The SEQDESIGN Procedure

Whitehead Methods

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

normal upper Pi Subscript k Baseline equals StartFraction upper I Subscript k Baseline Over upper I Subscript upper X Baseline EndFraction

where upper I Subscript k is the information available at stage k and upper I Subscript upper X is the maximum information, the information available at the end of the trial if the trial does not stop early.

One-Sided Symmetric Designs

A one-sided symmetric design is a one-sided design with identical Type I and Type II error probabilities. For a one-sided symmetric design with an upper alternative, alpha Subscript u Baseline equals beta Subscript u, the boundary values in the score scale from continuous monitoring are as follows:

  • upper S Subscript alpha u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals upper C Subscript u Baseline theta Subscript u Superscript negative 1 Baseline plus tau Subscript u Baseline theta Subscript u Baseline upper I Subscript k

  • upper S Subscript beta u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals theta Subscript u Baseline upper I Subscript k Baseline minus left-parenthesis upper C Subscript u Baseline theta Subscript u Superscript negative 1 Baseline minus tau Subscript u Baseline theta Subscript u Baseline upper I Subscript k Baseline right-parenthesis

where theta Subscript u is the upper alternative reference, tau Subscript u is a specified constant for the slope, 0 less-than-or-equal-to tau Subscript u Baseline less-than one-half, and upper C Subscript u is a constant, fixed for STOP=BOTH and derived for STOP=ACCEPT and STOP=REJECT.

The upper beta boundary value can also be expressed as

  • upper S Subscript beta u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals minus upper C Subscript u Baseline theta Subscript u Superscript negative 1 Baseline plus left-parenthesis 1 minus tau Subscript u Baseline right-parenthesis theta Subscript u Baseline upper I Subscript k

Thus, these straight-line 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:

  • upper S Subscript alpha u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals upper C Subscript u Baseline theta Subscript u Superscript negative 1 Baseline plus tau Subscript u Baseline theta Subscript u Baseline upper I Subscript k Baseline minus g Subscript k

  • upper S Subscript beta u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals minus upper C Subscript u Baseline theta Subscript u Superscript negative 1 Baseline plus left-parenthesis 1 minus tau Subscript u Baseline right-parenthesis theta Subscript u Baseline upper I Subscript k Baseline plus g Subscript k

where g 1 equals 0.583 StartRoot upper I 1 EndRoot and g Subscript k Baseline equals 0.583 StartRoot upper I Subscript k Baseline minus upper I Subscript left-parenthesis k minus 1 right-parenthesis Baseline EndRoot, k greater-than 1 are the adjustments.

Note that with the adjustment g Subscript k, 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).

One-Sided Asymmetric Designs

For a one-sided asymmetric design with an upper alternative, alpha Subscript u Baseline not-equals beta Subscript u, the boundary values computed using the score scale, are given by the following:

  • upper S Subscript alpha u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals upper C Subscript u Baseline theta overTilde Subscript u Baseline Superscript negative 1 Baseline plus tau Subscript u Baseline theta overTilde Subscript u Baseline upper I Subscript k Baseline minus g Subscript k

  • upper S Subscript beta u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals minus upper C Subscript u Baseline theta overTilde Subscript u Baseline Superscript negative 1 Baseline plus left-parenthesis 1 minus tau Subscript u Baseline right-parenthesis theta overTilde Subscript u Baseline upper I Subscript k Baseline plus g Subscript k

where theta Subscript u Baseline overTilde is the modified alternative reference

theta Subscript u Baseline overTilde equals StartFraction 2 normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus alpha Subscript u Baseline right-parenthesis Over normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus alpha Subscript u Baseline right-parenthesis plus normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus beta Subscript u Baseline right-parenthesis EndFraction theta Subscript u

The modified alternative reference theta overTilde Subscript u Baseline equals theta Subscript u if alpha Subscript u Baseline equals beta Subscript u.

For a design with early stopping to reject or accept the null hypothesis, upper S Subscript alpha u Baseline left-parenthesis 1 right-parenthesis equals upper S Subscript beta u Baseline left-parenthesis 1 right-parenthesis, the boundary values at the final stage are equal. The modified drift parameter d overTilde Subscript u is given by

d overTilde Subscript u Baseline equals theta overTilde Subscript u Baseline StartRoot upper I Subscript upper X Baseline EndRoot equals StartFraction 1 Over 1 minus 2 tau Subscript u Baseline EndFraction left-parenthesis StartRoot h Subscript upper K Baseline Superscript 2 Baseline plus 2 upper C Subscript u Baseline left-parenthesis 1 minus 2 tau Subscript u Baseline right-parenthesis EndRoot minus h Subscript upper K Baseline right-parenthesis

where h Subscript upper K Baseline equals g Subscript upper K Baseline upper I Subscript upper X Superscript negative one-half Baseline equals 0.583 StartRoot 1 minus normal upper Pi Subscript left-parenthesis upper K minus 1 right-parenthesis Baseline EndRoot.

A one-sided Whitehead design with early stopping to reject or accept the null hypothesis is illustrated in Example 110.7.

Two-Sided Designs

The boundary values for a two-sided design are generated by combining boundary values from two one-sided designs. With the STOP=BOTH option, this produces a double triangular design (Whitehead 1997, p. 98).

The boundary values for a two-sided design, using the score scale, are then given by the following:

  • upper S Subscript alpha u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals upper C Subscript u Baseline theta overTilde Subscript u Baseline Superscript negative 1 Baseline plus tau Subscript u Baseline theta overTilde Subscript u Baseline upper I Subscript k Baseline minus g Subscript k

  • upper S Subscript beta u Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals minus upper C Subscript u Baseline theta overTilde Subscript u Baseline Superscript negative 1 Baseline plus left-parenthesis 1 minus tau Subscript u Baseline right-parenthesis theta overTilde Subscript u Baseline upper I Subscript k Baseline plus g Subscript k

  • upper S Subscript beta l Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals minus upper C Subscript l Baseline theta overTilde Subscript l Baseline Superscript negative 1 Baseline plus left-parenthesis 1 minus tau Subscript l Baseline right-parenthesis theta overTilde Subscript l Baseline upper I Subscript k Baseline minus g Subscript k

  • upper S Subscript alpha l Baseline left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis equals upper C Subscript l Baseline theta overTilde Subscript l Baseline Superscript negative 1 Baseline plus tau Subscript l Baseline theta overTilde Subscript l Baseline upper I Subscript k Baseline plus g Subscript k

where the modified alternative references are

theta overTilde Subscript u Baseline equals StartFraction 2 normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus alpha Subscript u Baseline right-parenthesis Over normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus alpha Subscript u Baseline right-parenthesis plus normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus beta Subscript u Baseline right-parenthesis EndFraction theta Subscript u
theta overTilde Subscript l Baseline equals StartFraction 2 normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus alpha Subscript l Baseline right-parenthesis Over normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus alpha Subscript l Baseline right-parenthesis plus normal upper Phi Superscript negative 1 Baseline left-parenthesis 1 minus beta Subscript l Baseline right-parenthesis EndFraction theta Subscript l

The modified alternative reference theta overTilde Subscript u Baseline equals theta Subscript u if alpha Subscript u Baseline equals beta Subscript u and theta overTilde Subscript l Baseline equals theta Subscript l if alpha Subscript l Baseline equals beta Subscript l.

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, upper S Subscript alpha l Baseline left-parenthesis 1 right-parenthesis equals upper S Subscript beta l Baseline left-parenthesis 1 right-parenthesis and upper S Subscript alpha u Baseline left-parenthesis 1 right-parenthesis equals upper S Subscript beta u Baseline left-parenthesis 1 right-parenthesis. These modified drift parameters are then given by

d overTilde Subscript l Baseline equals theta overTilde Subscript l Baseline StartRoot upper I Subscript upper X Baseline EndRoot equals StartFraction 1 Over 1 minus 2 tau Subscript l Baseline EndFraction left-parenthesis StartRoot h Subscript upper K Baseline Superscript 2 Baseline plus 2 upper C Subscript l Baseline left-parenthesis 1 minus 2 tau Subscript l Baseline right-parenthesis EndRoot minus h Subscript upper K Baseline right-parenthesis
d overTilde Subscript u Baseline equals theta overTilde Subscript u Baseline StartRoot upper I Subscript upper X Baseline EndRoot equals StartFraction 1 Over 1 minus 2 tau Subscript u Baseline EndFraction left-parenthesis StartRoot h Subscript upper K Baseline Superscript 2 Baseline plus 2 upper C Subscript u Baseline left-parenthesis 1 minus 2 tau Subscript u Baseline right-parenthesis EndRoot minus h Subscript upper K Baseline right-parenthesis

where h Subscript upper K Baseline equals g Subscript upper K Baseline upper I Subscript upper X Superscript negative one-half Baseline equals 0.583 StartRoot 1 minus normal upper Pi Subscript left-parenthesis upper K minus 1 right-parenthesis Baseline EndRoot.

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 tau Subscript u and tau Subscript l in the TAU= option, and then the intercept parameters upper C Subscript u and upper C Subscript l, and the resulting boundary values are derived. If both the maximum information and alternative references are specified, the procedure derives upper C Subscript u and upper C Subscript l by maintaining either the overall alpha levels (BOUNDARYKEY=ALPHA) or the overall beta levels (BOUNDARYKEY=BETA). If the maximum information and alternative reference are not both specified, the procedure derives the boundary values upper C Subscript u and upper C Subscript l by maintaining both the overall alpha and overall beta levels.

For a design with early stopping to reject or accept the null hypothesis (STOP=BOTH), Whitehead’s triangular test uses tau Subscript u Baseline equals tau Subscript l Baseline equals 0.25 and compute upper C Subscript u Baseline equals minus 2 normal l normal o normal g left-parenthesis 2 alpha Subscript u Baseline right-parenthesis and upper C Subscript l Baseline equals minus 2 normal l normal o normal g left-parenthesis 2 alpha Subscript l Baseline right-parenthesis for the boundary values. If the maximum information and alternative reference are both specified, the BOUNDARYKEY=ALPHA option uses the specified alpha values to compute the beta values and boundary values. The final-stage boundary values are modified to maintain the overall alpha levels if they exist. Similarly, the BOUNDARYKEY=BETA option uses the specified beta values to compute the alpha values and boundary values. The final-stage boundary values are modified to maintain the overall beta levels if they exist.

If the maximum information and alternative reference are not both specified, the specified alpha and beta values are used to derive boundary values. The BOUNDARYKEY=NONE option uses these boundary values without adjustment. The BOUNDARYKEY=ALPHA option modifies the final-stage boundary values to maintain the overall alpha levels if they exist. Similarly, the BOUNDARYKEY=BETA option modifies the final-stage boundary values to maintain the overall beta levels if they exist.

Applicable Boundary Keys

Table 7 lists applicable boundary keys for a design that uses Whitehead methods.

Table 7: Applicable Boundary Keys for Whitehead Methods

Specified Parameters Boundary Keys
Early Stopping (Alt Ref – Max Info) Tau  Alpha   Beta   None   Both 
Reject upper H 0 X    X X X
Accept upper H 0 X    X X X
Reject/Accept upper H 0 X 0.25 X X
Reject upper H 0    X X
Accept upper H 0    X X
Reject/Accept upper H 0 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 tau Subscript u in the TAU= option, and then the intercept parameter upper C Subscript u and the resulting boundary values are derived. If both the maximum information and alternative reference are specified, the procedure derives upper C Subscript u by maintaining either the overall alpha levels (BOUNDARYKEY=ALPHA) or the overall beta levels (BOUNDARYKEY=BETA). If the maximum information and alternative reference are not both specified, the procedure derives the boundary values and upper C Subscript u by maintaining both the overall alpha and overall beta levels.

For a design with early stopping to reject or accept the null hypothesis (STOP=BOTH), Whitehead’s triangular test uses tau Subscript u Baseline equals 0.25 and solves upper C Subscript u Baseline equals 2 normal l normal o normal g left-parenthesis StartFraction 1 Over 2 alpha Subscript u Baseline EndFraction right-parenthesis for the boundary values. If the maximum information and alternative reference are both specified, the BOUNDARYKEY=ALPHA option uses the specified alpha value to compute the beta value and boundary values. The final-stage boundary value is modified to maintain the overall alpha level if it exists. Similarly, the BOUNDARYKEY=BETA option uses the specified beta value to compute the alpha value and boundary values. The final-stage boundary value is modified to maintain the overall beta level if it exists.

If the maximum information and alternative reference are not both specified, the specified alpha and beta values are used to derive boundary values. The BOUNDARYKEY=NONE option uses these boundary values without adjustment. The BOUNDARYKEY=ALPHA option modifies the final-stage boundary value to maintain the overall alpha level if it exists. Similarly, the BOUNDARYKEY=BETA option modifies the final-stage boundary value to maintain the overall beta level if it exists.

Last updated: December 09, 2022