The SEQDESIGN Procedure

Whitehead Methods

Subsections:

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

\[ \Pi _{k} = \frac{I_{k}}{I_{X}} \]

where $I_{k}$ is the information available at stage k and $I_{X}$ 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, $\alpha _{u}=\beta _{u}$, the boundary values in the score scale from continuous monitoring are as follows:

  • $S_{\alpha u}({\Pi }_{k}) = C_{u} {\theta }^{-1}_{u} + \tau _{u} {\theta }_{u} I_{k}$

  • $S_{\beta u}({\Pi }_{k}) = {\theta }_{u} \, {I_{k}} - (C_{u} {\theta }^{-1}_{u} - \tau _{u} {\theta }_{u} I_{k})$

where ${\theta }_{u}$ is the upper alternative reference, $\tau _{u}$ is a specified constant for the slope, $0 \leq \tau _{u} < \frac{1}{2}$, and $C_{u}$ 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

  • $S_{\beta u}({\Pi }_{k}) = - C_{u} {\theta }^{-1}_{u} + (1 - \tau _{u}) {\theta }_{u} I_{k}$

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:

  • $S_{\alpha u}({\Pi }_{k}) = C_{u} {\theta }^{-1}_{u} + \tau _{u} {\theta }_{u} I_{k} - g_{k}$

  • $S_{\beta u}({\Pi }_{k}) = - C_{u} {\theta }^{-1}_{u} + (1 - \tau _{u}) {\theta }_{u} I_{k} + g_{k}$

where $g_{1}= 0.583 \sqrt { I_{1}}$ and $g_{k}= 0.583 \sqrt { I_{k} - I_{(k-1)}}$, $k>1$ are the adjustments.

Note that with the adjustment $g_{k}$, 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, $\alpha _{u} \neq \beta _{u}$, the boundary values computed using the score scale, are given by the following:

  • $S_{\alpha u}({\Pi }_{k}) = C_{u} {\tilde{{\theta }}_{u}}^{-1} + \tau _{u} \tilde{{\theta }}_{u} I_{k} - g_{k}$

  • $S_{\beta u}({\Pi }_{k}) = - C_{u} {\tilde{{\theta }}_{u}}^{-1} + (1 - \tau _{u}) \tilde{{\theta }}_{u} I_{k} + g_{k}$

where $\tilde{{\theta }_{u}}$ is the modified alternative reference

\[ \tilde{{\theta }_{u}} = \frac{2 {\Phi }^{-1} (1-\alpha _ u)}{{\Phi }^{-1} (1-\alpha _ u) + {\Phi }^{-1} (1-\beta _ u)} \, {\theta }_{u} \]

The modified alternative reference $\tilde{\theta }_{u} = {\theta }_{u}$ if $\alpha _{u} = \beta _{u}$.

For a design with early stopping to reject or accept the null hypothesis, $S_{\alpha u}(1) = S_{\beta u}(1)$, the boundary values at the final stage are equal. The modified drift parameter $\tilde{d}_{u}$ is given by

\[ \tilde{d}_{u} = \tilde{\theta }_{u} \sqrt {I_{X}} = \frac{1}{1 - 2 \tau _{u}} \left( \sqrt { {h_{K}}^{2} + 2 C_{u} (1 - 2 \tau _{u}) } - h_{K} \right) \]

where $h_{K} = g_{K} \, I_{X}^{-\frac{1}{2}} = 0.583 \sqrt {1 - \Pi _{(K-1)}}$.

A one-sided Whitehead design with early stopping to reject or accept the null hypothesis is illustrated in Example 89.7.

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:

  • $S_{\alpha u}({\Pi }_{k}) = C_{u} {\tilde{\theta }_{u}}^{-1} + \tau _{u} \tilde{{\theta }}_{u} I_{k} - g_{k}$

  • $S_{\beta u}({\Pi }_{k}) = - C_{u} {\tilde{\theta }_{u}}^{-1} + (1 - \tau _{u}) \tilde{{\theta }}_{u} I_{k} + g_{k}$

  • $S_{\beta l}({\Pi }_{k}) = - C_{l} {\tilde{\theta }_{l}}^{-1} + (1 - \tau _{l}) \tilde{{\theta }}_{l} I_{k} - g_{k}$

  • $S_{\alpha l}({\Pi }_{k}) = C_{l} {\tilde{\theta }_{l}}^{-1} + \tau _{l} \tilde{{\theta }}_{l} I_{k} + g_{k}$

where the modified alternative references are

\[ \tilde{\theta }_{u} = \frac{2 {\Phi }^{-1} (1-\alpha _ u)}{{\Phi }^{-1} (1-\alpha _ u) + {\Phi }^{-1} (1-\beta _ u)} \, {\theta }_{u} \]
\[ \tilde{\theta }_{l} = \frac{2 {\Phi }^{-1} (1-\alpha _ l)}{{\Phi }^{-1} (1-\alpha _ l) + {\Phi }^{-1} (1-\beta _ l)} \, {\theta }_{l} \]

The modified alternative reference $\tilde{\theta }_{u}= {\theta }_{u}$ if $\alpha _{u} = \beta _{u}$ and $\tilde{\theta }_{l}= {\theta }_{l}$ if $\alpha _{l} = \beta _{l}$.

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, $S_{\alpha l}(1) = S_{\beta l}(1)$ and $S_{\alpha u}(1) = S_{\beta u}(1)$. These modified drift parameters are then given by

\[ \tilde{d}_{l} = \tilde{{\theta }}_{l} \sqrt {I_{X}} = \frac{1}{1 - 2 \tau _{l}} \left( \sqrt { {h_{K}}^{2} + 2 C_{l} (1 - 2 \tau _{l}) } - h_{K} \right) \]
\[ \tilde{d}_{u} = \tilde{{\theta }}_{u} \sqrt {I_{X}} = \frac{1}{1 - 2 \tau _{u}} \left( \sqrt { {h_{K}}^{2} + 2 C_{u} (1 - 2 \tau _{u}) } - h_{K} \right) \]

where $h_ K = g_ K I_{X}^{-\frac{1}{2}} = 0.583 \sqrt {1 - \Pi _{(K-1)}}$.

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 $\tau _{u}$ and $\tau _{l}$ in the TAU= option, and then the intercept parameters $C_{u}$ and $C_{l}$, and the resulting boundary values are derived. If both the maximum information and alternative references are specified, the procedure derives $C_{u}$ and $C_{l}$ 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 $C_{u}$ and $C_{l}$ 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 $\tau _{u}= \tau _{l}= 0.25$ and compute $C_{u} = -2 \, \mr{log}( 2 \alpha _{u} )$ and $C_{l} = -2 \, \mr{log}( 2 \alpha _{l} )$ 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 89.7 lists applicable boundary keys for a design that uses Whitehead methods.

Table 89.7: Applicable Boundary Keys for Whitehead Methods

   

Specified Parameters

 

Boundary Keys

Early Stopping

 

(Alt Ref – Max Info)

Tau

 

 Alpha 

 Beta 

 None 

 Both 

Reject $H_{0}$

 

X

   X

      

X

X

   

Accept $H_{0}$

 

X

   X

 

X

X

   

Reject/Accept $H_{0}$

 

X

0.25

 

X

X

   

Reject $H_{0}$

   

   X

       

X

Accept $H_{0}$

   

   X

       

X

Reject/Accept $H_{0}$

   

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 $\tau _{u}$ in the TAU= option, and then the intercept parameter $C_{u}$ and the resulting boundary values are derived. If both the maximum information and alternative reference are specified, the procedure derives $C_{u}$ 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 $C_{u}$ 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 $\tau _{u}= 0.25$ and solves $C_{u} = 2 \, \mr{log}( \frac{1}{2 \alpha _{u}} )$ 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.