The Annals of Applied Statistics

Backward estimation of stochastic processes with failure events as time origins

Kwun Chuen Gary Chan and Mei-Cheng Wang

Full-text: Open access

Abstract

Stochastic processes often exhibit sudden systematic changes in pattern a short time before certain failure events. Examples include increase in medical costs before death and decrease in CD4 counts before AIDS diagnosis. To study such terminal behavior of stochastic processes, a natural and direct way is to align the processes using failure events as time origins. This paper studies backward stochastic processes counting time backward from failure events, and proposes one-sample nonparametric estimation of the mean of backward processes when follow-up is subject to left truncation and right censoring. We will discuss benefits of including prevalent cohort data to enlarge the identifiable region and large sample properties of the proposed estimator with related extensions. A SEER–Medicare linked data set is used to illustrate the proposed methodologies.

Article information

Source
Ann. Appl. Stat., Volume 4, Number 3 (2010), 1602-1620.

Dates
First available in Project Euclid: 18 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1287409388

Digital Object Identifier
doi:10.1214/09-AOAS319

Mathematical Reviews number (MathSciNet)
MR2758343

Zentralblatt MATH identifier
1202.62108

Keywords
Marked process left truncation prevalent cohort recurrent event process recurrent marker process survival analysis

Citation

Chan, Kwun Chuen Gary; Wang, Mei-Cheng. Backward estimation of stochastic processes with failure events as time origins. Ann. Appl. Stat. 4 (2010), no. 3, 1602--1620. doi:10.1214/09-AOAS319. https://projecteuclid.org/euclid.aoas/1287409388


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References

  • Bang, H. and Tsiatis, A. A. (2000). Estimating medical costs with censored data. Biometrika 87 329–343.
  • Bilias, Y., Gu, M. and Ying, Z. (1997). Towards a general asymptotic theory for Cox model with staggered entry. Ann. Statist. 25 662–682.
  • Chan, I. S. F., Neaton, J. D., Saravolatz, L. D., Crane, L. R. and Osterberger, J. (1995). Frequencies of opportunistic diseases prior to death among HIV-infected persons. Aids 9 1145–1151.
  • Cook, R. J. and Lawless, J. F. (1997). Marginal analysis of recurrent events and a terminating event. Stat. Med. 16 911–924.
  • Ghosh, D. and Lin, D. Y. (2000). Nonparametric analysis of recurrent events and death. Biometrics 56 554–562.
  • Gross, S. T. and Lai, T. L. (1996). Nonparametric estimation and regression analysis with left-truncated and right-censored data. J. Amer. Statist. Assoc. 91 1166–1180.
  • Huang, Y. and Louis, T. A. (1998). Nonparametric estimation of the joint distribution of survival time and mark variables. Biometrika 85 785–798.
  • Lai, T. L. and Ying, Z. (1991). Estimating a distribution function with truncated and censored data. Ann. Statist. 19 417–442.
  • Lawless, J. F. and Nadeau, C. (1995). Some simple robust methods for the analysis of recurrent events. Technometrics 37 158–168.
  • Lin, D. Y. (2000). Proportional means regression for censored medical costs. Biometrics 56 775–778.
  • Lin, D. Y., Fleming, T. R. and Wei, L. J. (1994). Confidence bands for survival curves under the proportional hazards model. Biometrika 81 73–81.
  • Lin, D. Y., Feuer, E. J., Etzioni, R. and Wax, Y. (1997). Estimating medical costs from incomplete follow-up data. Biometrics 53 419–434.
  • Lin, D. Y., Wei, L. J., Yang, I. and Ying, Z. (2000). Semiparametric regression for the mean and rate functions of recurrent events. J. Roy. Statist. Soc. Ser. B Stat. Methodol. 62 711–730.
  • Nelson, W. (1988). Graphical analysis of system repair data. Journal of Quality Technology 20 24–35.
  • Pawitan, Y. and Self, S. (1993). Modeling disease marker processes in AIDS. J. Amer. Statist. Assoc. 88 719–726.
  • Pepe, M. S. and Cai, J. (1993). Some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates. J. Amer. Statist. Assoc. 88 811–820.
  • Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hayward, CA.
  • Strawderman, R. L. (2000). Estimating the mean of an increasing stochastic process at a censored stopping time. J. Amer. Statist. Assoc. 95.
  • Tsai, W.-Y., Jewell, N. P. and Wang, M.-C. (1987). A note on the product-limit estimator under right censoring and left truncation. Biometrika 74 883–886.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.
  • Wang, M.-C. (1991). Nonparametric estimation from cross-sectional survival data. J. Amer. Statist. Assoc. 86 130–143.
  • Wang, M.-C. and Chiang, C. T. (2002). Non-parametric methods for recurrent event data with informative and non-informative censorings. Stat. Med. 21 445–456.
  • Wang, M.-C., Qin, J. and Chiang, C. T. (2001). Analyzing recurrent event data with informative censoring. J. Amer. Statist. Assoc. 96 1057–1065.
  • Warren, J. L., Klabunde, C. N., Schrag, D., Bach, P. B. and Riley, G. F. (2002). Overview of the SEER–Medicare data: Content, research applications, and generalizability to the United States elderly population. Med. Care 40 3–18.
  • Woodroofe, M. (1985). Estimating a distribution function with truncated data. Ann. Statist. 13 163–177.
  • Zhao, H. and Tian, L. (2001). On estimating medical cost and incremental cost-effectiveness ratios with censored data. Biometrics 57 1002–1008.