The SEQDESIGN Procedure

Boundary Variables

The boundaries created in group sequential trials depend on the type of the alternative hypothesis and the early stopping criterion. Table 83.5 shows the boundaries created with various design specifications.

Table 83.5: Boundary Variables

Specifications		Boundary Variables
Alternative		Lower		Upper
Hypothesis	Early Stopping	Alpha	Beta	Beta	Alpha
Lower	Accept $H_{0}$		X
	Reject $H_{0}$	X
	Accept/Reject $H_{0}$	X	X
Upper	Accept $H_{0}$			X
	Reject $H_{0}$				X
	Accept/Reject $H_{0}$			X	X
Two-sided	Accept $H_{0}$		X	X
	Reject $H_{0}$	X			X
	Accept/Reject $H_{0}$	X	X	X	X

Up to four boundaries can be generated in a group sequential design:

the upper $\alpha$ boundary, to reject the null hypothesis for the upper alternative
the upper $\beta$ boundary, to accept the null hypothesis with an upper alternative
the lower $\beta$ boundary, to accept the null hypothesis with a lower alternative
the lower $\alpha$ boundary, to reject the null hypothesis for the lower alternative

For a two-sided design, the null hypothesis is accepted only if both the hypothesis is accepted with an upper alternative and the hypothesis is accepted with a lower alternative.

For a one-sided design with a lower alternative, only the lower boundaries are created. Similarly, for a one-sided design with an upper alternative, only the upper boundaries are created. For example, Figure 83.10 shows the boundary plot for a one-sided test with an upper alternative.

Figure 83.10: Boundary Plot for One-Sided Test

Figure 83.10 corresponds to a one-sided sequential design with early stopping to reject or accept the null hypothesis. For a sequential test with early stopping only to reject the null hypothesis, there are no acceptance boundary values at interim stages. The acceptance boundary value and its associated acceptance region are displayed only at the final stage. Similarly, for a sequential test with early stopping only to accept the null hypothesis, there are no rejection boundary values at interim stages. The rejection boundary value and its associated rejection region are displayed only at the final stage.

For a two-sided design, both the lower and upper boundaries are created. For a design with early stopping to reject the null hypothesis, $\alpha$ boundaries are created. Similarly, for a design with early stopping to accept the null hypothesis, $\beta$ boundaries are created. For a design with early stopping to accept or reject the null hypothesis, both the $\alpha$ and $\beta$ boundaries are created.

For example, Figure 83.11 shows the boundary plot for a two-sided test.

Figure 83.11: Boundary Plot for Two-Sided Test

Figure 83.11 corresponds to a two-sided sequential design with early stopping to reject or accept the null hypothesis. For a sequential test with early stopping only to reject the null hypothesis, there are no acceptance boundary values at interim stages. The acceptance boundary value and its associated acceptance region are displayed only at the final stage. Similarly, for a sequential test with early stopping only to accept the null hypothesis, there are no rejection boundary values at interim stages. The rejection boundary value and its associated rejection region are displayed only at the final stage.