= 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.]

Rearranging the equation given in Collett(2003)

Power = $$2 \Phi(\sqrt{np(1-p)hr^text{2}}-z_text{a/2})-1$$

where ''n'' is the total sample size, ''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{1-a} + z_text{1-b})^text{2}}{\sigma^text{2}\hr^text{2}}

with $$\sigma^text{2}$$ equal to the variance of the covariate.

dc = $$\frac{dc}{1-R^text{2}}$$ where $$R^text{2}$$ is the squared multiple correlation regression one covaraite on the others.

__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.