Stochastic Curtailment |
Lan, Simon, and Halperin (1982) introduce stochastic curtailment to stop a trial if, given current data, it is likely to predict the outcome of the trial with high probability. That is, a trial can be stopped to reject the null hypothesis if, given current data in the analyses, the conditional probability of rejecting under at the end of the trial is greater than , where the constant should be between and and values of or are recommended (Jennison and Turnbull 2000, p. 206). Similarly, a trial can be stopped to accept the null hypothesis if, given current data in the analyses, the conditional probability of rejecting under the alternative hypothesis at the end of the trial is less than .
The following two approaches for stochastic curtailment are available in the SEQTEST procedures: conditional power approach and predictive power approach. For each approach, the derived group sequential test is used as the reference test for rejection.
In the SEQTEST procedure, the conditional power at an interim stage is the probability that the test statistic at the final stage (stage ) would exceed the rejection critical value (Cui, Hung, and Wang 1999, p. 854; Emerson, Kittelson, and Gillen 2005, p. 13). If there exist interim stages between the th stage and the final stage, , the conditional power is not the conditional probability to reject the null hypothesis . In this case, you can set the next stage as the final stage, and the conditional power is the conditional probability to reject .
The conditional distribution of given the observed statistic at the th stage and the hypothetical reference is
where is the fraction of information at the th stage.
The power for the upper alternative, , is then given by
where is the cumulative distribution function of the standardized statistic and is the upper critical value at the final stage.
Similarly, the power for the lower alternative, , is
where is the lower critical value at the final stage.
A special case of the conditional power is the futility index (Ware, Muller, and Braunwald, 1985). It is one minus the conditional power under :
That is, it is the probability of accepting the null hypothesis under the alternative hypothesis given current data. A high futility index indicates a small probability of success (rejecting ) given the current data.
If , the maximum likelihood estimate at stage , the powers for the upper and lower alternatives can be simplified:
The conditional power depends on the specified reference , which might be supported by the current data (Jennison and Turnbull 2000, p. 210). An alternative is to use the predictive power (Herson 1979), which is a weighted average of the conditional power over values of . Without prior knowledge about , then with , the maximum likelihood estimate at stage , the posterior distribution for (Jennison and Turnbull 2000, p. 211) is
Thus, the predictive power at stage for the upper and lower alternatives can be derived as
where and are the upper and lower critical values at the final stage.