The SEQDESIGN Procedure

Type I and Type II Errors

The Type I error is the error of rejecting the null hypothesis when the null hypothesis is correct, and the Type II error is the error of not rejecting the null hypothesis when the null hypothesis is incorrect. The level of significance is the probability of making a Type I error. The Type II error depends on the hypothetical reference of the alternative hypothesis, and the Type II error probability is defined as the probability of not rejecting the null hypothesis when a specific alternative reference is true. The power is then defined as the probability of rejecting the null hypothesis at the alternative reference.

In a sequential design, if the maximum information and alternative reference are not both specified, the critical values are created such that both the specified Type I and the specified Type II error probability levels are maintained in the design. Otherwise, the critical values are created such that either the specified Type I error probability or the specified Type II error probability is maintained.

One-Sided Tests

For a K-stage group sequential design with an upper alternative hypothesis and early stopping to reject or accept the null hypothesis , the boundaries contain the upper critical values and upper critical values , . At each interim stage, , the null hypothesis is rejected if the observed statistic , is accepted if , or the process is continued to the next stage if . At the final stage , the hypothesis is either rejected or accepted.

The overall Type I error probability is given by

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

where is the 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 b Subscript j Baseline less-than-or-equal-to 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

Similarly, the Type II error probability

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

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

beta 1 equals upper P Subscript theta equals theta 1 Baseline left-parenthesis z 1 less-than b 1 right-parenthesis

At a subsequent stage k,

beta Subscript k Baseline equals upper P Subscript theta equals theta 1 Baseline left-parenthesis b Subscript j Baseline less-than-or-equal-to 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 less-than b Subscript k Baseline right-parenthesis

With an upper alternative hypothesis , the power is the probability of rejecting the null hypothesis for the upper alternative.

1 minus beta equals 1 minus sigma-summation Underscript k equals 1 Overscript upper K Endscripts beta Subscript k Baseline equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts upper P Subscript theta equals theta 1 Baseline left-parenthesis b Subscript j Baseline less-than-or-equal-to 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

For a design with early stopping to reject only, the interim upper critical values are set to , , and . For a design with early stopping to accept only, the interim upper critical values are set to , , and .

Similarly, the Type I and Type II error probabilities for a K-stage design with a lower alternative hypothesis can also be derived.

Two-Sided Tests

For a K-stage group sequential design with two-sided alternative hypotheses and , and early stopping to reject or accept the null hypothesis , the boundaries contain the upper critical values , upper critical values , lower critical values , and lower critical values , . At each interim stage, , the null hypothesis is rejected if the observed statistic or , is accepted if , or the process is continued to the next stage if or . At the final stage and , the hypothesis is either rejected or accepted.

The overall upper Type I error probability is given by

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

where is the spending at stage k for the upper alternative. That is, at stage 1,

alpha Subscript u Baseline 1 Baseline 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 u k Baseline equals upper P Subscript theta equals 0 Baseline left-parenthesis normal bar a Subscript j Baseline less-than z Subscript j Baseline less-than-or-equal-to normal bar b Subscript j Baseline normal o normal r b Subscript j Baseline less-than-or-equal-to 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

Similarly, the overall lower Type I error probability can also be derived, and the overall Type I error probability .

The overall upper Type II error probability is given by

beta Subscript u Baseline equals sigma-summation Underscript k equals 1 Overscript upper K Endscripts beta Subscript u k

where is the upper spending at stage k. That is, at stage 1,

beta Subscript u Baseline 1 Baseline equals upper P Subscript theta equals theta Sub Subscript 1 u Baseline left-parenthesis z 1 less-than normal bar a 1 normal o normal r normal bar b 1 less-than z 1 less-than b 1 right-parenthesis

At a subsequent stage k,

StartLayout 1st Row beta Subscript u k Baseline equals upper P Subscript theta equals theta Sub Subscript 1 u Baseline left-parenthesis normal bar a Subscript j Baseline less-than z Subscript j Baseline less-than-or-equal-to normal bar b Subscript j Baseline or b Subscript j Baseline less-than-or-equal-to 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 less-than normal bar a Subscript k Baseline or normal bar b Subscript k Baseline less-than z Subscript k Baseline less-than b Subscript k Baseline right-parenthesis EndLayout

With an upper alternative hypothesis , the power is the probability of rejecting the null hypothesis for the upper alternative:

1 minus beta Subscript u Baseline equals 1 minus sigma-summation Underscript k equals 1 Overscript upper K Endscripts beta Subscript u k

which is

upper P Subscript theta equals theta Sub Subscript 1 u Baseline left-parenthesis normal bar a Subscript j Baseline less-than z Subscript j Baseline less-than-or-equal-to normal bar b Subscript j Baseline normal o normal r b Subscript j Baseline less-than-or-equal-to 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

The overall lower Type II error probability and power can be similarly derived.

For a design with early stopping only to reject , both the interim lower and upper critical values are set to missing, , and , . For a design with early stopping only to accept , the interim upper critical values are set to , , and the interim lower critical values are set to , , , and , .

Last updated: December 09, 2022