The SEQTEST Procedure

Boundary Adjustments for Minimum Error Spending

In a group sequential clinical trial, boundary values created from a design such as an O’Brien-Fleming design might be too conservative in early stages. Thus the trial is unlikely to stop in early stages. Lan and DeMets (1983, p. 662) suggest truncating boundary values to a number such as 3.5 for the trial to have a reasonable probability of stopping at early stages. Instead of truncating boundary values by a specified number, the ERRSPENDMIN= option provides individual minimum error spending at each interim stage to stop the trial early.

For a K-stage trial, denote the derived cumulative error spending at stage k after adjusting for information levels by $e_{k}, k=1, 2, \ldots , K$ . Also denote the specified minimum error spending at interim stage k by $\epsilon _{k}, k=1, 2, \ldots , K-1$ . Then the cumulative error spending at stage 1 is $e’_{1} = \mr{max} ( e_{1}, \; \epsilon _{1})$ . If $e_{1} < e’_{1}$ , the error spending values at subsequent interim stages are adjusted proportionally by

$e’_{j} = e’_{1} + \frac{e_{j}-e_{1}}{e_{K}-e_{1}} \; (e_{K}-e’_{1})$

for $j=2, \ldots , K-1$ .

The process is repeated at each subsequent interim stage. That is, at stage $k, k=2, \ldots , K-1$ , denote the updated cumulative $\beta$ spending at stage j by $e_{j}$ , $j=k, k+1, \ldots , K$ . Then the cumulative error spending at stage k is $e’_{k} = \mr{max} ( e_{k}, \; e’_{k-1} + \epsilon _{k})$ . If $e_{k} < e’_{k}$ , the error spending values at subsequent interim stages are adjusted proportionally by

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

for $j=k+1, \ldots , K-1$ .

Note that the ERRSPENDMIN= option is applicable only to the boundaries specified in the BOUNDARYKEY= option. That is, the ERRSPENDMIN= option is applicable to the $\alpha$ boundaries with BOUNDARYKEY=ALPHA or BOUNDARYKEY=BOTH, and it is applicable to the $\beta$ boundaries with BOUNDARYKEY=BETA or BOUNDARYKEY=BOTH.