Cox regression model with randomly censored covariates. (English) Zbl 1429.62487
Summary: This paper deals with a Cox proportional hazards regression model, where some covariates of interest are randomly right-censored. While methods for censored outcomes have become ubiquitous in the literature, methods for censored covariates have thus far received little attention and, for the most part, dealt with the issue of limit-of-detection. For randomly censored covariates, an often-used method is the inefficient complete-case analysis (CCA) which consists in deleting censored observations in the data analysis. When censoring is not completely independent, the CCA leads to biased and spurious results. Methods for missing covariate data, including type I and type II covariate censoring as well as limit-of-detection do not readily apply due to the fundamentally different nature of randomly censored covariates. We develop a novel method for censored covariates using a conditional mean imputation based on either Kaplan-Meier estimates or a Cox proportional hazards model to estimate the effects of these covariates on a time-to-event outcome. We evaluate the performance of the proposed method through simulation studies and show that it provides good bias reduction and statistical efficiency. Finally, we illustrate the method using data from the Framingham Heart Study to assess the relationship between offspring and parental age of onset of cardiovascular events.
MSC:
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
Keywords:
censored covariate; complete-case analysis; Cox proportional hazards model; random censoring; survival analysisReferences:
| [1] | Atem, F., & Matsouaka, R. A. (2017). Linear regression model with a randomly censored predictor: Estimation procedures. arXiv:1710.08349. |
| [2] | Atem, F. D., Qian, J., Maye, J. E., Johnson, K. A., & Betensky, R. A. (2016). Multiple imputation of a randomly censored covariate improves logistic regression analysis. Journal of Applied Statistics, 43, 1-112886-2896. · Zbl 1516.62128 |
| [3] | Atem, F. D., Qian, J., Maye, J. E., Johnson, K. A., & Betensky, R. A. (2017). Linear regression with a randomly censored covariate: Application to an alzheimer’s study. Journal of the Royal Statistical Society: Series C (Applied Statistics), 66(2), 313-328. |
| [4] | Atem, F. D., Sampene, E., & Greene, T. J. (2017). Improved conditional imputation for linear regression with a randomly censored predictor. Statistical Methods in Medical Research, 28, 432-444. |
| [5] | Atkinson, K. E. (2008). An introduction to numerical analysis. New York: John Wiley & Sons. |
| [6] | Austin, P. C., & Hoch, J. S. (2004). Estimating linear regression models in the presence of a censored independent variable. Statistics in Medicine, 23(3), 411-429. |
| [7] | Bartlett, J. W., Seaman, S. R., White, I. R., Carpenter, J. R., & A. D. N. Initiative. (2015). Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model. Statistical Methods in Medical Research, 24(4), 462-487. |
| [8] | Beesley, L. J., Bartlett, J. W., Wolf, G. T., & Taylor, J. M. (2016). Multiple imputation of missing covariates for the cox proportional hazards cure model. Statistics in Medicine, 35, 4701-4717. |
| [9] | Bernhardt, P. W., Wang, H. J., & Zhang, D. (2014). Flexible modeling of survival data with covariates subject to detection limits via multiple imputation. Computational Statistics & Data Analysis, 69, 81-91. · Zbl 1471.62028 |
| [10] | Breslow, N. E. (1972). Contribution to the discussion of the paper by Dr Cox. Journal of the Royal Statistical Society, Series B, 34(2), 216-217. |
| [11] | Buck, S. F. (1960). A method of estimation of missing values in multivariate data suitable for use with an electronic computer. Journal of the Royal Statistical Society. Series B (Methodological), 22, 302-306. · Zbl 0101.12701 |
| [12] | Buonaccorsi, J. P. (2010). Measurement error: Models, methods, and applications. Boca Raton, FL: CRC Press. · Zbl 1277.62014 |
| [13] | Carroll, R. J., Ruppert, D., Stefanski, L. A., & Crainiceanu, C. M. (2006). Measurement error in nonlinear models: A modern perspective, Boca Raton, FL: CRC press. · Zbl 1119.62063 |
| [14] | Chen, G., & Levy, D. (2016). Contributions of the Framingham heart study to the epidemiology of coronary heart disease. JAMA Cardiology, 1(7), 825-830. |
| [15] | Chen, Q., Wu, H., Ware, L. B., & Koyama, T. (2014). A Bayesian approach for the cox proportional hazards model with covariates subject to detection limit. International Journal of Statistics in Medical Research, 3(1), 32. |
| [16] | Cole, S. R., Chu, H., Nie, L., & Schisterman, E. F. (2009). Estimating the odds ratio when exposure has a limit of detection. International Journal of Epidemiology, 38(6), 1674-1680. |
| [17] | Collins, L. M., Schafer, J. L., & Kam, C.‐M. (2001). A comparison of inclusive and restrictive strategies in modern missing data procedures. Psychological Methods, 6(4), 330. |
| [18] | D’Angelo, G., & Weissfeld, L. (2008). An index approach for the cox model with left censored covariates. Statistics in Medicine, 27(22), 4502-4514. |
| [19] | Datta, S. (2005). Estimating the mean life time using right censored data. Statistical Methodology, 2(1), 65-69. · Zbl 1248.62177 |
| [20] | Dinse, G. E., Jusko, T. A., Ho, L. A., Annam, K., Graubard, B. I., Hertz‐Picciotto, I., Miller, F. W., Gillespie, B. W., & Weinberg, C. R. (2014). Accommodating measurements below a limit of detection: A novel application of cox regression, American Journal of Epidemiology, 179(8), 1018-1024. |
| [21] | Fitzsimons, G. J. (2008). Death to dichotomizing. Journal of Consumer Research, 35(1), 5-8. |
| [22] | Helsel, D. R. (2006). Fabricating data: How substituting values for nondetects can ruin results, and what can be done about it. Chemosphere, 65(11), 2434-2439. |
| [23] | Helsel, D. R. (2011). Statistics for censored environmental data using Minitab and R, Vol. 77. New York: John Wiley & Sons. · Zbl 1280.62004 |
| [24] | Helsel, D. R. (2005). Nondetects and data analysis. Statistics for censored environmental data. New York: Wiley‐Interscience. · Zbl 1058.62111 |
| [25] | Kalbfleisch, J. D., & Prentice, R. L. (2011). The statistical analysis of failure time data, Vol. 360. New York: John Wiley & Sons. · Zbl 0504.62096 |
| [26] | Klein, J. P., & Moeschberger, M. L. (2005). Survival analysis: Techniques for censored and truncated data. New York: Springer Science & Business Media. |
| [27] | Kong, S., & Nan, B. (2016). Semiparametric approach to regression with a covariate subject to a detection limit. Biometrika, 103(1), 161-174. · Zbl 1452.62539 |
| [28] | Langohr, K., Gómez, G., & Muga, R. (2004). A parametric survival model with an interval‐censored covariate. Statistics in Medicine, 23(20), 3159-3175. |
| [29] | Lee, J. S., Cole, S. R., Richardson, D. B., Dittmer, D. P., Miller, W. C., Moore, R. D., … Eron, J. J.Jr; Center for AIDS Research Network of Integrated Clinical Systems (2017). Incomplete viral suppression and mortality in hiv patients after antiretroviral therapy initiation. Aids, 31(14), 1989-1997. |
| [30] | Lee, S., Park, S., & Park, J. (2003). The proportional hazards regression with a censored covariate. Statistics & Probability Letters, 61(3), 309-319. · Zbl 1038.62093 |
| [31] | Lipsitz, S., Parzen, M., Natarajan, S., Ibrahim, J., & Fitzmaurice, G. (2004). Generalized linear models with a coarsened covariate. Journal of the Royal Statistical Society: Series C (Applied Statistics), 53(2), 279-292. · Zbl 1111.62314 |
| [32] | Little, R. J. (1992). Regression with missing x’s: A review. Journal of the American Statistical Association, 87(420), 1227-1237. |
| [33] | Meng, X.‐L. (1994). Multiple‐imputation inferences with uncongenial sources of input. Statistical Science, 9, 538-558. |
| [34] | Rathouz, P. J. (2007). Identifiability assumptions for missing covariate data in failure time regression models. Biostatistics, 8(2), 345-356. · Zbl 1144.62084 |
| [35] | Rigobon, R., & Stoker, T. M. (2009). Bias from censored regressors. Journal of Business & Economic Statistics, 27(3), 340-353. |
| [36] | Royston, P., Altman, D. G., & Sauerbrei, W. (2006). Dichotomizing continuous predictors in multiple regression: A bad idea. Statistics in Medicine, 25(1), 127-141. |
| [37] | Rubin, D. B. (2004). Multiple imputation for nonresponse in surveys, Vol. 81. New York: John Wiley & Sons. · Zbl 1070.62007 |
| [38] | Sattar, A., Sinha, S. K., & Morris, N. J. (2012). A parametric survival model when a covariate is subject to left‐censoring. Journal of Biometrics & Biostatistics, S3. |
| [39] | Sattar, A., Sinha, S. K., Wang, X.‐F., & Li, Y. (2015). Frailty models for pneumonia to death with a left‐censored covariate. Statistics in Medicine, 34(14), 2266-2280. |
| [40] | Schafer, J. L. (1999). Multiple imputation: A primer. Statistical Methods in Medical Research, 8(1), 3-15. |
| [41] | Tsimikas, J. V., Bantis, L. E., & Georgiou, S. D. (2012). Inference in generalized linear regression models with a censored covariate. Computational Statistics & Data Analysis, 56(6), 1854-1868. · Zbl 1368.62217 |
| [42] | Tu, Y.‐K., & Greenwood, D. C. (2012), Modern methods for epidemiology. New York: Springer Science & Business Media. |
| [43] | Wang, H. J., & Feng, X. (2012). Multiple imputation for m‐regression with censored covariates. Journal of the American Statistical Association, 107(497), 194-204. · Zbl 1261.62039 |
| [44] | White, I. R., & Royston, P. (2009). Imputing missing covariate values for the cox model. Statistics in Medicine, 28(15), 1982-1998. |
| [45] | Wu, H., Chen, Q., Ware, L. B., & Koyama, T. (2012). A Bayesian approach for generalized linear models with explanatory biomarker measurement variables subject to detection limit: An application to acute lung injury. Journal of Applied Statistics, 39(8), 1733-1747. · Zbl 1473.62377 |
| [46] | Yi, G. Y. (2017). Statistical analysis with measurement error or misclassification: Strategy, method and application, New York: Springer Science & Business Media. · Zbl 1377.62012 |
| [47] | Yue, Y. R., & Wang, X.‐F. (2016). Bayesian inference for generalized linear mixed models with predictors subject to detection limits: An approach that leverages information from auxiliary variables. Statistics in Medicine, 35(10), 1689-1705. |
| [48] | Zhou, H., & Pepe, M. S. (1995). Auxiliary covariate data in failure time regression. Biometrika, 82(1), 139-149. · Zbl 0823.62100 |
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