Survival data hazard rate

Each trial that is to be included in the meta-analysis will provide a measure of treatment effect (difference). For continuous data this could be the mean response on the active treatment minus the mean response in the placebo arm. Alternatively, for binary data the treatment effect could be captured by the difference in the cure rates, for example, or by the odds ratio. For survival data, the hazard ratio would often be the measure of treatment difference, but equally well it could be the difference in the two-year survival rates. 

Power calculations for survival data are more complex due to the nature of the analyses as well as factors that are involved in the accrual of participants (i.e., follow-up time, prevalence of risk factor, etc.). The following example is based on the method discussed by Simon and Altman (41) using an 18-month overall survival rate of 40%, two-sided alpha level of 5%, and no attrition for varying levels of risk factor prevalence and hazard ratios. 

The initial idea is to use the differential equations of a probabilistic transfer model with hazard rates varying with the age of the molecules, i.e., to enlarge the limiting hypothesis (9.2). The objective is to find nonexponential families of survival distributions that are mathematically tractable and yet sufficiently flexible to fit the observed data. In the simplest case, the differential equation (9.7) links hazard rates and survival distributions. Nevertheless, this relation was at the origin of an erroneous use of the hazard function. In fact, substituting in this relation the age a by the exogenous time t, we obtain... 

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... 

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