Cox Regression with Covariate Measurement Error

This article deals with parameter estimation in the Cox proportional hazards model when covariates are measured with error. We consider both the classical additive measurement error model and a more general model which represents the mis-measured version of the covariate as an arbitrary linear function of the true covariate plus random noise. Only moment conditions are imposed on the distributions of the covariates and measurement error. Under the assumption that the covariates are measured precisely for a validation set, we develop a class of estimating equations for the vector-valued regression parameter by correcting the partial likelihood score function. The resultant estimators are proven to be consistent and asymptotically normal with easily estimated variances. Furthermore, a corrected version of the Breslow estimator for the cumulative hazard function is developed, which is shown to be uniformly consistent and, upon proper normalization, converges weakly to a zero-mean Gaussian process. Simulation studies indicate that the asymptotic approximations work well for practical sample sizes. The situation in which replicate measurements (instead of a validation set) are available is also studied.

[1]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[2]  Robert V. Foutz,et al.  On the Unique Consistent Solution to the Likelihood Equations , 1977 .

[3]  R. Gill,et al.  Cox's regression model for counting processes: a large sample study : (preprint) , 1982 .

[4]  R. Prentice.  Covariate measurement errors and parameter estimation in a failure time regression model , 1982 .

[5]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[6]  Wayne A. Fuller,et al.  Measurement Error Models , 1988 .

[7]  L. Stefanski.  Unbiased estimation of a nonlinear function a normal mean with application to measurement err oorf models , 1989 .

[8]  Tsuyoshi Nakamura.  Corrected score function for errors-in-variables models : Methodology and application to generalized linear models , 1990 .

[9]  T. Nakamura,et al.  Proportional hazards model with covariates subject to measurement error. , 1992, Biometrics.

[10]  Response Variable Error , 1995 .

[11]  M. Pepe,et al.  Auxiliary covariate data in failure time regression , 1995 .

[12]  D. Ruppert,et al.  Measurement Error in Nonlinear Models , 1995 .

[13]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[14]  R. Prentice,et al.  Regression calibration in failure time regression. , 1997, Biometrics.

[15]  J. Buzas.  Unbiased scores in proportional hazards regression with covariate measurement error , 1998 .

[16]  M. Kenward,et al.  Contribution to the discussion of the paper by Diggle, Tawn and Moyeed , 1998 .

[17]  M. Davidian,et al.  Estimating the parameters in the Cox model when covariate variables are measured with error. , 1998, Biometrics.

[18]  F. Kong,et al.  CONSISTENT ESTIMATION IN COX PROPORTIONAL HAZARDS MODEL WITH COVARIATE MEASUREMENT ERRORS , 1999 .

[19]  Zhiliang Ying,et al.  Semiparametric regression for the mean and rate functions of recurrent events , 2000 .

[20]  Yi Li,et al.  Survival Analysis With Heterogeneous Covariate Measurement Error , 2004 .