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| Power = $$2 \Phi(\sqrt{dp(1-p)hr^text{2}}-z_text{a/2})-1$$ | Power = $$\Phi(\sqrt{dp(1-p)hr^text{2}}-z_text{a/2})$$ |
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| d = $$\frac{(z_text{a/2} + z_text{b/2})^text{2}}{\sigma^text{2}\log(hr)^text{2}} | d = $$\frac{(z_text{a/2} + z_text{b})^text{2}}{\sigma^text{2}\log(hr)^text{2}} |
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 Collett (2003) [http://stats.stackexchange.com/questions/7508/power-analysis-for-survival-analysis illustrated here] corresponding to a group regression estimate (ratio of hazards) in a Cox regression model. 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 ratio for a continuous covariate could be comparing rates at one sd above the mean to that at the mean.
Rearranging the equation given in Collett(2003)
Power = $$\Phi(\sqrt{dp(1-p)hr^text{2}}-z_text{a/2})$$
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) 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}$$ equal to the variance of the covariate.
dc = $$\frac{dc}{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].
References
Collett, D (2003) Modelling Survival Data in Medical Research. Second Edition. Chapman and Hall:London
Hsieh FY and Lavori PW (2000) Sample size calculations for the Cox proportional hazards regression models with nonbinary covariates Controlled Clinical Trials 21 552-560.