Boundary Adjustments for Overlapping Lower and Upper Boundaries

In the SEQTEST procedure, the and spending values at the stages are used to derive the boundary values for the trial. For a two-sided design with early stopping to accept , or to either reject or accept , a zero spending at an interim stage sets the boundary values to missing. A small spending at the current or subsequent interim stage might result in overlapping of the lower and upper boundaries for the two corresponding one-sided tests. Specifically, this form of overlapping occurs at an interim stage if the upper boundary value that is derived from the one-sided test for the upper alternative is less than the lower boundary value that is derived from the one-sided test for the lower alternative (Kittelson and Emerson 1999, pp. 881–882; Rudser and Emerson 2007, p. 6). You can use the BETAOVERLAP= option to specify how this type of overlapping is to be handled.

If BETAOVERLAP=ADJUST (which is the default) is specified, the procedure derives the boundary values for the two-sided design and then checks for overlapping of the two one-sided boundaries at the current and subsequent interim stages. If overlapping occurs at a particular stage, the boundary values for the two-sided design are set to missing (so the trial does not stop to accept the null hypothesis at this stage), and the spending values at subsequent stages are adjusted proportionally as follows.

If the boundary values are set to missing at stage in a -stage trial, the adjusted spending value at stage , , is updated for these missing boundary values, and then the spending values at subsequent stages are adjusted proportionally by

for , where and are cumulative spending values at stage before and after the adjustment, respectively.

After all these adjusted spending values are computed, the boundary values are then further modified for these adjusted spending values.

If you specify BETAOVERLAP=NOADJUST, no adjustment is made when overlapping of one-sided boundaries occurs.