The Proportional Hazards Model

Let T be the random variable the time until death of something. Suppose its density is given by the exponential distribution 

The probability of death by time t is given by the cumulative distribution function (CDF) of the random variable and is 

The survival Junction is the probability of surviving to time t and is given by 

The hazard Junction gives the instantaneous probability of death at time t given survival up until time t. It is given by 

when time until death follows the exponential distribution, the hazard function will be constant. We will only use this exponential model for survival times. Survival models with non-constant hazard functions are discussed in McCullagh and Nelder (1989). 

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The most popular method for analysis of covariance is the proportional hazards model. This model, originally developed by Cox (1972), is now used extensively in the analysis of survival data to incorporate and adjust for both centre and covariate effects. The model assumes that the hazard ratio for the treatment effect is constant. 

The proportional hazards model, as the name suggests, assumes that the hazard ratio is a constant. As such it provides a direct extension of the logrank test, which is a simple two treatment group comparison. Indeed if the proportional hazards model is fitted to data without the inclusion of baseline factors then the p-value for the test Hg c = 0 will be essentially the same as the p-value arising out of the logrank test. 

Cox regression. Another name for the proportional hazards model originally proposed by Sir David Cox in 1972. 

Samrout, M., Chatelet, E., Kouta, R. Chebbo, N. 2009. Optimization of maintenance policy using the proportional hazard model. Reliability Engineering and System Safety 94, 44-52. 

Finally, let us mention the proportional hazard model for software vulnerability developed similarly as described by Pham (2003) for software reliability. 

The literature offers several methods and models to analyze survival data under the circumstance of different usage conditions. Cox (1972) introduced the Proportional Hazards Model (PHM) to allow for the influence of covariates. This approach is one of the most widely used, particularly in survival analysis. Bendell et al. (1991) discuss the application of this model especially in the context of reliability data. Case studies involving the PHM in the area of reliability were done by Kumar et al. (1992), Bendell et al. (1986) and Jardine et al. (1989). Alternatives to the PHM can be found in Oakes (1995), Kordonsky Gertsbakh (1997) and Duchesne (1999) which follow the idea of multiple times scales to take more than one explanatory variable into account. Another possibility arises if there is some knowledge about the physical-failure mechanism. This approach is usually used with acceler-... 

Kumar D., Klefsjo, B., Kuma 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. 

Vlok, R, Coetzee, J., Banjevic, D., Jardine, A. Makis, V 2002. Optimal component replacement decisions using vibration monitoring and the proportional-hazards model. Journal of the operational research society, 193-202. 

The observations of the censoring variable Wi come from the Poisson distribution, a member of the exponential family. The logarithm of the parameter /r, is linked to the linear predictor rj. The observations are independent. Clearly the proportional hazards model is a generalized linear model and can be analyzed the same way as the Poisson regression model. 

The proportional hazards model is used to regress censored survival data on when the values of a set of predictor variables are known for each observation. For each observation we record f j, its time of death, Wi the censoring variable which equals 1 if we observe the true death ti, and 0 when the observation is censored and all we know the true lifetime is greater than ti, and the values of the predictors Xji,..., Xip for that observation. 

When we have eensored survival times data, and we relate the linear predictor to the hazard function we have the proportional hazards model. The function BayesCPH draws a random sample from the posterior distribution for the proportional hazards model. First, the function finds an approximate normal likelihood function for the proportional hazards model. The (multivariate) normal likelihood matches the mean to the maximum likelihood estimator found using iteratively reweighted least squares. Details of this are found in Myers et al. (2002) and Jennrich (1995). The covariance matrix is found that matches the curvature of the likelihood function at its maximum. The approximate normal posterior by applying the usual normal updating formulas with a normal conjugate prior. If we used this as the candidate distribution, it may be that the tails of true posterior are heavier than the candidate distribution. This would mean that the accepted values would not be a sample from the true posterior because the tails would not be adequately represented. Assuming that y is the Poisson censored response vector, time is time, and x is a vector of covariates then... 

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