One Arm Survival Sample Size and Power

Program Code

The program is written in JavaScript.

References

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

Description

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.

Input Items

The user is prompted to enter values for the following items. For items that have initial default values set, the values are given in parentheses.

Output items


Note: If this page does not display properly, you may need to add https://cdn.mathjax.org to your browser's list of trusted sites.