The SEQDESIGN Procedure

Boundary Scales

The boundaries computed by the SEQDESIGN procedure are applied to test statistics computed during the analysis, and so generally, the scale you select for the boundaries is determined by the scale of the statistics that you will be using.

The following scales are available in the SEQDESIGN procedure:

  • MLE, maximum likelihood estimate

  • standardized Z

  • score statistic S

  • p-value

These scales are all equivalent for a given set of boundary values—that is, there exists a unique transformation between any two of these scales. If you know the boundary values in terms of statistics from one scale, you can uniquely derive the boundary values of statistics for other scales. You can specify the scale with the BOUNDARYSCALE= option; the default is BOUNDARYSCALE=STDZ, the standardized Z scale.

You can also select the boundary scale to better examine the features of an individual group sequential design or to compare features among multiple designs. For example, with the standardized Z scale, the boundary values for the Pocock design are identical across all stages, and the O’Brien-Fleming design has boundary values (in absolute value) that decrease over the stages.

The remaining section demonstrates the transformations from one scale to the other scales. If the maximum likelihood estimate ModifyingAbove theta With caret is computed by the analysis, then

ModifyingAbove theta With caret tilde upper N left-parenthesis theta comma StartFraction 1 Over upper I EndFraction right-parenthesis

where I is the Fisher information if it does not depend on theta. Otherwise, I is either the expected Fisher information evaluated at ModifyingAbove theta With caret or the observed Fisher information. See the section Maximum Likelihood Estimator for a detailed description of these statistics.

With the MLE statistic ModifyingAbove theta With caret, the corresponding standardized Z statistic is computed as

upper Z equals ModifyingAbove theta With caret StartRoot upper I EndRoot tilde upper N left-parenthesis theta StartRoot upper I EndRoot comma 1 right-parenthesis

and the corresponding score statistic is computed as

upper S equals ModifyingAbove theta With caret upper I tilde upper N left-parenthesis theta upper I comma upper I right-parenthesis

Similarly, if a score statistic S is computed by the analysis, then with

upper S tilde upper N left-parenthesis theta upper I comma upper I right-parenthesis

where I is the information, either an expected Fisher information (upper E Subscript theta equals 0 Baseline left-parenthesis upper I left-parenthesis theta right-parenthesis right-parenthesis or upper E Subscript theta equals ModifyingAbove theta With caret Baseline left-parenthesis upper I left-parenthesis theta right-parenthesis right-parenthesis) or an observed Fisher information (upper I left-parenthesis 0 right-parenthesis or upper I left-parenthesis ModifyingAbove theta With caret right-parenthesis).

The corresponding standardized Z statistic is computed as

upper Z equals StartFraction upper S Over StartRoot upper I EndRoot EndFraction tilde upper N left-parenthesis theta StartRoot upper I EndRoot comma 1 right-parenthesis

and the corresponding MLE-scaled statistic is computed as

ModifyingAbove theta With caret equals StartFraction upper S Over upper I EndFraction tilde upper N left-parenthesis theta comma StartFraction 1 Over upper I EndFraction right-parenthesis

With a standardized normal Z statistic, the corresponding fixed-sample nominal p-value depends on the type of alternative hypothesis. With an upper alternative, the nominal p-value is defined as the one-sided p-value under the null hypothesis upper H 0 colon theta equals 0 with an upper alternative:

p Subscript k Baseline equals 1 minus normal upper Phi left-parenthesis upper Z right-parenthesis

With a lower alternative or a two-sided alternative, the nominal p-value is defined as the one-sided p-value under the null hypothesis upper H 0 colon theta equals 0 with a lower alternative:

p Subscript k Baseline equals normal upper Phi left-parenthesis upper Z right-parenthesis

which is an increasing function of the standardized Z statistic (Emerson, Kittelson, and Gillen 2005, p. 12).

The BOUNDARYSCALE= MLE, STDZ, SCORE, and PVALUE options display the boundary values in the MLE, standardize Z, score, and p-value scales, respectively. For example, suppose y Subscript k Baseline 1 Baseline comma y Subscript k Baseline 2 Baseline comma ellipsis comma y Subscript k n Sub Subscript k Subscript Baseline are n Subscript k observations of a response variable Y in a data set from a normal distribution with an unknown mean mu and a known variance sigma squared. Then

y Subscript k j Baseline tilde upper N left-parenthesis mu comma sigma squared right-parenthesis

for k equals 1 comma 2 comma ellipsis comma upper K, where K is the number of groups and n Subscript k is the number of observations at group k.

If upper N Subscript k is the cumulative number of observations for the first k groups, then the sample mean from these upper N Subscript k observations

y overbar Subscript k Baseline equals StartFraction 1 Over upper N Subscript k Baseline EndFraction sigma-summation Underscript j equals 1 Overscript upper N Subscript k Baseline Endscripts y Subscript k j

has a normal distribution with mean theta and variance sigma squared slash upper N Subscript k:

y overbar Subscript k Baseline tilde upper N left-parenthesis theta comma StartFraction sigma squared Over upper N Subscript k Baseline EndFraction right-parenthesis

To test the null hypothesis mu equals mu 0, upper H 0 colon theta equals 0, where theta equals mu minus mu 0 can be used. The MLE of theta is ModifyingAbove theta With caret Subscript k Baseline equals y overbar Subscript k Baseline minus mu 0 and

ModifyingAbove theta With caret Subscript k Baseline tilde upper N left-parenthesis theta comma StartFraction 1 Over upper I Subscript k Baseline EndFraction right-parenthesis

where the information is the inverse of the variance of y overbar Subscript k,

upper I Subscript k Baseline equals StartFraction upper N Subscript k Baseline Over sigma squared EndFraction

The corresponding standardized Z statistic is

upper Z Subscript k Baseline equals ModifyingAbove theta With caret Subscript k Baseline upper I Subscript k Superscript one-half Baseline tilde upper N left-parenthesis theta upper I Subscript k Superscript one-half Baseline comma 1 right-parenthesis

The score statistic in the SEQDESIGN procedure is then given by

upper S Subscript k Baseline equals ModifyingAbove theta With caret Subscript k Baseline upper I Subscript k Baseline equals upper Z Subscript k Baseline upper I Subscript k Superscript one-half Baseline tilde upper N left-parenthesis theta upper I Subscript k Baseline comma upper I Subscript k Baseline right-parenthesis

For a null hypothesis upper H 0 colon theta equals 0 with an upper alternative, the nominal p-value of the standardized Z statistic is p Subscript k Baseline equals 1 minus normal upper Phi left-parenthesis upper Z Subscript k Baseline right-parenthesis. For a null hypothesis upper H 0 colon theta equals 0 with a lower alternative or a two-sided alternative, the nominal p-value of the standardized Z statistic is p Subscript k Baseline equals normal upper Phi left-parenthesis upper Z Subscript k Baseline right-parenthesis.

Last updated: December 09, 2022