Survival analysis power calculations
Power may be evaluated for comparing hazard rates (per unit time) using this [attachment:coxpow.xls spreadsheet] which uses a simple formula taken from Schoenfeld (1983), Hsieh and Lavori (2000) and Collett (2003) corresponding to a group regression estimate (ratio of hazards) in a Cox regression model.
In particular from Schoenfeld (1983) the total number of events, d, required is
d = $$\frac{(z_text{a/2} + z_text{b})text{2}}{p(1-p)[log(hr)]text{2}}$$
Rearranging the above equation
Power = $$\Phi(\sqrt{dp(1-p)[log hr]^text{2}}-z_text{a/2})$$
where d is the total number of events, p the probability of occurrence of the event in the population, hr the hazard ratio, a the two-sided type I error, $$\Phi$$ the inverse normal function and z the Standard Normal (or probit) function.
Hsieh and Lavori (2000) further give sample size formulae for the number of deaths using continuous covariates in the Cox regression.
d = $$\frac{(z_text{a/2} + z_text{b})text{2}}{\sigmatext{2}\log(hr)^text{2}}
with $$\sigma^text{2}$$ equal to the variance of the covariate.
The ratio for a continuous covariate could be comparing rates at one sd above the mean to that at the mean.
dc = $$\frac{d}{1-Rtext{2}}$$ where $$Rtext{2}$$ is the squared multiple correlation regression of the covariate of interest with the others in the case of more than one continuous covariate. This method is computed using this [attachment:powcoxc.xls spreadsheet.]
This approach is similar to Hsieh's approach to computing power in logistic regression (see [:FAQ/power/llogPow:here].) This method may also be computed using the powerEpiCont function in R as illustrated [:FAQ/power/hazNR: here].
Alternatively the effect size can be expressed in terms of ratios of group survival rates as used by the power calculator given [http://www.stattools.net/SSizSurvival_Pgm.php here.] The free downloadable software [http://www.brixtonhealth.com/pepi4windows.html WINPEPI] also computes this sample size and power for comparing two survival functions.
References
Collett, D (2003) Modelling Survival Data in Medical Research. Second Edition. Chapman and Hall:London
Hsieh FY and Lavori PW (2000) [http://www.sciencedirect.com/science/article/pii/S0197245600001045 Sample size calculations for the Cox proportional hazards regression models with nonbinary covariates] Controlled Clinical Trials 21 552-560.
Schoenfeld DA (1983) Sample size formulae for the proportional hazards regression model. Biometrics 39 499-503.