Advertisement

Copula Models for Dependent Censoring

  • Takeshi Emura
  • Yi-Hau Chen
Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

This chapter provides mathematical infrastructures for copula models, focusing on applications to survival analysis involving dependent censoring. After reviewing the concept of copulas, we introduce measures of dependence, including Kendall’s tau and the cross-ratio function. We also introduce the idea of residual dependence that explains how dependence between event times arises and how it can be modeled by copulas. Finally, we apply copulas for modeling the effect of dependent censoring and analyze the bias of the Cox regression analysis owing to dependent censoring.

Keywords

Archimedean copula Clayton’s copula Cox regression Cross-ratio function Gumbel’s copula Kendall’s tau Residual dependence Univariate Cox regression 

References

  1. Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65(1):141–151MathSciNetCrossRefzbMATHGoogle Scholar
  2. Day R, Bryant J, Lefkopoulou M (1997) Adaptation of bivariate frailty models for prediction, with application to biological markers as prognostic indicators. Biometrika 84(1):45–56MathSciNetCrossRefzbMATHGoogle Scholar
  3. Emura T, Chen YH (2016) Gene selection for survival data under dependent censoring, a copula-based approach. Stat Methods Med Res 25(6):2840–2857MathSciNetCrossRefGoogle Scholar
  4. Emura T, Nakatochi M, Murotani K, Rondeau V (2017a) A joint frailty-copula model between tumour progression and death for meta-analysis. Stat Methods Med Res 26(6):2649–2666MathSciNetCrossRefGoogle Scholar
  5. Emura T, Nakatochi M, Matsui S, Michimae H, Rondeau V (2017b) Personalized dynamic prediction of death according to tumour progression and high-dimensional genetic factors: meta-analysis with a joint model. Stat Methods Med Res,  https://doi.org/10.1177/0962280216688032
  6. Emura T, Pan CH (2017). Parametric likelihood inference and goodness-of-fit for dependently left-truncated data, a copula-based approach. Stat Pap,  https://doi.org/10.1007/s00362-017-0947-z
  7. Emura T, Wang W, Hung HN (2011) Semi-parametric inference for copula models for dependently truncated data. Stat Sinica 21:349–367zbMATHGoogle Scholar
  8. Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, New YorkzbMATHGoogle Scholar
  9. Frank MJ (1979) On the simultaneous associativity of f(x, y) and x + y - f(x, y). Aequationes Matbematicae 19:194–226MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gumbel EJ (I960). Distributions de valeurs extremes en plusieurs dimensions. PubL Inst Statist. Parids 9: 171–173Google Scholar
  11. Joe H (1993) Parametric families of multivariate distributions with given margins. J Multivar Anal 46:262–282MathSciNetCrossRefzbMATHGoogle Scholar
  12. Kalbfleisch JD, Prentice RL (2002) The statistical analysis of failure time data, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  13. Morgenstern D (1956) Einfache Beispiele zweidimensionaler Verteilungen. Mitteilungsblatt für Mathematishe Statistik. 8:234–235MathSciNetzbMATHGoogle Scholar
  14. Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  15. Oakes D (1989) Bivariate survival models induced by frailties. J Am Stat Assoc 84:487–493MathSciNetCrossRefzbMATHGoogle Scholar
  16. Rivest LP, Wells MT (2001) A martingale approach to the copula-graphic estimator for the survival function under dependent censoring. J Multivar Anal 79:138–155MathSciNetCrossRefzbMATHGoogle Scholar
  17. Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris. 8:229–231zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Takeshi Emura
    • 1
  • Yi-Hau Chen
    • 2
  1. 1.Graduate Institute of StatisticsNational Central UniversityTaoyuanTaiwan
  2. 2.Institute of Statistical ScienceAcademia SinicaTaipeiTaiwan

Personalised recommendations