= 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 = $$[ ( z(a/2) + z(b) )^2 ^ ] / [ p(1-p)[log(hr)]^2 ^ ]$$

Rearranging the above equation 

Power = $$\Phi(\sqrt{dp(1-p)[log hr]^2 ^-z(a/2))$$

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 = $$[ (z(a/2) + z(b) )^2 ^] / [\sigma^2 ^\log(hr)^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 = $$\frac{d}{1-R^text{2}}$$ 
where $$R^text{2}$$ 
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.]] The same equation for computing the power above, for a binary covariate, is used with $$\sigma^text{2}$$ replacing p(1-p) as the variance of the covariate in the denominator.

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.