Bayesian Interim Analysis of Weibull Regression Models with Gamma Frailty

  • George D. Papandonatos
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 114)


This paper considers the problem of planning prospective clinical studies where the primary endpoint is a terminal event and the response variable is a survival time. It is assumed that the lifetimes of the individuals in the study display extra-Weibull variability that causes the usual proportional hazards assumption to fail. The introduction of a Gamma-distributed frailty term to accommodate the between-subject heterogeneity leads to a logarithmic F accelerated failure time model to which the second-order expansions of Papandonatos & Geisser [1] can be applied. The predictive simulation approach of Papandonatos & Geisser [2] can then be used to evaluate the length of the study period needed for a Bayesian hypothesis testing procedure to achieve a conclusive result.

Key words

Bayesian Inference Stochastic Curtailment Weibull Frailty 


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  1. [1]
    G.D. Papandonatos And S. Geisser, Laplace Approximations for Censored Regression Models, Canad. J. Statist., 25 (1997), pp. 337–358.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    G.D. Papandonatos And S. Geisser, Bayesian Interim Analysis of Lifetime Data, Canad. J. Statist., to appear.Google Scholar
  3. [3]
    P. Armitage, Discussion of the paper by Jennison and Turnbull, J. R. Statist. Soc. B, 51 (1989), pp. 333–335.Google Scholar
  4. [4]
    D.R. Cox, Regression models and life tables (with discussion), J. R. Statist. Soc. B, 34 (1972), pp. 187–220.zbMATHGoogle Scholar
  5. [5]
    J.D. Kalbfleisch And R.L. Prentice, The Statistical Analysis of Failure Time Data, Wiley, New York (1980).zbMATHGoogle Scholar
  6. [6]
    N. Reid, A Conversation with Sir David Cox, Statist. Sci., 9 (1994), pp. 439–455.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    A. Ciampi, S.A. Hogg And L. Kates, Regression analysis of censored survival data with the generalized.F family-an alternative to the proportional hazards model, Statist. in Med., 5 (1986), pp. 85–96.CrossRefGoogle Scholar
  8. [8]
    D.R. Cox And D. Oakes, Analysis of Survival Data, Chapman & Hall, London (1984).Google Scholar
  9. [9]
    J. Neyman And E.L. Scott, Consistent estimates based on partially consistent observations, Econometrica, 16 (1948), pp. 1–32.MathSciNetCrossRefGoogle Scholar
  10. [10]
    C. Elbers And G. Ridder, True and spurious duration dependence: The identifiability of the proportional hazard model, Review of Economic Studies, 49 (1982), pp. 403–410.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    T. Lancaster, The econometric analysis of transition data, Econometric Society Monographs, Cambridge University Press, Cambridge (1992).Google Scholar
  12. [12]
    J.L. Hodges, Jr. And E.L. Lehmann,Testing the approximate validity of statistical hypotheses,J. R. Statist. Soc. B,16(1954),pp.261–268MathSciNetzbMATHGoogle Scholar
  13. [13]
    S. Geisser, Predictive Inference: An Introduction, Chapman & Hall, London (1993).zbMATHGoogle Scholar
  14. [14]
    S. Geisser, On the curtailment of sampling, Canad. J. Statist., 20 (1992), pp. 297–309.MathSciNetzbMATHGoogle Scholar
  15. [15]
    S. Geisser, Bayesian interim analysis of censored exponential observations, Statistics & Probability Letters, 18 (1993), pp. 163–168.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    D.J. Spiegelhalter And L.S. Freedman, Bayesian approaches to Clinical Trials, Bayesian Statistics 3, J.M. Bernardo, M.H. DeGroot, D.V. Lindley and A.F.M. Smith (eds.), Clarendon Press, Oxford (1988), pp. 453–477.Google Scholar
  17. [17]
    D.J. Spiegelhalter, L.S. Freedman And M.K.B. Parmar, Applying Bayesian ideas in drug development and clinical trials, Statist. in Med., 12 (1993), pp. 1501–1511.CrossRefGoogle Scholar
  18. [18]
    D.J. Spiegelhalter, L.S. Freedman And M.K.B. Parmar, Bayesian approaches to randomized trials, J. R. Statist. Soc. A, 157 (1994), pp. 357–416.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    S.L. Zeger, K.-Y. Liang And P.S. Albert, Models for Longitudinal Data: A Generalized Estimating Equation Approach,Biometrics, 44 (1988), pp. 1049–1060.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    P.K. Andersen, O. Borgan, R.D. Gill And N. Keiding, Statistical Models Based on Counting Processes, Springer-Verlag, New York (1993).zbMATHCrossRefGoogle Scholar
  21. [21]
    N. Keiding, P.K. Andersen And J.P. Klein, The role of frailty and accelerated failure time models in describing heterogeneity due to omitted covariates, Statist. in Med., 16 (1997), pp. 215–224.CrossRefGoogle Scholar
  22. [22]
    S.K. Bar-Lev And P. Enis, Reproducibility and natural exponential families with power variance functions, Ann. Statist., 14 (1986), pp. 1507–1522.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    R.L. Prentice, Discrimination among some parametric models, Biometrika, 61 (1975), 607–614.MathSciNetCrossRefGoogle Scholar
  24. [24]
    H. Jeffreys, Theory of Probability, Clarendon Press, Oxford (1961).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • George D. Papandonatos
    • 1
  1. 1.Department of StatisticsState University of New York at BuffaloBuffaloUSA

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