Unified family methods (Kittelson and Emerson, 1999) derive boundary values with a specified boundary shape. For example, Pocock’s method (Pocock, 1977) derives equal boundary values for all stages in the standardized Z scale. In addition to Pocock’s method, the unified family methods include the O’BrienFleming, power family, and unified family triangular methods.
The boundary values at each stage 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.
With the unified family method, the boundary values for the upper boundary , upper boundary , lower boundary , and lower boundary , using the standardized normal scale, are given by the following:
where and are the lower and upper alternative references, , , , and are the specified shape functions, and , , , and are the critical values derived to achieve the specified and levels.
If a derived lower boundary value is greater than its corresponding upper boundary value , then both values are set to missing.
Note that the drift parameters and are derived in the SEQDESIGN procedure. The boundary values in standardized Z scale can be derived without specifying the maximum information and alternative reference.
The shape function in the SEQDESIGN procedure is given by
where the parameters and . can be specified for each boundary separately.
The parameters and determine the shape of the boundaries. Special cases of the unified family methods also include power family methods and triangular methods. Table 83.6 summarizes the corresponding parameter values in the unified family for these methods.
Table 83.6: Parameters in the Unified Family for Various Methods
Unified Family 


Method 
Option 
Rho 
Tau 

Pocock 
POC 
0 
0 

O’BrienFleming 
OBF 
0.5 
0 

Power family 
POW (RHO=) 

0 

Triangular 
TRI (TAU=) 
0.5 

Note that the power parameter , where is the power parameter used in Jennison and Turnbull (2000) and Wang and Tsiatis (1987) and is the power parameter used in Kittelson and Emerson (1999).
Also note that instead of the three parameters used in the unified family methods by Kittelson and Emerson (1999), only two parameters are used in the SEQDESIGN procedure. The other parameter is fixed at zero.
If the maximum information is available, the boundary values derived from a unified family method can also be displayed in the MLE scale:
These MLE scale boundary values are computed by multiplying by the standardized Z scale boundary values at stage k.
If the maximum information is available, the boundary values derived from a unified family method can also be displayed in the score scale:
These MLE scale boundary values are computed by multiplying by the standardized Z scale boundary values at stage k.
For a design with a lower alternative or a twosided alternative, the pvalue scale boundary values are the cumulative normal distribution function values of the standardized Z boundary values:
These nominal pvalues are the onesided fixedsample pvalues under the null hypothesis with a lower alternative.
For a onesided design with an upper alternative, the pvalue scale boundary values are the onesided fixedsample pvalues under the null hypothesis with an upper alternative:
The shape function for Pocock’s method (Pocock, 1977) is given by
The resulting boundary values for a twosided design with an early stopping to reject the null hypothesis are as follows:
That is, the rejection boundary values are constant over all stages of different information levels in the standardized Z scale.
Note that compared with other designs, Pocock’s design tends to stop the trials early with a larger pvalue. For a new treatment, Pocock’s design to stop a trial early with a large pvalue might not be persuasive enough to make a new treatment widely accepted (Pocock and White, 1999). A Pocock design is illustrated in Example 83.3.
The shape function for the O’BrienFleming method (O’Brien and Fleming, 1979) is given by
The resulting boundary values for a twosided design with early stopping to reject the null hypothesis are as follows:
That is, the rejection boundaries are inversely proportional to the square root of the information levels in the standardized Z scale.
In the score scale, these boundaries can be displayed as follows:
which are constants over all stages in the score scale. An O’BrienFleming design is illustrated in Example 83.2.
The shape function for a power family method (Wang and Tsiatis, 1987; Emerson and Fleming, 1989; Pampallona and Tsiatis, 1994) is given by
The resulting boundary values for a twosided design with early stopping to reject the null hypothesis are as follows:
The rejection boundaries depend on the power parameter . The power family includes the Pocock and O’BrienFleming methods, and the power parameter is used to allow continuous movement between these two methods.
The shape function for a triangular method (Kittelson and Emerson, 1999) in the unified family is given by
The resulting boundary values for a twosided design with early stopping to reject the null hypothesis are as follows:
In the score scale, these boundaries are as follows:
Thus, in the score scale, the boundary function is a linear function of the information . With these straightline boundaries, a triangular method for a onesided trial with early stopping to reject or accept the null hypothesis produces a triangular continuation region. Similarly, for a twosided design, the continuation region is a union of two separate triangular regions. A triangular method is illustrated in Example 83.6.