Covariates survival data

In the next section we will discuss Kaplan-Meier curves, which are used both to display the data and also to enable the calculation of summary statistics. We will then cover the logrank and Gehan-Wilcoxon tests which are simple two group comparisons for censored survival data (akin to the unpaired t-test), and then extend these ideas to incorporate centre effects and also allow the inclusion of baseline covariates. 

In Chapter 6 we covered methods for adjusted analyses and analysis of covariance in relation to continuous (ANOVA and ANCOVA) and binary and ordinal data (CMH tests and logistic regression). Similar methods exist for survival data. As with these earlier methods, particularly in relation to binary and ordinal data, there are numerous advantages in accounting for such factors in the analysis. If the randomisation has been stratified, then such factors should be incorporated into the analysis in order to preserve the properties of the resultant p-values. 

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-... 

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).

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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