The SEQDESIGN Procedure

Error Spending Methods

For each sequential design, the alpha and beta errors spent at each stage can be computed from the boundary values. For example, for a K-stage design with an upper alternative hypothesis upper H 1 colon theta equals theta 1 and early stopping to reject the null hypothesis upper H 0 colon theta equals 0, the boundary values in a standardized Z scale are the upper alpha critical values a Subscript k, k equals 1 comma 2 comma ellipsis comma upper K. At each interim stage, the null hypothesis upper H 0 is rejected if the observed standardized Z statistic z Subscript k Baseline greater-than-or-equal-to a Subscript k. Otherwise, the process continues to the next stage. At the final stage, the hypothesis is rejected if z Subscript upper K Baseline greater-than-or-equal-to a Subscript upper K. Otherwise, the null hypothesis is accepted.

The boundary values a Subscript k are derived such that the overall Type I error probability

alpha equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts alpha Subscript k

where alpha Subscript k is the alpha spending at stage k. That is, at stage 1,

alpha 1 equals upper P Subscript theta equals 0 Baseline left-parenthesis z 1 greater-than-or-equal-to a 1 right-parenthesis

At a subsequent stage k,

alpha Subscript k Baseline equals upper P Subscript theta equals 0 Baseline left-parenthesis z Subscript j Baseline less-than a Subscript j Baseline comma j equals 1 comma 2 comma ellipsis comma k minus 1 comma z Subscript k Baseline greater-than-or-equal-to a Subscript k Baseline right-parenthesis

Since each design can be uniquely identified by the alpha and beta errors spent at each stage, a design can then be derived by specifying the alpha and beta errors to be used at each stage. The error spending method (Lan and DeMets 1983) distributes the error to be used at each stage and then derives the boundary values. Numerous forms of the error spending function are available. Kim and DeMets (1987) examine the functions f left-parenthesis t right-parenthesis equals t, f left-parenthesis t right-parenthesis equals t Superscript three-halves, and f left-parenthesis t right-parenthesis equals t squared, where t is the information fraction. Jennison and Turnbull (2000, p. 148) generalize these functions to the power functions f left-parenthesis t semicolon rho right-parenthesis equals t Superscript rho Baseline comma rho greater-than 0.

The ERRFUNCPOC option uses the cumulative error spending function (Lan and DeMets 1983)

upper E left-parenthesis t right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column normal i normal f t greater-than-or-equal-to 1 2nd Row 1st Column normal l normal o normal g left-parenthesis 1 plus left-parenthesis e minus 1 right-parenthesis t right-parenthesis 2nd Column normal i normal f 0 less-than t less-than 1 3rd Row 1st Column 0 2nd Column normal o normal t normal h normal e normal r normal w normal i normal s normal e EndLayout

With a specified error of alpha or beta, the cumulative error spending at stage k is alpha upper E left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis or beta upper E left-parenthesis normal upper Pi Subscript k Baseline right-parenthesis, where normal upper Pi Subscript k Baseline equals upper I Subscript k Baseline slash upper I Subscript upper X is the information fraction at stage k. The method produces boundaries similar to those produced with Pocock’s method.

The ERRFUNCOBF option uses the cumulative error spending function (Lan and DeMets 1983)

upper E left-parenthesis t semicolon a right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column normal i normal f t greater-than-or-equal-to 1 2nd Row 1st Column StartFraction 1 Over a EndFraction 2 left-parenthesis 1 minus normal upper Phi left-parenthesis StartFraction z Subscript left-parenthesis 1 minus a slash 2 right-parenthesis Baseline Over StartRoot t EndRoot EndFraction right-parenthesis right-parenthesis 2nd Column normal i normal f 0 less-than t less-than 1 3rd Row 1st Column 0 2nd Column normal o normal t normal h normal e normal r normal w normal i normal s normal e EndLayout

where a is either alpha for the alpha spending function or beta for the beta spending function. That is, with a specified error of alpha or beta, the cumulative error spending at stage k is alpha upper E left-parenthesis normal upper Pi Subscript k Baseline semicolon alpha right-parenthesis or beta upper E left-parenthesis normal upper Pi Subscript k Baseline semicolon beta right-parenthesis. The method produces boundaries similar to those produced with the O’Brien-Fleming method.

The ERRFUNCGAMMA option uses the gamma cumulative error spending function (Hwang, Shih, and DeCani 1990)

upper E left-parenthesis t semicolon gamma right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column normal i normal f t greater-than-or-equal-to 1 2nd Row 1st Column StartFraction 1 minus e Superscript minus gamma t Baseline Over 1 minus e Superscript negative gamma Baseline EndFraction 2nd Column normal i normal f 0 less-than t less-than 1 comma gamma not-equals 0 3rd Row 1st Column t 2nd Column normal i normal f 0 less-than t less-than 1 comma gamma equals 0 4th Row 1st Column 0 2nd Column normal o normal t normal h normal e normal r normal w normal i normal s normal e EndLayout

where gamma is the parameter gamma specified in the GAMMA= option. That is, with a specified error of alpha or beta, the cumulative error spending at stage k is alpha upper E left-parenthesis normal upper Pi Subscript k Baseline semicolon gamma right-parenthesis or beta upper E left-parenthesis normal upper Pi Subscript k Baseline semicolon gamma right-parenthesis.

The ERRFUNCPOW option uses the cumulative error spending function (Jennison and Turnbull 2000, p. 148)

upper E left-parenthesis t semicolon rho right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column 1 2nd Column normal i normal f t greater-than-or-equal-to 1 2nd Row 1st Column t Superscript rho Baseline 2nd Column normal i normal f 0 less-than t less-than 1 3rd Row 1st Column 0 2nd Column normal o normal t normal h normal e normal r normal w normal i normal s normal e EndLayout

where rho is the power parameter specified in the RHO= option. That is, with a specified error of alpha or beta, the cumulative error spending at stage k is alpha upper E left-parenthesis normal upper Pi Subscript k Baseline semicolon rho right-parenthesis or beta upper E left-parenthesis normal upper Pi Subscript k Baseline semicolon rho right-parenthesis.

Error spending methods derive boundary values at each stage sequentially and require much more computation than other types of methods for group sequential trials with a large number of stages, especially for a two-sided asymmetric design with early stopping to accept upper H 0, or to reject or accept upper H 0.

Note that for a two-sided design with the STOP=BOTH or STOP=ACCEPT option, at each interim stage, the SEQDESIGN procedure first produces the lower and upper beta boundary values based on the one-sided beta spending. If the lower beta boundary value is greater than or equal to its corresponding upper beta boundary value, there is no early stopping to accept the null hypothesis at this stage, and the corresponding beta spending is distributed proportionally to the remaining stages.

For the error spending functions not available in the SEQDESIGN procedure, you can first compute the corresponding error spending at each stage explicitly, then use the SEQDESIGN procedure with the ERRSPEND= option to specify these errors directly.

For example, if the information levels are equally spaced in a five-stage design, the option ERRFUNCPOW (RHO=2) produces relative cumulative errors of left-parenthesis 1 slash 5 right-parenthesis squared, left-parenthesis 2 slash 5 right-parenthesis squared, left-parenthesis 3 slash 5 right-parenthesis squared, left-parenthesis 4 slash 5 right-parenthesis squared, and 1. This is equivalent to using the option ERRSPEND (1 4 9 16 25).

A one-sided error spending design is illustrated in Example 110.8 and a two-sided asymmetric error spending design is illustrated in Example 110.11.

Last updated: December 09, 2022