Survival data proportional hazards model

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 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-... 

Proportional Hazards Model Likelihood for Censored Survival Data... 

One of the advantages of using the parametric proportional hazards model is that estimated survival curves for individuals given their predictor values can be calculated from the data. In this model, each individual has his/her own constant hazard rate. Let hi be the hazard rate for individual i. For that individual, the time until death follows the exponential distribution given by Equation 9.3 with parameter A = /i. From Equation 9.4, the survival function for individual i is given by... 

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... 

Figure 1 The relative 6-year mortality hazard ratios are shown for reported usual sleep hr from 2-3 hr/night to 10 or more hr/night, relative to 1.0 assigned to the hazard for 7 hr/night as the reference standard. The solid line with 95% confidence interval bars shows results from a 32-covariate Cox proportional hazards survival model, as reported previously (3). The dotted lines show data from models that excluded subjects who were not initially healthy, i.e., who died within the first year or whose questionnaires reported any cancer, heart disease, stroke, chronic bronchitis, emphysema, asthma, or current illness (a yes answer to the question are you sick at the present time ). The dot-dash lines with X symbols show models controlling only for age, insomnia, and use of sleeping pills. Data were from 635,317 women and 478,619 men. The thin solid lines with diamonds show the percent of subjects with each reported sleep duration (right axis).

---