Modeling Frailty in Manufacturing Processes

  • James T. Wassell
  • Gregory W. Kulczycki
  • Ernest S. Moyer


The expected service life of respirator safety devices produced by different manufacturers is determined using frailty models to account for unobserved differences in manufacturing process and raw materials. The gamma and positive stable frailty distributions are used to obtain survival distribution estimates when the baseline hazard is assumed to be Weibull. Frailty distributions are compared using laboratory test data of the failure times for 104 respirator cartridges produced by 10 different manufacturers. Likelihood ratio tests results indicate that both frailty models provide a significant improvement over a Weibull model assuming independence. Results are compared to fixed effects approaches for analysis of this data.


Frailty Model Cumulative Hazard Function Gamma Frailty Frailty Distribution Weibull Regression 
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  1. 30 CFR part 11. “30 Mineral Resources.” Code of Federal Regulations Title 30, Part 11. 1980. pp 7–70.Google Scholar
  2. Chambers J.M., Mallows C.L., and Stuck B.W. (1976), “A method for simulating stableGoogle Scholar
  3. random variables,“ Journal of American Statistical Association, 71, 340–344.Google Scholar
  4. Clayton D.G. (1978), “A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic heart disease,” Biometrika, 65, 141–151.MathSciNetMATHCrossRefGoogle Scholar
  5. Costigan TM and JP Klein (1993), “Multivariate survival analysis based on frailty models,”Google Scholar
  6. Advances in Reliability,A.P. Basu Ed. 43–58.Google Scholar
  7. Elsevier Science Publishers B.V. Hougaard P. (1986a), “Survival models for heterogeneous populations derived from stable distributions,” Biometrika 73, 387–96.MathSciNetMATHCrossRefGoogle Scholar
  8. Hougaard P. (1986b), “A class of multivariate failure time distributions,” Biometrika 73, 671–8.MathSciNetMATHGoogle Scholar
  9. Klein J.P. and Moeschberger M.L. (1988), “Bounds on net survival probabilities for dependent competing risks,” Biometrics 44, 529–538.MathSciNetMATHCrossRefGoogle Scholar
  10. Moeschberger M.L. and Klein J.P. (1984), “Consequences of departures from independence in exponential series systems,” Technometrics 26, 277–284.MathSciNetCrossRefGoogle Scholar
  11. Moyer E.S., Peterson J.A., and Calvert C. (1994), “Evaluation of carbon tetrachloride replacement agents for use in testing non-powered organic vapor cartridges,” Submitted to Applied Occupational and Environmental Hygiene.Google Scholar
  12. Oakes D. (1989), “Bivariate survival models induced by frailties,” Journal of the American Statistical Association 84, 487–493.MathSciNetMATHCrossRefGoogle Scholar
  13. Wassell J.T. and Moeschberger M.L., (1993), “A bivariate survival model with modified gamma frailty for assessing the impact of interventions,” Statistics in Medicine 12, 241–248.CrossRefGoogle Scholar
  14. Whitmore G.A. and Lee M.T. (1991), “A multivariate survival distribution generated by an inverse gaussian mixture of exponentials,” Technometrics 33, 39–50.MathSciNetMATHCrossRefGoogle Scholar
  15. Wolfram S. (1991), Mathematica: a system for doing mathematics by computer, 2nd edition, 1991, Addison-Wesley Publishing Company, Inc.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • James T. Wassell
    • 1
  • Gregory W. Kulczycki
    • 1
  • Ernest S. Moyer
    • 1
  1. 1.National Institute for Occupational Safety and HealthMorgantownUSA

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