The Whitehead methods (Whitehead and Stratton, 1983; Whitehead, 1997, 2001) derive boundary values by adjusting the boundary values generated from continuous monitoring. With continuous monitoring, the boundary values are on a straight line in the score scale for each boundary. For a group sequential design, the boundary values at an interim stage k depend on the information fractions
where is the information available at stage k and is the maximum information, the information available at the end of the trial if the trial does not stop early.
A onesided symmetric design is a onesided design with identical Type I and Type II error probabilities. For a onesided symmetric design with an upper alternative, , the boundary values in the score scale from continuous monitoring are as follows:
where is the upper alternative reference, is a specified constant for the slope, , and is a constant, fixed for STOP=BOTH and derived for STOP=ACCEPT and STOP=REJECT.
The upper boundary value can also be expressed as
Thus, these straightline boundaries form a triangle in the score statistic scale.
To adjust for the nature of discrete monitoring, the group sequential boundary values are given by the following:
where and , are the adjustments.
Note that with the adjustment , the resulting boundaries form a Christmas tree shape within the original triangle and are referred to as the Christmas tree boundaries (Whitehead, 1997, p. 73).
For a onesided asymmetric design with an upper alternative, , the boundary values computed using the score scale, are given by the following:
where is the modified alternative reference
The modified alternative reference if .
For a design with early stopping to reject or accept the null hypothesis, , the boundary values at the final stage are equal. The modified drift parameter is given by
where .
A onesided Whitehead design with early stopping to reject or accept the null hypothesis is illustrated in Example 83.7.
The boundary values for a twosided design are generated by combining boundary values from two onesided designs. With the STOP=BOTH option, this produces a double triangular design (Whitehead, 1997, p. 98).
The boundary values for a twosided design, using the score scale, are then given by the following:
where the modified alternative references are
The modified alternative reference if and if .
For a design with early stopping to reject or accept the null hypothesis, the two upper boundary values at the final stage are identical and the two lower boundary values at the final stage are identical. That is, and . These modified drift parameters are then given by
where .
For a design with early stopping to reject the null hypothesis, or a design with early stopping to accept the null hypothesis, you can specify the slope parameters and in the TAU= option, and then the intercept parameters and , and the resulting boundary values are derived. If both the maximum information and alternative references are specified, the procedure derives and by maintaining either the overall levels (BOUNDARYKEY=ALPHA) or the overall levels (BOUNDARYKEY=BETA). If the maximum information and alternative reference are not both specified, the procedure derives the boundary values and by maintaining both the overall and overall levels.
For a design with early stopping to reject or accept the null hypothesis (STOP=BOTH), Whitehead’s triangular test uses and compute and for the boundary values. If the maximum information and alternative reference are both specified, the BOUNDARYKEY=ALPHA option uses the specified values to compute the values and boundary values. The finalstage boundary values are modified to maintain the overall levels if they exist. Similarly, the BOUNDARYKEY=BETA option uses the specified values to compute the values and boundary values. The finalstage boundary values are modified to maintain the overall levels if they exist.
If the maximum information and alternative reference are not both specified, the specified and values are used to derive boundary values. The BOUNDARYKEY=NONE option uses these boundary values without adjustment. The BOUNDARYKEY=ALPHA option modifies the finalstage boundary values to maintain the overall levels if they exist. Similarly, the BOUNDARYKEY=BETA option modifies the finalstage boundary values to maintain the overall levels if they exist.
Table 83.7 lists applicable boundary keys for a design that uses Whitehead methods.
Table 83.7: Applicable Boundary Keys for Whitehead Methods
Specified Parameters 
Boundary Keys 


Early Stopping 
(Alt Ref – Max Info) 
Tau 
Alpha 
Beta 
None 
Both 

Reject 
X 
X 

X 
X 

Accept 
X 
X 
X 
X 

Reject/Accept 
X 
0.25 
X 
X 

Reject 
X 
X 

Accept 
X 
X 

Reject/Accept 
0.25 
X 
X 
X 
Note that the symbol “X” under “(Alt Ref – Max Info)” indicates that both alternative reference and maximum information are specified.
For a design with early stopping to reject the null hypothesis, or a design with early stopping to accept the null hypothesis, you can specify the slope parameter in the TAU= option, and then the intercept parameter and the resulting boundary values are derived. If both the maximum information and alternative reference are specified, the procedure derives by maintaining either the overall levels (BOUNDARYKEY=ALPHA) or the overall levels (BOUNDARYKEY=BETA). If the maximum information and alternative reference are not both specified, the procedure derives the boundary values and by maintaining both the overall and overall levels.
For a design with early stopping to reject or accept the null hypothesis (STOP=BOTH), Whitehead’s triangular test uses and solves for the boundary values. If the maximum information and alternative reference are both specified, the BOUNDARYKEY=ALPHA option uses the specified value to compute the value and boundary values. The finalstage boundary value is modified to maintain the overall level if it exists. Similarly, the BOUNDARYKEY=BETA option uses the specified value to compute the value and boundary values. The finalstage boundary value is modified to maintain the overall level if it exists.
If the maximum information and alternative reference are not both specified, the specified and values are used to derive boundary values. The BOUNDARYKEY=NONE option uses these boundary values without adjustment. The BOUNDARYKEY=ALPHA option modifies the finalstage boundary value to maintain the overall level if it exists. Similarly, the BOUNDARYKEY=BETA option modifies the finalstage boundary value to maintain the overall level if it exists.