Boundary Adjustments for Overlapping Lower and Upper Boundaries

For the fixed boundary shape methods and Whitehead methods, the boundary values for all stages are derived simultaneously for each boundary. For a two-sided design with STOP=ACCEPT or STOP=BOTH, simultaneous derivation might result in overlapping of the lower and upper boundaries. That is, at an interim stage , the lower boundary value might be greater than its corresponding upper boundary value. In this case, these two boundary values are set to missing and the design does not stop at stage to accept the null hypothesis (Jennison and Turnbull 2000, p. 113).

For the error spending methods, the boundary values are derived sequentially for the stages. For a two-sided design with STOP=ACCEPT or STOP=BOTH, a small spending at an 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 derived from the one-sided test for the upper alternative is less than the lower boundary value 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 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 the cumulative spending values st 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. The BETAOVERLAP= option is illustrated in Example 77.10.