The SEQTEST Procedure

Boundary Adjustments for Information Levels

In a group sequential clinical trial, if the information level for the observed test statistic does not match the corresponding information level in the BOUNDARY= data set, the INFOADJ=PROP option (which is the default) can be used to modify information levels at future stages to accommodate this observed information level. With the adjusted information levels, the ERRSPENDADJ= option provides various methods to compute error spending values at the current and future interim stages. These error spending values are then used to derive boundary values in the SEQTEST procedure. See the section Error Spending Methods in Chapter 110, The SEQDESIGN Procedure, for more information about how to use these error spending values to derive boundary values.

The ERRSPENDADJ=NONE option keeps the error spending the same at each stage. The ERRSPENDADJ=ERRLINE option uses a linear interpolation on the cumulative error spending in the design stored in the BOUNDARY= data set to derive the error spending for each unmatched information level (Kittelson and Emerson 1999, p. 882). That is, the cumulative error spending for an information level I is computed as

e left-parenthesis upper I right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column e 1 left-parenthesis StartFraction upper I Over upper I 1 EndFraction right-parenthesis 2nd Column normal i normal f upper I less-than upper I 1 2nd Row 1st Column e Subscript j Baseline plus left-parenthesis alpha Subscript j plus 1 Baseline minus alpha Subscript j Baseline right-parenthesis left-parenthesis StartFraction upper I minus upper I Subscript j Baseline Over upper I Subscript j plus 1 Baseline minus upper I Subscript j Baseline EndFraction right-parenthesis 2nd Column normal i normal f upper I Subscript j Baseline less-than-or-equal-to upper I less-than upper I Subscript j plus 1 Baseline 3rd Row 1st Column e Subscript upper K Baseline 2nd Column normal i normal f upper I greater-than-or-equal-to upper I Subscript upper K EndLayout

where e 1, e 2, …, e Subscript upper K are the cumulative errors at the K stages of the design that is stored in the BOUNDARY= data set.

The ERRSPENDADJ=ERRFUNCPOC option uses Pocock-type 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 an error level of alpha or beta, the cumulative error spending for an information level I is e left-parenthesis upper I right-parenthesis equals alpha upper E left-parenthesis upper I slash upper I Subscript upper K Baseline right-parenthesis or e left-parenthesis upper I right-parenthesis equals beta upper E left-parenthesis upper I slash upper I Subscript upper K Baseline right-parenthesis.

The ERRSPENDADJ=ERRFUNCOBF option uses O’Brien-Fleming-type 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, and normal upper Phi is the cumulative distribution function of the standardized Z statistic. That is, with an error level of alpha or beta, the cumulative error spending for an information level I is e left-parenthesis upper I right-parenthesis equals alpha upper E left-parenthesis upper I slash upper I Subscript upper K Baseline semicolon alpha right-parenthesis or e left-parenthesis upper I right-parenthesis equals beta upper E left-parenthesis upper I slash upper I Subscript upper K Baseline semicolon beta right-parenthesis.

The ERRSPENDADJ=ERRFUNCGAMMA option uses 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 an error level of alpha or beta, the cumulative error spending for an information level I is e left-parenthesis upper I right-parenthesis equals alpha upper E left-parenthesis upper I slash upper I Subscript upper K Baseline semicolon gamma right-parenthesis or e left-parenthesis upper I right-parenthesis equals beta upper E left-parenthesis upper I slash upper I Subscript upper K Baseline semicolon gamma right-parenthesis.

The ERRSPENDADJ=ERRFUNCPOW option uses power 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= suboption. With an error level of alpha or beta, the cumulative error spending for an information level I is e left-parenthesis upper I right-parenthesis equals alpha upper E left-parenthesis upper I slash upper I Subscript upper K Baseline semicolon rho right-parenthesis or e left-parenthesis upper I right-parenthesis equals beta upper E left-parenthesis upper I slash upper I Subscript upper K Baseline semicolon rho right-parenthesis.

If the BOUNDARYKEY=BOTH option is specified, the maximum information required for the trial might not be the same as the maximum information level stored in the BOUNDARY= data set. In this case, the information levels at future stages are adjusted proportionally, and the same error spending values that were computed based on the maximum information level stored in the BOUNDARY= data set are used to derive boundary values for the trial.

If an error spending function is used to create boundaries for the design in the SEQDESIGN procedure, then in order to better maintain the design features throughout the group sequential trial, the same error spending function to create boundaries for the design in the SEQDESIGN procedure should be used to modify boundaries in the SEQTEST procedure at each subsequent stage.

Last updated: December 09, 2022