One Arm Exponential Survival Sample Size and Power

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Lawless, Jerald F. Statistical Models and Methods for Lifetime Data. 2nd ed. John Wiley & Sons, 2003.


The formulas are based on the assumptions of uniform accrual over time, no loss to follow-up, exponentially distributed death times, and use of the exponential MLE test. The important normal approximation used in the sample size and power calculations is equation 4.1.5 (page 149) of Lawless:

$$ Z = \frac{\hat{\phi} - \phi}{I_1(\hat{\phi})^{-1/2}} \sim N(0,1) $$ Here, \(\phi\) is the cube root transformed hazard ratio under the null hypothesis, \(\hat{\phi}\) is the cube root transformed hazard ratio under the alternative, and \(I_1(\hat{\phi}) = 9r/\hat{\phi}^2\) (where \(r\) is the expected number of deaths) is the observed Fisher information. This leads to the following sample size formula: $$ n = \left( \frac{\sigma_0 \, z_{1-\alpha/s} + \sigma_1 \, z_{1-\beta}}{|\hat{\phi} - \phi|} \right)^2 + \frac{1}{2} $$ as well as the following power formula: $$ power = P \! \left( Z \leq \frac{\sigma_0 \, z_{\alpha/s} + |\hat{\phi} - \phi| \sqrt{n}}{\sigma_1} \right) $$ Here, \(\sigma_0\) and \(\sigma_1\) represent the null and alternative standard deviations, respectively, and \(z_p\) is the \(p\)th quantile of a standard normal distribution.

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