Modeling left-truncated and right-censored survival data with longitudinal covariates. (English) Zbl 1257.62114

Summary: There is a surge in medical follow-up studies that include longitudinal covariates in the modeling of survival data. So far, the focus has been largely on right-censored survival data. We consider survival data that are subject to both left truncation and right censoring. Left truncation is well known to produce biased sample. The sampling bias issue has been resolved in the literature for the case which involves baseline or time-varying covariates that are observable. The problem remains open, however, for the important case where longitudinal covariates are present in survival models. A joint likelihood approach has been shown in the literature to provide an effective way to overcome those difficulties for right-censored data, but this approach faces substantial additional challenges in the presence of left truncation.
Here we thus propose an alternative likelihood to overcome these difficulties and show that the regression coefficient in the survival component can be estimated unbiasedly and efficiently. Issues about the bias for the longitudinal component are discussed. The new approach is illustrated numerically through simulations and data from a multi-center AIDS cohort study.


62P10 Applications of statistics to biology and medical sciences; meta analysis
62N01 Censored data models
62G05 Nonparametric estimation
92C50 Medical applications (general)
62N02 Estimation in survival analysis and censored data
62E20 Asymptotic distribution theory in statistics
65C05 Monte Carlo methods
65C60 Computational problems in statistics (MSC2010)
Full Text: DOI arXiv Euclid


[1] Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models Based on Counting Processes. Springer Series in Statistics . Springer, New York. · Zbl 0769.62061
[2] Cox, D. R. (1972). Regression models and life-tables (with discussion). J. Roy. Statist. Soc. Ser. B 34 187-220. · Zbl 0243.62041
[3] Dafni, U. G. and Tsiatis, A. A. (1998). Evaluating surrogate markers of clinical outcome when measured with error. Biometrics 54 1445-1462. · Zbl 1058.62597 · doi:10.2307/2533670
[4] DeGruttola, V. and Tu, X. (1994). Modeling progression of CD4-lymphocyte count and its relationship to survival time. Biometrics 50 1003-1014. · Zbl 0825.62792 · doi:10.2307/2533439
[5] Dupuy, J.-F., Grama, I. and Mesbah, M. (2006). Asymptotic theory for the Cox model with missing time-dependent covariate. Ann. Statist. 34 903-924. · Zbl 1092.62100 · doi:10.1214/009053606000000038
[6] Henderson, R., Diggle, P. and Dobson, A. (2000). Joint modelling of longitudinal measurements and event time data. Biostatistics 1 465-480. · Zbl 1089.62519 · doi:10.1093/biostatistics/1.4.465
[7] Hsieh, F., Tseng, Y.-K. and Wang, J.-L. (2006). Joint modeling of survival and longitudinal data: Likelihood approach revisited. Biometrics 62 1037-1043. · Zbl 1116.62105 · doi:10.1111/j.1541-0420.2006.00570.x
[8] Klein, J. P. and Moeschberger, M. L. (2003). Survival Analysis : Techniques for Censored and Truncated Data . Springer, New York. · Zbl 1011.62106
[9] Laird, N. M. and Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics 38 963-974. · Zbl 0512.62107 · doi:10.2307/2529876
[10] Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 226-233. · Zbl 0488.62018
[11] Lynden-Bell, D. (1971). A method of allowing for known observational selection in small samples applied to 3CR quasars. Monthly Notices of the Royal Astronomy Society 155 95-118.
[12] Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95 449-485. · Zbl 0995.62033 · doi:10.2307/2669386
[13] Rezza, G., Lazzarin, A., Angarano, G., Sinicco, A., Pristerá, R., Tirelli, U., Salassa, B., Ricchi, E., Aiuti, F. and Menniti-lppolito, F. (1989). Tje natural history of HIV infection in intravenous drug users: Risk of disease progression in a cohort of serconverters. AIDS 3 87-90.
[14] Song, X., Davidian, M. and Tsiatis, A. A. (2002). A semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data. Biometrics 58 742-753. · Zbl 1210.62132 · doi:10.1111/j.0006-341X.2002.00742.x
[15] The-Italian-Seroconversion-Study (1992). Disease progression and early predictors of AIDS in HIV-seroconverted injecting drug users. AIDS 6 421-426.
[16] Tseng, Y.-K., Hsieh, F. and Wang, J.-L. (2005). Joint modelling of accelerated failure time and longitudinal data. Biometrika 92 587-603. · Zbl 1152.62380 · doi:10.1093/biomet/92.3.587
[17] Tsiatis, A. A. and Davidian, M. (2004). Joint modeling of longitudinal and time-to-event data: An overview. Statist. Sinica 14 809-834. · Zbl 1073.62087
[18] Tsiatis, A. A., DeGruttola, V. and Wulfsohn, M. (1995). Modeling the relationship of survival to longitudinal data measured with error: Applications to survival and CD4 counts in patients with AIDS. J. Amer. Statist. Assoc. 90 23-37. · Zbl 0818.62102 · doi:10.2307/2291126
[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
[20] Vardi, Y. (1985). Empirical distributions in selection bias models. Ann. Statist. 13 178-205. · Zbl 0578.62047 · doi:10.1214/aos/1176346585
[21] Wang, M.-C. (1987). Product limit estimates: A generalized maximum likelihood study. Comm. Statist. Theory Methods 16 3117-3132. · Zbl 0651.62027 · doi:10.1080/03610928708829561
[22] Wang, C. Y. (2006). Corrected score estimator for joint modeling of longitudinal and failure time data. Statist. Sinica 16 235-253. · Zbl 1087.62112
[23] Woodroofe, M. (1985). Estimating a distribution function with truncated data. Ann. Statist. 13 163-177. · Zbl 0574.62040 · doi:10.1214/aos/1176346584
[24] Wulfsohn, M. S. and Tsiatis, A. A. (1997). A joint model for survival and longitudinal data measured with error. Biometrics 53 330-339. · Zbl 0874.62140 · doi:10.2307/2533118
[25] Zeng, D. and Cai, J. (2005). Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time. Ann. Statist. 33 2132-2163. · Zbl 1086.62034 · doi:10.1214/009053605000000480
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