The SEQDESIGN Procedure

Applicable One-Sample Tests and Sample Size Computation

The SEQDESIGN procedure provides sample size computation for two one-sample tests: normal mean and binomial proportion. The required sample size depends on the variance of the response variable—that is, the sample proportion for a binomial proportion test.

In a typical clinical trial, a hypothesis is designed to reject, not accept, the null hypothesis to show the evidence for the alternative hypothesis. Thus, in most cases, the proportion under the alternative hypothesis is used to derive the required sample size. For a test of the binomial proportion, the REF=NULLPROP and REF=PROP options use proportions under the null and alternative hypotheses, respectively.

Test for a Normal Mean

The MODEL=ONESAMPLEMEAN option in the SAMPLESIZE statement derives the sample size required to test a normal mean by using the sample mean statistic for the null hypothesis mu equals mu 0. At stage k, the sample mean is computed as

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

where y Subscript k j is the value of the jth observation available in the kth stage and upper N Subscript k is the cumulative sample size at stage k.

An equivalent hypothesis is upper H 0 colon theta equals 0, where theta equals mu minus mu 0.

The MLE statistic for theta,

ModifyingAbove theta With caret Subscript k Baseline equals y overbar Subscript k Baseline minus mu 0 tilde upper N left-parenthesis theta comma upper I Subscript k Baseline Superscript negative 1 Baseline right-parenthesis

where the information

upper I Subscript k Baseline equals StartFraction 1 Over normal upper V normal a normal r left-parenthesis ModifyingAbove theta With caret right-parenthesis EndFraction equals StartFraction 1 Over normal upper V normal a normal r left-parenthesis y overbar Subscript k Baseline right-parenthesis EndFraction equals StartFraction upper N Subscript k Baseline Over sigma squared EndFraction

is the inverse of the variance.

That is, the standardized statistic

upper Z Subscript k Baseline equals ModifyingAbove theta With caret Subscript k Baseline StartRoot upper I Subscript k Baseline EndRoot equals left-parenthesis y overbar Subscript k Baseline minus mu 0 right-parenthesis StartRoot upper I Subscript k Baseline EndRoot tilde upper N left-parenthesis theta StartRoot upper I Subscript k Baseline EndRoot comma 1 right-parenthesis

Thus, to test the hypothesis upper H 0 colon theta equals 0 against a two-sided alternative upper H 1 colon theta equals theta 1, upper H 0 is rejected at stage k if the statistic upper Z Subscript k is less than or equal to the lower alpha boundary value or if upper Z Subscript k is greater than or equal to the upper alpha boundary value at stage k.

If the variance sigma squared is unknown, the sample variance can be used if it is assumed that the sample variance is computed from a large sample such that the test statistic has an approximately normal distribution.

The maximum information is needed to derive the required sample size. If the maximum information is not specified or derived with the ALTREF= option in the procedure, the MEAN=theta 1 option in the SAMPLESIZE statement is used to specify the alternative reference and thus to derive the maximum information.

In the SEQDESIGN procedure, the computed total sample size

upper N Subscript upper K Baseline equals sigma squared upper I Subscript upper X

where upper I Subscript upper X is the maximum information and sigma is the specified standard deviation. With an available maximum information, you can specify the MODEL=ONESAMPLEMEAN( STDDEV= sigma) option in the SAMPLESIZE statement to compute the required total sample size and individual sample size at each stage. A procedure such as PROC MEANS can be used to derive a one-sample Z test for a normal mean.

Test for a Binomial Proportion

The MODEL=ONESAMPLEFREQ option in the SAMPLESIZE statement derives the sample size required to test a binomial proportion by using the null hypothesis p equals p 0, where p is the proportion of a binomial population. At stage k, the MLE for p is computed as

ModifyingAbove p With caret 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

where y Subscript k j is the value of the jth observation available in the kth stage and upper N Subscript k is the cumulative sample size at stage k.

An equivalent hypothesis is upper H 0 colon theta equals 0, where theta equals p minus p 0. If p 0 is not close to 0 or 1, then for a large sample, ModifyingAbove theta With caret Subscript k Baseline equals ModifyingAbove p With caret Subscript k Baseline minus p 0 has an approximately normal distribution

ModifyingAbove theta With caret Subscript k Baseline tilde upper N left-parenthesis theta comma upper I Subscript k Superscript negative 1 Baseline right-parenthesis

where the information upper I Subscript k Baseline equals left-parenthesis p left-parenthesis 1 minus p right-parenthesis slash upper N Subscript k Baseline right-parenthesis Superscript negative 1 is the inverse of the variance normal upper V normal a normal r left-parenthesis ModifyingAbove theta With caret right-parenthesis.

Then the standardized statistic

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

In practice, the estimated sample proportion ModifyingAbove p With caret at stage k can be used to derive the information upper I Subscript k and test statistic upper Z Subscript k. Thus, to test the hypothesis upper H 0 against an upper alternative upper H 1 colon theta equals theta 1 greater-than 0, upper H 0 is rejected at stage k if the statistic upper Z Subscript k is greater than or equal to the upper alpha boundary at stage k.

The maximum information upper I Subscript upper X is needed to derive the required sample size. If the maximum information is not specified or derived with the ALTREF= option in the procedure, the PROP= option in the SAMPLESIZE statement is used to specify the alternative reference and to derive the maximum information for the sample size calculation.

It is assumed that the sample size is sufficiently large such that the test statistic has an approximately normal distribution. With the hypotheses upper H 0 colon p equals p 0 and upper H 1 colon p equals p 1, the SEQDESIGN procedure derives the total sample size

upper N Subscript upper X Baseline equals p Superscript asterisk Baseline left-parenthesis 1 minus p Superscript asterisk Baseline right-parenthesis upper I Subscript upper X

where p Superscript asterisk Baseline equals p 0 if REF=NULLPROP is specified. Otherwise, p Superscript asterisk Baseline equals p 1.

If the PROP= option in the SAMPLESIZE statement is not specified, then the alternative reference theta 1 derived in the SEQDESIGN procedure is used to compute p 1 equals p 0 plus theta 1.

The ALTREF= option in the PROC statement can be used to specify theta 1. Otherwise, the PROP= option in the SAMPLESIZE statement must be specified.

For example, with upper H 0 colon p equals 0.5, upper H 1 colon p equals 0.6, and REF=PROP (which is the default),

upper N Subscript upper K Baseline equals p Superscript asterisk Baseline left-parenthesis 1 minus p Superscript asterisk Baseline right-parenthesis upper I Subscript upper X Baseline equals left-parenthesis 0.6 times 0.4 right-parenthesis upper I Subscript upper X Baseline equals 0.24 upper I Subscript upper X

You can specify the MODEL=ONESAMPLEFREQ option in the SAMPLESIZE statement to compute the required total sample size and individual sample size at each stage. A procedure such as PROC GENMOD with the default DIST=NORMAL option in the MODEL statement can be used to derive the Z test for a binomial proportion.

Last updated: December 09, 2022