A review on reliability models with covariates

  • Nima Gorjian
  • Lin Ma
  • Murthy Mittinty
  • Prasad Yarlagadda
  • Yong Sun


Modern Engineering Asset Management (EAM) requires the accurate assessment of current and the prediction of future asset health condition. Suitable mathematical models that are capable of predicting Time-to-Failure (TTF) and the probability of failure in future time are essential. In traditional reliability models, the lifetime of assets is estimated using failure time data. However, in most real-life situations and industry applications, the lifetime of assets is influenced by different risk factors, which are called covariates. The fundamental notion in reliability theory is the failure time of a system and its covariates. These covariates change stochastically and may influence and/or indicate the failure time. Research shows that many statistical models have been developed to estimate the hazard of assets or individuals with covariates. An extensive amount of literature on hazard models with covariates (also termed covariate models), including theory and practical applications, has emerged. This paper is a state-of-the-art review of the existing literature on these covariate models in both the reliability and biomedical fields. One of the major purposes of this expository paper is to synthesise these models from both industrial reliability and biomedical fields and then contextually group them into non-parametric and semiparametric models. Comments on their merits and limitations are also presented. Another main purpose of this paper is to comprehensively review and summarise the current research on the development of the covariate models so as to facilitate the application of more covariate modelling techniques into prognostics and asset health management.


Failure Time Covariate Model Baseline Hazard Function Accelerate Failure Time Model Failure Time Data 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Cox DR. (1972) Regression models and life-tables. Journal of Royal Statistical Society, 34(2), 187-220.MATHGoogle Scholar
  2. 2.
    Kumar D & KlefsjÖ B. (1994) Proportional hazards model: a review. Reliability Engineering & System Safety, 44(2), 177-188.CrossRefGoogle Scholar
  3. 3.
    Kumar D & Westberg U. (1996) Some reliability models for analysing the effects of operating conditions. International Journal of Reliability, Quality and Safety Engineering, 4(2), 133-148.CrossRefGoogle Scholar
  4. 4.
    Ma L. (2007) Condition monitoring in engineering asset management. APVC. p. 16.Google Scholar
  5. 5.
    Ma Z & Krings AW. (2008) Survival analysis approach to reliability, survivability and prognostics and health management. IEEE Aerospace Conference. pp. 1-20.Google Scholar
  6. 6.
    Liao H. (2004) Degradation models and design of accelerated degradation testing plans. United States -- New Jersey: Rutgers The State University of New Jersey - New Brunswick.Google Scholar
  7. 7.
    Luxhoj JT & Shyur H-J. (1997) Comparison of proportional hazards models and neural networks for reliability estimation. Intelligent Manufacturing, 8(3), 227-234.CrossRefGoogle Scholar
  8. 8.
    Cox DR & Oakes D. (1984) Analysis of survival data. London ; ; New York: Chapman and Hall.Google Scholar
  9. 9.
    Crowder MJ. (1991) Statistical analysis of reliability data (1st ed.). London ; ; New York: Chapman & Hall.Google Scholar
  10. 10.
    Kumar D & Klefsjoe B. (1994) Proportional hazards model-an application to power supply cables of electric mine loaders. International Journal of Reliability, Quality and Safety Engineering, 1(3), 337-352.CrossRefGoogle Scholar
  11. 11.
    Landers TL & Kolarik WJ. (1986) Proportional hazards models and MIL-HDBK-217. Microelectronics and Reliability, 26(4), 763-772.CrossRefGoogle Scholar
  12. 12.
    Kalbfleisch JD & Prentice RL. (2002) The statistical analysis of failure time data (Second ed.). New Jersey: Wiley.MATHGoogle Scholar
  13. 13.
    Cox DR. (1975) Partial likelihood. Biometrika, 62(2), 269-276.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kalbfleisch JD & Prentice RL. (1973) Marginal likelihoods based on Cox's regression and life model. Biometrika, 60(2), 267-278.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Booker J, Campbell K, Goldman AG, Johnson ME & Bryson MC. (1981) Applications of Cox's proportional hazards model to light water reactor component failure data (No. LA-8834-SR; Other: ON: DE81023991).Google Scholar
  16. 16.
    Breslow N. (1974) Covariance analysis of censored survival data. Biometrics, 30(1), 89-99.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Arjas E. (1988) A graphical method for assessing goodness of fit in Cox's proportional hazards model. American Statistical Association, 83(401), 204-212.CrossRefGoogle Scholar
  18. 18.
    Bendell A, Walley M, Wightman DW & Wood LM. (1986) Proportional hazards modelling in reliability analysis - an application to brake discs on high speed trains. Quality and Reliability Engineering International, 2(1), 45-52.CrossRefGoogle Scholar
  19. 19.
    Dale CJ. (1985) Application of the proportional hazards model in the reliability field. Reliability Engineering, 10(1), 1-14.CrossRefGoogle Scholar
  20. 20.
    Ansell RO & Ansell JI. (1987) Modelling the reliability of sodium sulphur cells. Reliability engineering, 17(2), 127-137.CrossRefGoogle Scholar
  21. 21.
    Jardine AKS, Anderson PM & Mann DS. (1987) Application of the Weibull proportional hazads model to aircraft and marine engine failure data. Quality and Reliability Engineering International, 3(2), 77-82.CrossRefGoogle Scholar
  22. 22.
    Landers TL & Kolarik WJ. (1987) Proportional hazards analysis of field warranty data. Reliability engineering, 18(2), 131-139.CrossRefGoogle Scholar
  23. 23.
    Baxter MJ, Bendell A, Manning PT & Ryan SG. (1988) Proportional hazards modelling of transmission equipment failures. Reliability Engineering and System Safety, 21, 129–144.CrossRefGoogle Scholar
  24. 24.
    Chan CK. (1990) A proportional hazards approach to correlate SiO2-breakdown voltage and time distributions. IEEE Transactions on Reliability, 39(2), 147-150.MATHCrossRefGoogle Scholar
  25. 25.
    Drury MR, Walker EV, Wightman DW & Bendell A. (1988) Proportional hazards modelling in the analysis of computer systems reliability. Reliability Engineering and System Safety, 21, 197-214.CrossRefGoogle Scholar
  26. 26.
    Jardine AKS, Ralston P, Reid N & Stafford J. (1989) Proportional hazards analysis of diesel engine failure data. Quality and Reliability Engineering International 5(3), 207-16.CrossRefGoogle Scholar
  27. 27.
    Leitao ALF & Newton DW. (1989) Proportional hazards modelling of aircraft cargo door complaints. Quality and Reliability Engineering International, 5(3), 229-238.CrossRefGoogle Scholar
  28. 28.
    Mazzuchi TA & Soyer R. (1989) Assessment of machine tool reliability using a proportional hazards model. Naval Research Logistics, 36(6), 765-777.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Mazzuchi TA, Soyer R & Spring RV. (1989) The proportional hazards model in reliability. Reliability and Maintainability Symposium, 1989. Proceedings., Annual. pp. 252-256.Google Scholar
  30. 30.
    Elsayed EA & Chan CK. (1990) Estimation of thin-oxide reliability using proportional hazards models. IEEE Transactions on Reliability 39(3), 329-335.CrossRefGoogle Scholar
  31. 31.
    Kumar D. (1995) Proportional hazards modelling of repairable systems. Quality and Reliability Engineering International, 11(5), 361-369.CrossRefGoogle Scholar
  32. 32.
    Kumar D, KlefsjÖ B & Kumar U. (1992) Reliability analysis of power transmission cables of electric mine loaders using the proportional hazards model. Reliability Engineering & System Safety, 37(3), 217-222.CrossRefGoogle Scholar
  33. 33.
    Love CE & Guo R. (1991) Using proportional hazard modelling in plant maintenance. Quality and Reliability Engineering International, 7(1), 7-17.CrossRefGoogle Scholar
  34. 34.
    Love CE & Guo R. (1991) Application of weibull proportional hazards modelling to bad-as-old failure data. Quality and Reliability Engineering International, 7(3), 149-157.CrossRefGoogle Scholar
  35. 35.
    Pettitt AN & Daud IB. (1990) Investigating time dependence in Cox's proportional hazards model. Applied Statistics, 39(3), 313-329.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Gasmi S, Love CE & Kahle W. (2003) A general repair, proportional-hazards, framework to model complex repairable systems. IEEE Transactions on Reliability, 52(1), 26-32.CrossRefGoogle Scholar
  37. 37.
    Jardine AKS, Banjevic D, Wiseman M, Buck S & Joseph T. (2001) Optimizing a mine haul truck wheel motors' condition monitoring program: Use of proportional hazards modeling. Quality in Maintenance Engineering, 7(4), 286.CrossRefGoogle Scholar
  38. 38.
    Krivtsov VV, Tananko DE & Davis TP. (2002) Regression approach to tire reliability analysis. Reliability Engineering & System Safety, 78(3), 267-273.CrossRefGoogle Scholar
  39. 39.
    Martorell S, Sanchez A & Serradell V. (1999) Age-dependent reliability model considering effects of maintenance and working conditions. Reliability Engineering & System Safety, 64(1), 19-31.CrossRefGoogle Scholar
  40. 40.
    Park S. (2004) Identifying the hazard characteristics of pipes in water distribution systems by using the proportional hazards model: 2. Applications. KSCE Journal of Civil Engineering, 8(6), 669-677.CrossRefGoogle Scholar
  41. 41.
    Prasad PVN & Rao KRM. (2002) Reliability models of repairable systems considering the effect of operating conditions. Annual Reliability and Maintainability Symposium pp. 503-510.Google Scholar
  42. 42.
    Prasad PVN & Rao KRM. (2003) Failure analysis and replacement strategies of distribution transformers using proportional hazard modeling. Annual Reliability and Maintainability Symposium pp. 523-527.Google Scholar
  43. 43.
    Wallace JM, Mavris DN & Schrage DP. (2004) System reliability assessment using covariate theory. Annual Symposium Reliability and Maintainability pp. 18-24.Google Scholar
  44. 44.
    Kumar D & Westberg U. (1996) Proportional hazards modeling of time-dependent covariates using linear regression: a case study. IEEE Transactions on Reliability, 45(3), 386-392.CrossRefGoogle Scholar
  45. 45.
    Kay R. (1977) Proportional hazard regression models and the analysis of censored survival data. Applied Statistics, 26(3), 227-237.CrossRefGoogle Scholar
  46. 46.
    Anderson JA & Senthilselvan A. (1982) A two-step regression model for hazard functions. Applied Statistics, 31(1), 44-51.CrossRefGoogle Scholar
  47. 47.
    Mau J. (1986) On a graphical method for the detection of time-dependent effects of covariates in survival data. Applied Statistics, 35(3), 245-255.CrossRefMathSciNetGoogle Scholar
  48. 48.
    Pijnenburg M. (1991) Additive hazards models in repairable systems reliability. Reliability Engineering and System Safety, 31(3), 369-390.CrossRefGoogle Scholar
  49. 49.
    Wightman D & Bendell T. (1995) Comparison of proportional hazards modeling, additive hazards modeling and proportional intensity modeling when applied to repairable system reliability. International Journal of Reliability, Quality and Safety Engineering, 2(1), 23-34.CrossRefGoogle Scholar
  50. 50.
    Newby M. (1994) Why no additive hazards models? IEEE Transactions on Reliability, 43(3), 484-488.CrossRefGoogle Scholar
  51. 51.
    Newby M. (1992) A critical look at some point-process models for repairable systems. IMA Journal of Management Mathematics, 4(4), 375-394.MATHCrossRefGoogle Scholar
  52. 52.
    Andersen PK & Væth M. (1989) Simple parametric and nonparametric models for excess and relative mortality. Biometrics, 45(2), 523-535.MATHCrossRefGoogle Scholar
  53. 53.
    Badía FG, Berrade MD & Campos CA. (2002) Aging properties of the additive and proportional hazard mixing models. Reliability Engineering & System Safety, 78(2), 165-172.CrossRefGoogle Scholar
  54. 54.
    Lin DY & Ying Z. (1995) Semiparametric analysis of general additive-multiplicative hazard models for counting processes. The Annals of Statistics, 23(5), 1712-1734.MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Shyur H-J, Elsayed EA & Luxhøj JT. (1999) A general model for accelerated life testing with time-dependent covariates. Naval Research Logistics, 46(3), 303-321.MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Lin Z & Fei H. (1991) A nonparametric approach to progressive stress accelerated life testing. IEEE Transactions on Reliability 40(2), 173-176.MATHCrossRefGoogle Scholar
  57. 57.
    Mettas A. (2000) Modeling and analysis for multiple stress-type accelerated life data. Annual Reliability and Maintainability Symposium. pp. 138-143.Google Scholar
  58. 58.
    Newby M. (1988) Accelerated failure time models for reliability data analysis. Reliability Engineering & System Safety, 20(3), 187-197.CrossRefGoogle Scholar
  59. 59.
    Ciampi A & Etezadi-Amoli J. (1985) A general model for testing the proportional hazards and the accelerated failure time hypotheses in the analysis of censored survival data with covariates. Communications in Statistics - Theory and Methods, 14(3), 651 - 667.CrossRefGoogle Scholar
  60. 60.
    Etezadi-Amoli J & Ciampi A. (1987) Extended Hazard Regression for Censored Survival Data with Covariates: A Spline Approximation for the Baseline Hazard Function. Biometrics, 43(1), 181-192.MATHCrossRefGoogle Scholar
  61. 61.
    Landers TL & Soroudi HE. (1991) Robustness of a semi-parametric proportional intensity model. IEEE Transactions on Reliability, 40(2), 161-164.MATHCrossRefGoogle Scholar
  62. 62.
    Lugtigheid D, Jardine AKS & Jiang X. (2007) Optimizing the performance of a repairable system under a maintenance and repair contract. Quality and Reliability Engineering International, 23(8), 943-960.CrossRefGoogle Scholar
  63. 63.
    Vlok P-J, Wnek M & Zygmunt M. (2004) Utilising statistical residual life estimates of bearings to quantify the influence of preventive maintenance actions. Mechanical Systems and Signal Processing, 18(4), 833-847.CrossRefGoogle Scholar
  64. 64.
    Jiang S-T, Landers TL & Rhoads TR. (2006) Proportional intensity models robustness with overhaul intervals. Quality and Reliability Engineering International, 22(3), 251-263.CrossRefGoogle Scholar
  65. 65.
    Prentice RL, Williams BJ & Peterson AV. (1981) On the regression analysis of multivariate failure time data. Biometrika, 68(2), 373-379.MATHCrossRefMathSciNetGoogle Scholar
  66. 66.
    Percy DF, Kobbacy KAH & Ascher HE. (1998) Using proportional-intensities models to schedule preventivemaintenance intervals. IMA J Management Math, 9(3), 289-302.MATHCrossRefGoogle Scholar
  67. 67.
    McCullagh P. (1980) Regression models for ordinal data. Journal of the Royal Statistical Society. Series B (Methodological), 42(2), 109-142.MATHMathSciNetGoogle Scholar
  68. 68.
    Bennett S. (1983) Analysis of survival data by the proportional odds model. Statistics in Medicine, 2(2), 273-277.CrossRefGoogle Scholar
  69. 69.
    Bennett S. (1983) Log-logistic regression models for survival data. Applied Statistics, 32(2), 165-171.CrossRefGoogle Scholar
  70. 70.
    Sun Y, Ma L, Mathew J, Wang W & Zhang S. (2006) Mechanical systems hazard estimation using condition monitoring. Mechanical Systems and Signal Processing, 20(5), 1189-1201.CrossRefGoogle Scholar
  71. 71.
    Sun Y & Ma L. (2007) Notes on "mechanical systems hazard estimation using condition monitoring"--Response to the letter to the editor by Daming Lin and Murray Wiseman. Mechanical Systems and Signal Processing, 21(7), 2950-2955.CrossRefMathSciNetGoogle Scholar
  72. 72.
    Kobbacy KAH, Fawzi BB, Percy DF & Ascher HE. (1997) A full history proportional hazards model for preventive maintenance scheduling. Quality and Reliability Engineering International, 13(4), 187-198.CrossRefGoogle Scholar
  73. 73.
    Jardine AKS. (1983) Component and system replacement decisions. J.K S (Ed.). Image Sequence Processing and Dynamic Scene Analysis. pp. 647-654.Google Scholar
  74. 74.
    Jozwiak IJ. (1997) An introduction to the studies of reliability of systems using the Weibull proportional hazards model. Microelectronics and Reliability, 37(6), 915-918.CrossRefGoogle Scholar
  75. 75.
    Jardine AKS & Anders M. (1985) Use of concomitant variables for reliability estimation. Maintenance Management International, 5, 135-140.Google Scholar
  76. 76.
    Jardine AKS, Banjevic D & Makis V. (1997) Optimal replacement policy and the structure of software for conditionbased maintenance. Quality in Maintenance Engineering, 3(2), 1355-2511.Google Scholar
  77. 77.
    Jardine AKS, Joseph T & Banjevic D. (1999) Optimizing condition-based maintenance decisions for equipment subject to vibration monitoring. Quality in Maintenance Engineering, 5(3), 192-202.CrossRefGoogle Scholar
  78. 78.
    Liao H, Zhao W & Guo H. (2006) Predicting remaining useful life of an individual unit using proportional hazards model and logistic regression model. Annual Reliability and Maintainability Symposium pp. 127-132.Google Scholar
  79. 79.
    McKeague IW. (1986) Estimation for a semimartingale regression model using the method of sieves. The Annals of Statistics, 14(2), 579-589.MATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    Aalen OO. (1989) A linear regression model for the analysis of life times. Statistics in Medicine 8(8), 907-925.CrossRefGoogle Scholar
  81. 81.
    Mau J. (1988) A comparison of counting process models for complicated life histories. Applied Stochastic Models and Data Analysis, 4, 283–298.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Nima Gorjian
    • 1
    • 2
  • Lin Ma
    • 1
    • 2
  • Murthy Mittinty
    • 3
  • Prasad Yarlagadda
    • 2
  • Yong Sun
    • 1
    • 2
  1. 1.Cooperative Research Centre for Integrated Engineering Asset Management (CIEAM)BrisbaneAustralia
  2. 2.School of Engineering SystemsQueensland University of Technology (QUT)BrisbaneAustralia
  3. 3.School of Mathematical SciencesQueensland University of Technology (QUT)BrisbaneAustralia

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