One Arm Exponential 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.

Length of the accrual period
Length of the follow-up period, i.e. the time from end of accrual to analysis
$\alpha$, the significance level (0.05)
One-sided or two-sided test (1)
$M_0$ and $M_a$, the median survival times for the null and alternative hypotheses, respectively. The outcome is assumed to be exponentially distributed. Note, there is a simple connection between median survival ($M$) and hazard rate ($\lambda$) for exponential data. The hazard rate is computed as follows: $$ \lambda = - \frac{\log(0.5)}{M} $$
Survival probabilities for the null and alternative at time $t$. The outcome is assumed to be exponentially distributed. Note, there is a simple connection between survival probabilities and hazard rates for exponential data. The hazard rate is computed as follows: $$ \lambda = - \frac{\log(S(t))}{t} $$ where $S(t)$ is the survivor function at time $t$.
$1 - \beta$, the desired power for accrual rate estimation (0.9), or $n$, the sample size for power estimation.

Output items

Sample size or power, depending on the selected calculation.
Upper and lower critical values for either median survival or for survival probabilities from the parametric exponential model. Wider intervals are needed for estimates based on the Kaplan-Meier estimate of the survival function.

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