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| 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. | 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. |
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| d = $$[ (z(a/2) + z(b) )^2 ^] / [\sigma^2 ^\log(hr)^2 ^] | d = $$[ (z(a/2) + z(b) )^2 ^] / [\sigma^2 ^ log(hr)^2 ^] |
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| with $$\sigma^text{2}$$ equal to the variance of the covariate. | with $$\sigma^2 ^$$ \mbox{equal to the variance of the covariate}. |
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| 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. | 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}. |
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| 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. | 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]] 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. |
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| Collett, D (2003) Modelling Survival Data in Medical Research. Second Edition. Chapman and Hall:London | [[attachment:collett-ch10.pdf | Collett, D (2003) Modelling Survival Data in Medical Research. Second Edition. Chapman and Hall:London]] |
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| [[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. | [[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. |
Survival analysis power calculations
Power may be evaluated for comparing hazard rates (per unit time) using this spreadsheet which uses a simple formula taken from Schoenfeld (1983), Hsieh and Lavori (2000) and 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 = $$[ (z(a/2) + z(b) )2 ] / [\sigma2 log(hr)2 ]
with $$\sigma2 $$ \mbox{equal to the variance of the covariate}.
The ratio for a continuous covariate could be comparing rates at one sd above the mean to that at the mean.
dc = $$\frac{d}{1-R^text{2}}$$ where $$R^text{2}$$ is the squared multiple correlation regression of the covariate of interest with the others in the case of more than one continuous covariate. This method is computed using this spreadsheet. The same equation for computing the power above, for a binary covariate, is used with $$\sigma2 $$ \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 here.) This method may also be computed using the powerEpiCont function in R as illustrated here.
Alternatively the effect size can be expressed in terms of ratios of group survival rates as used by the power calculators given here and here which uses results from Machin et al. (1997, 2009). The free downloadable software WINPEPI also computes this sample size and power for comparing two survival functions.
References
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. A downloaded pdf of this paper is 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.