# 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.

• 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.