= 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 [[attachment:powersa.pdf | Collett (2003, Chapter 10)]] 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 $$\sigma^2 ^$$ \mbox{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 $$\sigma^2 ^$$ \mbox{ 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 calculators given  [[http://powerandsamplesize.com/Calculators/Test-Time-To-Event-Data/Cox-PH-2-Sided-Equality | here]], which has R code as given below, and [[http://www.statstodo.com/SSizSurvival_Pgm.php|here]] which uses results from Machin et al. (1997, 2009). The free downloadable software [[http://www.brixtonhealth.com/pepi4windows.html|WINPEPI]] also computes this sample size and power for comparing two survival functions.

The R code mentioned above for Machin et al. example taken from [[http://powerandsamplesize.com/Calculators/Test-Time-To-Event-Data/Cox-PH-2-Sided-Equality | here]] where hr is the hazard ratio, hr0 is 1 (hr under the null hypothesis), pE is the overall probability of the event occurring within the study period, pA is the proportions of the sample size allotted to group 'A', alpha is type I error and beta is 1-power.

{{{
hr=2
hr0=1
pE=0.8
pA=0.5
alpha=0.05
beta=0.20
(n=((qnorm(1-alpha/2)+qnorm(1-beta))/(log(hr)-log(hr0)))^2/(pA*(1-pA)*pE))
ceiling(n) # 82
(Power=pnorm((log(hr)-log(hr0))*sqrt(n*pA*(1-pA)*pE)-qnorm(1-alpha/2)))
}}}

For the Collett example above we have hr=0.5729, pE=0.495, pA=0.5, alpha=0.05 and beta=0.1 which gives a total sample size of 274 (n) using the R code above which agrees with the sample size worked out by Collett in his example.

__References__

[[attachment:collett-ch10.pdf | 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. A downloaded pdf of this paper is [[attachment:hazsurv.pdf | here.]]

Machin D, Campbell M, Fayers, P and Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 176-177. 

Machin D, Campbell MJ, Tan SB and Tan SH (2009) Sample size tables for clinical studies. 3rd ed. Chichester: Wiley-Blackwell. 

Schoenfeld DA (1983) Sample size formulae for the proportional hazards regression model. ''Biometrics'' '''39''' 499-503.