The SEQTEST Procedure

Boundary Adjustments for Overlapping Lower and Upper $\bbeta $ Boundaries

In the SEQTEST procedure, the $\alpha $ and $\beta $ 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 $H_0$, or to either reject or accept $H_0$, a zero $\beta $ spending at an interim stage sets the $\beta $ boundary values to missing. A small $\beta $ spending at the current or subsequent interim stage might result in overlapping of the lower and upper $\beta $ boundaries for the two corresponding one-sided tests. Specifically, this form of overlapping occurs at an interim stage k if the upper $\beta $ boundary value that is derived from the one-sided test for the upper alternative is less than the lower $\beta $ 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 $\beta $ boundaries at the current and subsequent interim stages. If overlapping occurs at a particular stage, the $\beta $ 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 $\beta $ spending values at subsequent stages are adjusted proportionally as follows.

If the $\beta $ boundary values are set to missing at stage k in a K-stage trial, the adjusted $\beta $ spending value at stage k, $e’_{k}$, is updated for these missing $\beta $ boundary values, and then the $\beta $ spending values at subsequent stages are adjusted proportionally by

\[ e’_{j} = e’_{k} + \frac{e_{j}-e_{k}}{e_{K}-e_{k}} \; (e_{K}-e’_{k}) \]

for $j=k+1, \ldots , K$, where $e_{j}$ and $e’_{j}$ are cumulative $\beta $ spending values at stage j before and after the adjustment, respectively.

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

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