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| Total sample size may be evaluated for comparing hazard rates (per unit time) using this [attachment:coxsamp.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.] | 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 Collett (2003) corresponding to a group regression estimate (ratio of hazards) in a Cox regression model. |
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| In particular Collett(2003) gives the total sample size, d, required as | The ratio for a continuous covariate could be comparing rates at one sd above the mean to that at the mean. |
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| d = $$\frac{[z(\alpha/2) + z(\beta/2)]^text{2}}{p(1-p)log(hr)^text{2}}$$ | In particular from Schoenfeld (1983) the total number of events, d, required is |
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| for two-sided type I error, $$\alpha$$, power 1-$$\beta$$, event rate in population p and hazard ratio, hr. | d = $$\frac{(z_text{a/2} + z_text{b})^text{2}}{p(1-p)[log(hr)]^text{2}}$$ |
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| __Reference__ | 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. |
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| Collett, D (2003) Modelling Survival Data in Medical Research Second Edition. Chapman and Hall:London | Hsieh and Lavori (2000) further give sample size formulae for the number of deaths using continuous covariates in the Cox regression. dc = $$\frac{(z_text{a/2} + z_text{b})^text{2}}{\sigma^text{2}\[log(hr)]^text{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.] 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.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 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. Schoenfeld DA (1983) Sample size formulae for the proportional hazards regression model. ''Biometrics'' '''39''' 499-503. |
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 Collett (2003) corresponding to a group regression estimate (ratio of hazards) in a Cox regression model.
The ratio for a continuous covariate could be comparing rates at one sd above the mean to that at the mean.
In particular from Schoenfeld (1983) the total number of events, d, required is
d = $$\frac{(z_text{a/2} + z_text{b})text{2}}{p(1-p)[log(hr)]text{2}}$$
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 = $$\frac{(z_text{a/2} + z_text{b})text{2}}{\sigmatext{2}\[log(hr)]^text{2}}
with $$\sigma^text{2}$$ equal to the variance of the covariate.
dc2 = $$\frac{dc}{1-Rtext{2}}$$ where $$Rtext{2}$$ 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.]
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.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 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.
Schoenfeld DA (1983) Sample size formulae for the proportional hazards regression model. Biometrics 39 499-503.