Joint Modeling of Survival and Nonignorable Missing Longitudinal Quality-of-Life Data

  • Jean-François Dupuy


A problem that frequently arises in clinical trials where quality-of-life values are repeatedly measured on each individual under study, is that of dropout. In some settings, dropout may depend on unobserved components of the longitudinal process. Dropout is then termed nonignorable. Recently, several approaches have been proposed that accommodate nonignorable dropout in the modeling of a longitudinal process. However, most of these approaches rest upon the assumption that dropout can only occur at one of the pre-specified measurement times of quality-of-life. In this paper, we review some of these approaches and we propose a new joint model for longitudinal data with nonignorable dropout and time to dropout, which allows dropout to occur at any point in time. This model combines a first-order Markov model for the longitudinal quality-of-life data with a time-dependent Cox model for the dropout process. We discuss nonparametric maximum likelihood estimation in the suggested joint model and rely on an EM algorithm to calculate estimates of the parameters. The method is illustrated through analysis of quality-of-life data among patients involved in a cancer clinical trial. Comparison of its results to the ones obtained by fitting Diggle and Kenward’s model for nonignorable dropout to the same data is provided.


Frailty Model Nonparametric Maximum Likelihood Longitudinal Process Nonparametric Maximum Likelihood Estimator Dropout Model 
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  1. Chen, H.Y. and Little, R.J.A. (1999). Proportional hazards regression with missing covariates. Journal of the American Statistical Association 94, 896–908.CrossRefGoogle Scholar
  2. Cox, D.R. (1972). Regression models and life-tables (with discussion). Journal of the Royal Statistical Society, Series B 34, 187–220.Google Scholar
  3. DeGruttola, V. and Tu, X.M. (1994). Modeling the progression of CD4-lymphocyte count and its relationship to survival time. Biometrics 50, 1003–1014.CrossRefGoogle Scholar
  4. Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B 39, 1–38.Google Scholar
  5. Diggle, R.J. and Kenward, M.G. (1994). Informative dropout in longitudinal data analysis (with discussion). Applied Statistics 43, 49–93.CrossRefGoogle Scholar
  6. Dupuy, J.-F., Grama, I. and Mesbah, M. (2001). Identifiability in a Cox model with nonignorable missing time-dependent covariate. Technical Report SABRES/ University of South-Brittany, France.Google Scholar
  7. Dupuy, J.-F. and Mesbah, M. (2002). Joint modelling of event time and nonignorable missing longitudinal data. Lifetime Data Analysis (to appear).Google Scholar
  8. Heutte, N. and Huber-Carol, C. (2002). Semi-Markov models for quality of life data with censoring. In: Mesbah, M., Cole, B.F. and Lee, M.-L. (eds.), Statistical Methods for Quality of Life Studies: Design, Methods and Analysis. Boston: Kluwer.Google Scholar
  9. Hogan, J.W. and Laird, N.M. (1997a). Mixture models for the joint distribution of repeated measures and event times. Statistics in Medicine 16, 239–257.PubMedCrossRefGoogle Scholar
  10. Hogan, J.W. and Laird, N.M. (1997b). Model-based approaches to analysing incomplete longitudinal and failure time data. Statistics in Medicine 16, 259–272.PubMedCrossRefGoogle Scholar
  11. Hu, P., Tsiatis, A.A. and Davidian, M. (1998). Estimating the parameters in the Cox model when covariate variables are measured with error. Biometrics 54, 1407–1419.PubMedCrossRefGoogle Scholar
  12. Li, Y. and Lin, X. (2000). Covariate measurement errors in frailty models for clustered survival data. Biometrika 87, 849–866.CrossRefGoogle Scholar
  13. Little, R.J.A. and Rubin, D.B. (1987). Statistical Analysis with Missing Data. New York: Wiley.Google Scholar
  14. Little, R.J.A. (1995). Modeling the dropout mechanism in repeated-measures studies. Journal of the American Statistical Association 90, 1112–1121.CrossRefGoogle Scholar
  15. Molenberghs, G., Kenward, M.G. and Lesaffre, E. (1997). The analysis of longitudinal ordinal data with nonrandom dropout. Biometrika 84, 33–44.CrossRefGoogle Scholar
  16. Neider, J.A. and Mead, R. (1965). A simplex method for function minimisation. The Computer Journal 7, 303–313.CrossRefGoogle Scholar
  17. Nielsen, G.G., Gill, R.D., Andersen, P.K. and Sørensen, T.I.A. (1992). A counting process approach to maximum likelihood estimation in frailty models. Scandinavian Journal of Statistics 19, 25–43.Google Scholar
  18. Ribaudo, H.J., Thompson, S.G. and Allen-Mersh, T.G. (2000). A joint analysis of quality of life and survival using a random effect selection model, Statistics in Medicine 19, 3237–3250.PubMedCrossRefGoogle Scholar
  19. Scharfstein, D.O., Rotnitzky, A. and Robins, J.M. (1999). Adjusting for nonig-norable drop-out using semiparametric nonresponse models. Journal of the American Statistical Association 94, 1096–1120.CrossRefGoogle Scholar
  20. Schluchter, M.D. (1992). Methods for the analysis of informatively censored longitudinal data. Statistics in Medicine 11, 1861–1870.PubMedCrossRefGoogle Scholar
  21. Scilab Group (1998). Introduction to Scilab, User’s Guide. INRIA Meta2 Project/ ENPC Cergrene.Google Scholar
  22. Smith, D.M., Robertson, W.H. and Diggle, P.J. (1996). Oswald: object-oriented software for the analysis of longitudinal data in S. Technical Report MA 96/192, Department of Mathematics and Statistics, University of Lancaster, LAI 4YF, United Kingdom.Google Scholar
  23. Troxel, A.B., Harrington, D.P. and Lipsitz, S.R. (1998). Analysis of longitudinal data with non-ignorable non-monotone missing values. Applied Statistics 47, 425–438.Google Scholar
  24. Troxel, A.B., Lipsitz, S.R. and Harrington, D.P. (1998). Marginal models for the analysis of longitudinal measurements with nonignorable non-monotone missing data. Biometrika 85, 671–672.CrossRefGoogle Scholar
  25. Verbeke G. and Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. New York: Springer-Verlag.Google Scholar
  26. Wu, M.C. and Carroll, R.J. (1988). Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process. Biometrics 44, 175–188.CrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2002

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  • Jean-François Dupuy

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