The proportional hazards model with covariate measurement error. (English) Zbl 1074.62065

Summary: The proportional hazards regression model is commonly used to evaluate the relationship between survival and covariates. Covariates are frequently measured with error. Substituting mismeasured values for the true covariates leads to biased estimation. P. Hu et al. [Biometrics 54, 1407–1419 (1998; Zbl 1058.62557)] have proposed to base estimation in the proportional hazards model with covariate measurement error on a joint likelihood for survival and the covariate variable. Nonparametric maximum likelihood estimation (NPMLE) was used and simulations were conducted to assess the asymptotic validity of this approach. In this paper, we derive a rigorous proof of asymptotic normality of the NPML estimators.


62N02 Estimation in survival analysis and censored data
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62N01 Censored data models
62G08 Nonparametric regression and quantile regression


Zbl 1058.62557
Full Text: DOI


[1] Andersen, P.K.; Gill, R.D., Cox’s regression model for counting processesa large sample study, Ann. statist., 10, 1100-1120, (1982) · Zbl 0526.62026
[2] Bickel, P.J.; Klaassen, C.A.J.; Ritov, Y.; Wellner, J.A., Efficient and adaptive estimation for semiparametric models, (1993), The Johns Hopkins University Press Baltimore · Zbl 0786.62001
[3] Breslow, N.E., Contribution to the discussion on the paper by D.R. Cox, J. roy. statist. soc. ser. B, 34, 187-220, (1972)
[4] Buzas, J.S., Unbiased scores in proportional hazards regression with covariate measurement error, J. statist. plann. inference, 67, 247-257, (1998) · Zbl 0932.62109
[5] Chen, H.Y., Double-semiparametric method for missing covariates in Cox regression models, J. amer. statist. assoc., 97, 565-576, (2002) · Zbl 1073.62526
[6] Cox, D.R., Regression models and life-tables (with discussion), J. roy. statist. soc. ser. B, 34, 187-220, (1972) · Zbl 0243.62041
[7] Cox, D.R., Partial likelihood, Biometrika, 62, 269-276, (1975) · Zbl 0312.62002
[8] Dupuy, J.-F., 2002. Modélisation conjointe de données longitudinales et de durées de vie. Ph.D. Dissertation, Université Paris V-René Descartes.
[9] Dupuy, J.-F., 2004. Nonparametric maximum likelihood estimation in the proportional hazards model with covariate measurement error. In: Nikulin, M., Balakrishnan, N., Mesbah, M., Limnios, N. (Eds.), Parametric and Semiparametric Models and Applications to Reliability, Survival Analysis and Quality of Life. Birkhauser, Boston, 13-25.
[10] Dupuy, J.-F.; Mesbah, M., Joint modeling of survival and informatively missing continuous longitudinal data, (), 357-360
[11] Dupuy, J.-F.; Mesbah, M., Joint modeling of event time and nonignorable missing longitudinal data, Lifetime D. anal., 8, 99-115, (2002) · Zbl 1030.62081
[12] Dupuy, J.-F.; Mesbah, M., Estimation of the asymptotic variance of semiparametric maximum likelihood estimators in the Cox model with a missingtime-dependent covariate, Comm. statist. theory methods, 33, 1385-1401, (2004) · Zbl 1114.62325
[13] Henderson, R.; Diggle, P.; Dobson, A., Joint modelling of longitudinal measurements and event time data, Biostatistics, 1, 465-480, (2000) · Zbl 1089.62519
[14] Hu, C.; Lin, D.Y., Cox regression with covariate measurement error, Scand. J. statist., 29, 637-655, (2002) · Zbl 1035.62102
[15] Hu, P.; Tsiatis, A.A.; Davidian, M., Estimating the parameters in the Cox model when covariate variables are measured with error, Biometrics, 88, 447-458, (1998) · Zbl 1058.62557
[16] Huang, Y.; Wang, C.Y., Cox regression with accurate covariates unascertainablea nonparametric-correction approach, J. amer. statist. assoc., 45, 1209-1219, (2000) · Zbl 1008.62040
[17] Johansen, S., An extension of Cox’s regression model, Internat. statist. rev., 51, 258-262, (1983)
[18] Li, Y.; Lin, X., Covariate measurement errors in frailty models for clustered survival data, Biometrika, 81, 61-71, (2000) · Zbl 1028.62078
[19] Murphy, S.A., Consistency in a proportional hazards model incorporating a random effect, Ann. statist., 22, 712-731, (1994) · Zbl 0827.62033
[20] Murphy, S.A., Asymptotic theory for the frailty model, Ann. statist., 23, 182-198, (1995) · Zbl 0822.62069
[21] Nakamura, T., Proportional hazards model with covariates subject to measurement error, Biometrics, 48, 829-838, (1992)
[22] Nielsen, G.G.; Gill, R.D.; Andersen, P.K.; Sørensen, T.I.A., A counting process approach to maximum likelihood estimation in frailty models, Scand. J. statist., 19, 25-43, (1992) · Zbl 0747.62093
[23] Parner, E., Asymptotic theory for the correlated gamma-frailty model, Ann. statist., 26, 183-214, (1998) · Zbl 0934.62101
[24] Prentice, R.L., Covariate measurement errors and parameter estimation in a failure time regression model, Biometrika, 69, 331-342, (1982) · Zbl 0523.62083
[25] Rudin, W., Analyse fonctionnelle, (1995), Ediscience International Paris
[26] Scharfstein, D.O.; Tsiatis, A.A.; Gilbert, P.B., Semiparametric efficient estimation in the generalized odds-rate class of regression models for right-censored time-to-event data, Lifetime D. anal., 4, 355-391, (1998) · Zbl 0941.62043
[27] Song, X.; Davidian, D.; Tsiatis, A.A., An estimator for the proportional hazards model with multiple longitudinal covariates measured with error, Biostatistics, 3, 511-528, (2002) · Zbl 1138.62360
[28] Tsiatis, A.A.; Tsiatis, A.A., Applications to survival and CD4 counts in patients with AIDS, Ann. statist., J. amer. statist. assoc., 90, 27-37, (1981) · Zbl 0818.62102
[29] Van Der Vaart, A.W., Asymptotic statistics, (1998), Cambridge University Press New York · Zbl 0910.62001
[30] Van Der Vaart, A.W.; Wellner, J.A., Weak convergence and empirical processes, (1996), Springer New York · Zbl 0862.60002
[31] Wulfsohn, M.S.; Tsiatis, A.A., A joint model for survival and longitudinal data measured with error, Biometrics, 53, 330-339, (1997) · Zbl 0874.62140
[32] Zhou, H.; Pepe, M.S., Auxiliary covariate data in failure time regression, Biometrika, 82, 139-149, (1995) · Zbl 0823.62100
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