PROC SEQTEST: Repeated Confidence Intervals :: SAS/STAT(R) 9.22 User's Guide

The SEQTEST Procedure

Repeated Confidence Intervals

In a group sequential test, repeated confidence intervals for a parameter $\text{[math]}$ are defined as a sequence of intervals $\text{[math]}$ , $\text{[math]}$ , for which a simultaneous coverage probability is maintained (Jennison and Turnbull 2000, p. 189). That is, a $\text{[math]}$ sequence of repeated confidence intervals has

$\text{[math]}$

These confidence limits $\text{[math]}$ and $\text{[math]}$ can be created from observed statistic and boundary values at each stage.

Two-Sided Repeated Confidence Intervals

Two sequences of repeated confidence intervals can be derived for a two-sided test. One is a $\text{[math]}$ rejection repeated confidence intervals $\text{[math]}$ , $\text{[math]}$ , and the other is a $\text{[math]}$ acceptance repeated confidence intervals $\text{[math]}$ , $\text{[math]}$ , where $\text{[math]}$ and $\text{[math]}$ are the lower and upper Type I error probabilities for the test and $\text{[math]}$ and $\text{[math]}$ are the lower and upper Type II error probabilities for the test (Jennison and Turnbull 2000, p. 196).

The rejection lower and upper repeated confidence limits at stage $\text{[math]}$ are

$\text{[math]}$

The hypothesis is rejected for upper alternative if the lower limit $\text{[math]}$ and is rejected for lower alternative if the upper limit $\text{[math]}$ . That is, the hypothesis is rejected if both $\text{[math]}$ and $\text{[math]}$ are not in a rejection repeated confidence interval $\text{[math]}$ .

The acceptance lower and upper repeated confidence limits at stage $\text{[math]}$ are

$\text{[math]}$

The hypothesis is accepted if the lower limit $\text{[math]}$ and the upper limit $\text{[math]}$ . That is, a repeated confidence interval is contained in the interval $\text{[math]}$ .

One-Sided Repeated Confidence Intervals

Like the two-sided repeated confidence intervals, two sequences of repeated confidence intervals can be derived for a one-sided test. Suppose the one-sided test has an upper alternative $\text{[math]}$ . Then one sequence of repeated confidence intervals is a $\text{[math]}$ rejection repeated confidence intervals $\text{[math]}$ , $\text{[math]}$ , and the other is a $\text{[math]}$ acceptance repeated confidence intervals $\text{[math]}$ , $\text{[math]}$ , where $\text{[math]}$ and $\text{[math]}$ are the upper Type I and Type II error probabilities for the test. Thus, a sequence of repeated confidence intervals with confidence level greater than or equal to $\text{[math]}$ is given by $\text{[math]}$ .

The rejection lower repeated confidence limit and the acceptance upper repeated confidence limit at stage $\text{[math]}$ are

$\text{[math]}$

The hypothesis is rejected if the lower limit $\text{[math]}$ . and it is accepted if the upper limit $\text{[math]}$ .

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