## Conditional Power Calculator

Calculators are provided here to compute the conditional power given the interim results of a study for binomial, survival and normal endpoints.

There are three tabs to the calculator with the three kinds of endpoints. Each calculator has four boxes with calculators. The first box contains planned final sample size of the study. The sample size calculator adds no penalty for alpha spent as it assumes that there is no possibility of stopping for efficacy and a possibility for stopping for futility as a consequence of the conditional power assessment. There would be a penalty if one seeks to control the power as well and a penalty if there is a possibility of stopping for efficacy. Under such circumstances the sample size calculated (or number of events for the survival case) in the first box would be underestimated. When the sample size includes a penalty for early testing then one should overwrite the calculated sample size (or number of events in the survival case) with the one which accounts for the penalty for early looks. This will provide better results in the boxes that follow. The calculator is accurate when there is no alpha spending penalty (no early stopping for efficacy) to the sample size and is a close approximation to the case where there is an alpha spending penalty.

For the binomial calculator we have, as a default in the first box, a study seeking to establish with 80% power (100-Beta =20%) that ARM A has a superior response rate to ARM B using a two sided test for difference in proportions at a significance level (alpha) of 5% when anticipated responses in the two arms are 60% and 45%. With these inputs we would need 171 patients per arm and the Z-score test statistic is expected to be about 2.802.

The second box in the calculator allows us to enter the interim results. Presuming we get 37 responders in 60 patients in ARM A and 31 patients in 52 in ARM B, the interim test statistic and something called a B-value work out to 1.307 and 0.781 respectively. The B-value (computed as root of the information fraction at the interim multiplied by the interim test statistic), using developments in Proschan, Lan and Wittes (2006), can be used to get the projected distribution of the final test statistic and through that the conditional power given the interim results.

The third box in the calculator calculates the conditional power under three assumptions. Under the first assumption there is a null effect after the interim analysis or subsequent data is assumed to reflect equality of the two groups. Under the second assumption the differential effects in the two arms after the interim are expected to be the same as that before the interim. Under the third assumption we continue to have faith in the a-priori hypothesized differences (the 60% vs 45% assumed earlier) for the period after the interim despite the interim data. One would do this if the interim analysis is too early leading to early erratic patterns possibly due to incomplete training and learning of those conducting the study. Under each of these assumptions the projected test statistic is calculated and the corresponding conditional power is obtained. For our interim data the conditional power (the probability that we will see a statistically significant result when we finish the study) is 7.07%, 61.06% and 78.02% under null effects, continued trend effects and hypothesized effects respectively after the interim. Also calculated here is the information fraction which goes from 0 for the start of the study to 1 for the finish. The interim data entered computes as a 38.5% completion of the study at interim.

The fourth box in the calculator calculates the Bayesian predictive power. Proschan, Lan and Wittes (2006) recommend a prior with a mean consistent with the alternate hypothesis and a standard deviation chosen to weight the alternate hypothesis appropriately as compared to the empirical data. This mixes the empirical effect with the hypothesized effect. For a weight of 20% on the hypothesized effect we have a predictive power 63.21%. The choice of a weight inversely proportional to the information fraction is interesting as that emphasizes the empirical data as we get closer to the end of the study. Under this weight the predictive power is 72.85%.

The other two calculator tabs are similar. For the survival calculator, the calculator computes an approximate interim test statistic based on the interim number of events and the hazard ratio. The user could compute their Z-statistic based on the log rank U statistic divided by it’s standard error and enter that in the third cell of the interim results box instead of going with the calculator’s estimate. For the survival calculator invert the hazard ratio when necessary in order to provide a hazard ratio greater than 1.0. In the normal calculator set the group with the larger mean as ARM A.

For details on the calculations see the following attachment.

**Edit the blue cells in the spreadsheet and enter your data and the calculations in the bottom box of the spreadsheet will refresh. If you appreciate our efforts and would like to make a voluntary contribution of any amount to help us maintain and improve this site please do so at this link.**

Contours of constant LP Norms.