# Cox regression with missing covariate data using a modified partial likelihood method

- 426 Downloads
- 1 Citations

## Abstract

Missing covariate values is a common problem in survival analysis. In this paper we propose a novel method for the Cox regression model that is close to maximum likelihood but avoids the use of the EM-algorithm. It exploits that the observed hazard function is multiplicative in the baseline hazard function with the idea being to profile out this function before carrying out the estimation of the parameter of interest. In this step one uses a Breslow type estimator to estimate the cumulative baseline hazard function. We focus on the situation where the observed covariates are categorical which allows us to calculate estimators without having to assume anything about the distribution of the covariates. We show that the proposed estimator is consistent and asymptotically normal, and derive a consistent estimator of the variance–covariance matrix that does not involve any choice of a perturbation parameter. Moderate sample size performance of the estimators is investigated via simulation and by application to a real data example.

## Keywords

Cox model Missing covariate data Recursive estimation Survival data## References

- Andersen PK, Borgan Ø, Gill RD, Keiding N (1993) Statistical models based on counting processes. Springer-Verlag, New YorkCrossRefMATHGoogle Scholar
- Andersen PK, Gill RD (1982) Cox’s regression model for counting processes: a large sample study. Ann Stat 10:1100–1120MathSciNetCrossRefMATHGoogle Scholar
- Asgharian M (2014) On the singularities of the information matrix and multipath change-point problems. Theory Probab Appl 58:546–561MathSciNetCrossRefMATHGoogle Scholar
- Bagdonavicius V, Nikulin M (1999) Generalised proportional hazards model based on modified parital likelihood. Lifetime Data Anal 5:329–350MathSciNetCrossRefMATHGoogle Scholar
- Chen H (2002) Double-semiparametric method for missing covariates in Cox regression models. J Am Stat Assoc 97:565–576MathSciNetCrossRefMATHGoogle Scholar
- Chen H, Little R (1999) Proportional hazards regression with missing covariates. J Am Stat Assoc 94:896–908MathSciNetCrossRefMATHGoogle Scholar
- Cox DR (1972) Regression models and life-tables (with discussion). J R Stat Soc: Ser B 34:187–220Google Scholar
- Herring AH, Ibrahim JG (2001) Likelihood-based methods for missing covariates in the Cox proportional hazards model. J Am Stat Assoc 96:292–302MathSciNetCrossRefMATHGoogle Scholar
- Lin DY, Wei LJ, Ying Z (1993) Checking the Cox model with cumulative sums of martingale-based residuals. Biometrika 80:557–572MathSciNetCrossRefMATHGoogle Scholar
- Martinussen T (1999) Cox regression with incomplete covariate measurements using the EM-algorithm. Scand J Stat 26:479–491MathSciNetCrossRefMATHGoogle Scholar
- Martinussen T, Scheike TH (2006) Dynamic regression models for survival data. Springer-Verlag, New YorkMATHGoogle Scholar
- Pugh M, Robins J, Lipsitz S, Harrington D (1994) Inference in the Cox proportional hazards model with missing covariate data. Technical report, Harvard School og Public Health, Dept. of BiostatisticsGoogle Scholar
- Qi L, Wang CY, Prentice RL (2005) Weighted estimators for proportional hazards regression with missing covariates. J Am Stat Assoc 100:1250–1263MathSciNetCrossRefMATHGoogle Scholar
- Robins JM, Rotnitzky A, Zhao LP (1994) Estimation of regression coefficients when some regressors are not always observed. J Am Stat Assoc 89:846–866MathSciNetCrossRefMATHGoogle Scholar
- Sherman M (2006) Complex step derivatives: how did i miss this? Biomed Comput Rev 2(3):27Google Scholar
- Sterne JAC, White IR, Carlin JB, Spratt M, Royston P, Kenward MG, Wood AM, Carpenter JR (2009) Multiple imputation for missing data in epidemiological and clinical research: potential and pitfalls. BMJ 338:157–160CrossRefGoogle Scholar
- Tierney L, Kass RE, Kadane JB (1989) Fully exponential laplace approximations to expectations and variances of nonpositive functions. J Am Stat Assoc 84:710–716MathSciNetCrossRefMATHGoogle Scholar
- Wang CY, Chen HY (2001) Augmented inverse probability weighted estimator for Cox missing covariate regression. Biometrics 57:414–419MathSciNetCrossRefMATHGoogle Scholar
- White IR, Royston P (2009) Imputing missing covariate values for the Cox model. Stat Med 28:1982–98MathSciNetCrossRefGoogle Scholar
- Xu Q, Paik MC, Luo X, Tsai W-Y (2009) Reweighting estimators for Cox regression with missing covariates. J Am Stat Assoc 104:1155–1167MathSciNetCrossRefMATHGoogle Scholar
- Zucker D (2005) A pseudo partial likelihood method for semi-parametric survival regression with covariate errors. J Am Stat Assoc 100:1264–1277MathSciNetCrossRefMATHGoogle Scholar