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| Rearranging the equation given in Collett(2003) Power = $$2 \Phi(\sqrt{dp(1-p)hr^text{2}}-z_text{a/2})-1$$ 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. |
The ratio for a continuous covariate could be comparing rates at one sd above the mean to that at the mean. |
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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}} |
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Rearranging the above equation 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. |
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.
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}$$ equal to the variance of the covariate.
Rearranging the above equation
Power = $$\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$$ the inverse normal function and z the Standard Normal (or probit) function.
dc = $$\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.