 0938.62116 · doi:10.1111/1467-9469.00164
[4]	Breslow, N. E. (1996). Statistics in epidemiology: The case-control study. J. Amer. Statist. Assoc. 91 14–28. · Zbl 0870.62082 · doi:10.2307/2291379
[5]	Chen, K. (2001). Generalized case-cohort sampling. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 791–809. · Zbl 0988.62063 · doi:10.1111/1467-9868.00313
[6]	Chen, K. and Lo, S.-H. (1999). Case-cohort and case-control analysis with Cox’s model. Biometrika 86 755–764. · Zbl 0940.62108 · doi:10.1093/biomet/86.4.755
[7]	Goldstein, L. and Langholz, B. (1992). Asymptotic theory for nested case-control sampling in the Cox regression model. Ann. Statist. 20 1903–1928. JSTOR: · Zbl 0776.62024 · doi:10.1214/aos/1176348895
[8]	Langholz, B. and Goldstein, L. (1996). Risk set sampling in epidemiologic cohort studies. Statist. Sci. 11 35–53.
[9]	Langholz, B. and Thomas, D. C. (1990). Nested case-control and case-cohort methods of sampling from a cohort: A critical comparison. Amer. J. Epidemiology 131 169–176.
[10]	Langholz, B. and Thomas, D. C. (1991). Efficiency of cohort sampling designs: Some surprising results. Biometrics 47 1563–1571.
[11]	Oakes, D. (1981). Survival times: Aspects of partial likelihood (with discussion). Internat. Statist. Review 49 235–264. JSTOR: · Zbl 0479.62080 · doi:10.2307/1402606
[12]	Pollard, D. (1990). Empirical Processes : Theory and Applications. IMS, Hayward, CA. · Zbl 0741.60001
[13]	Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed. J. Amer. Statist. Assoc. 89 846–866. · Zbl 0815.62043 · doi:10.2307/2290910
[14]	Samuelsen, S. O. (1997). A pseudo-likelihood approach to analysis of nested case-control studies. Biometrika 84 379–394. · Zbl 0882.62107 · doi:10.1093/biomet/84.2.379
[15]	Sasieni, P. (1993a). Some new estimators for Cox regression. Ann. Statist. 21 1721–1759. JSTOR: · Zbl 0797.62020 · doi:10.1214/aos/1176349395
[16]	Sasieni, P. (1993b). Maximum weighted partial likelihood estimators for the Cox model. J. Amer. Statist. Assoc. 88 144–152. · Zbl 0771.62078 · doi:10.2307/2290707
[17]	Suissa, S., Edwardes, M. and Biovin, J.-F. (1998). External comparisons from nested case-control designs. Epidemiology 9 72–78.
[18]	Thomas, D. C. (1977). Addendum to “Methods of cohort analysis: Appraisal by application to asbestos mining,” by F. D. K. Liddell, J. C. McDonald and D. C. Thomas. J. Roy. Statist. Soc. Ser. A 140 483–485.
[19]	van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes : With Applications to Statistics. Springer, New York. · Zbl 0862.60002