Survival data hazard ratio

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. 

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. 

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).

This hypothesis has now been supported in a study of over 7000 patients who were discharged after a first admission for cardiovascular disease between 1989 and 1997 and who took low-dose aspirin and survived for at least 1 month (29). The adjusted hazard ratios for allcause mortality (HR = 1.93 95% Cl = 1.30, 287) and for cardiovascular mortality (HR = 1.73 Cl = 1.05, 2.84) were significantly raised in patients who took ibuprofen (mean dose 1210 mg/day) in addition to the aspirin. There was no increase in hazard in patients who combined aspirin with diclofenac, which is consistent with the in vitro data. However, this study had many limitations (30), and further epidemiological studies are needed to address this potentially important interaction. In the meantime, when patients taking low-dose aspirin for car-dioprotection also require long-term treatment with an NSAID, diclofenac would be preferable to ibuprofen. 

A practical challenge of Bayesian meta-analysis for rare AE data is that noninformative priors may lead to convergence failure due to very sparse data. Weakly informative priors may be used to solve this issue. In the example of the previous Bayesian meta-analysis with piecewise exponential survival models, the following priors for log hazard ratio (HR) (see Table 14.1) were considered. Prior 1 assumes a nonzero treatment effect with a mean log(HR) of 0.7 and a standard deviation of 2. This roughly translates to that the 95% confidence interval (Cl) of HR is between 0.04 and 110, with an estimate of HR to be 2.0. Prior 2 assumes a 0 treatment effect, with a mean log(HR) of 0 and a standard deviation of 2. This roughly translates to the assumption that we are 95% sure that the HR for treatment effect is between 0.02 and 55, with an estimate of the mean hazard of 1.0. Prior 3 assumes a nonzero treatment effect that is more informative than that of Prior 1, with a mean log(HR) of 0.7 and a standard deviation of 0.7. This roughly translates to the assumption that we are 95% sure that the HR... 

---