Journal of the Operational Research Society (2009) 60, 1621--1636  2009 Operational Research Society Ltd. All rights reserved. 0160-5682/09 www.palgrave-journals.com/jors/ Estimating parameters of proportional hazards model based on expert knowledge and statistical data A Zuashkiani ∗ , D Banjevic and AKS Jardine University of Toronto, Toronto, Ontario, Canada Proportional hazards model (PHM) is a convenient statistical tool that can be successfully applied in industrial problems, such as in accelerated life testing and condition-based maintenance, or in biomedical sciences. Estimation of PHM requires lifetime data, as well as condition monitoring data, which often is incomplete or missing, and necessitates the use of expert knowledge to compensate for it. This paper describes the methodology for elicitation of expert’s beliefs and experience necessary to estimate the parameters of a PHM with time-dependent covariates. The paper gives a background of PHM and review of the literature related to the knowledge elicitation problem and gives a foundation for the proposed methodology. The knowledge elicitation process is based on case analyses and comparisons. This method results in a set of inequalities, which in turn define a feasible space for the parameters of the PHM. By sampling from the feasible space an empirical prior distribution of the parameters can be estimated. Then, using Bayes rule and statistical data the posterior distribution can be obtained. This technique can also provide reliable outcomes when no statistical data are available. The technique has been tested several times in laboratory experiments and in a real industrial case and has shown promising results. Journal of the Operational Research Society (2009) 60, 1621 – 1636. doi:10.1057/jors.2008.119 Published online 19 November 2008 Keywords: condition-based maintenance; expert knowledge; proportional hazards model; Bayesian statistics Introduction and motivation The present research was motivated by collaboration with the steel and mining industries where it was necessary to estimate the proportional hazards model (PHM) in cases with little, incomplete or no data. The main purpose of the collaboration was to utilize PHM for maintenance decisions depending on condition monitoring (CM) variables. With some adjustments, the presented methodology can be used in other fields where PHM is used extensively, such as for lifetime data analysis in biomedical sciences. There has been an increasing demand for condition-based maintenance (CBM) techniques in industry due to the impor- tance of reliability and productivity in today’s industrial envi- ronment. CBM is typically applied to critical and complex equipment by taking into account monitoring diagnostic vari- ables (such as metal particles in the engine circulating oil, vibration intensity, temperature, etc, and age of the system; Collacott, 1977) and factors that can influence the state/health of the system. A common approach in CBM is to quan- tify or qualify imminent risk of failure in a short interval of time, depending on conditions and measurements. The ‘risk of failure’ is often not clearly defined, but is well understood and widely used by the practitioners. Some describe risk as ∗ Correspondence: A Zuashkiani, Apt. #PH207, 35 Finch Avenue East, North York, Ontario, Canada M2N 6Z8. E-mail: ali.zuashkiani@utoronto.ca the ‘catastrophic potential’ (Fischoff et al, 2000), others as the ‘possibility of loss and injury’, and quantitatively as the ‘probability of such loss’ (for a discussion, see Kaplan and Garrick, 1981). In our case we will consider risk as the ‘failure potential’, and quantify it by probability. A convenient statis- tical tool for quantification of risk of failure is the hazard rate function. For CBM, it is of great importance to have a model for the hazard function that depends on conditions and measurements. One of the most popular models is the PHM, introduced by Cox in the early 1970s (Cox, 1972), due to its relative simplicity and good flexibility. For an early review of PHM and its applications, see Kumar and Klefsj¨ o (1994) and for further discussion and more recent applications in CBM, see Jardine and Banjevic (2005). In this paper, we are partic- ularly interested in the Weibull PHM with time-dependent stochastic covariates. A practical difficulty in developing advanced statistical models is their requirement for large data sets in order to produce a reliable outcome. In the absence of such data or before enough data are collected, a methodology that can make use of information such as experts’ knowledge is needed. The structure of the Bayesian method allows the incorporation of both experts’ knowledge and collected data in model building. When the model becomes complicated, and also when it deals with not directly observable quantities, as is the case with PHM, it becomes increasingly difficult to extract experts’ knowledge in form of prior probability