= Survival analysis sample size calculations =

The total number of events may be evaluated for comparing hazard rates (per unit time) using this [[attachment:coxsamp.xls|spreadsheet]] which uses a simple formula taken from Schoenfeld (1983), Hsieh and Lavori (2000) and [[attachment:power.pdf | 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 = $$[ ( z(a/2) + z(b) )^2 ^] / [ p(1-p)[log(hr)]^2 ^ ]$$

for a two-sided type I error, ''a'', power ''1-b'', event rate in population ''p'', hazard ratio, ''hr'' 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.

dc = $$[ (z(a/2) + z(b) )^2 ^ ][\sigma^2 ^\[log(hr)]^2 ^]

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

dc2 = $$\frac{dc}{1-R^text{2}}$$ 
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:coxcN.xls|spreadsheet.]] The ratio for a continuous covariate could be comparing rates at one sd above the mean to that at the mean.

This approach is similar to Hsieh's approaches to sample size calculations for the odds ratio in a 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.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 survival curves.


__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. A downloaded pdf of this paper is [[attachment:powersurvival.pdf | here.]]

Machin D, Campbell M, Fayers, P, 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, 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.