The GLMPOWER Procedure

Contrasts in Fixed-Effect Univariate Models

The univariate linear model has the form

bold y equals bold upper X bold-italic beta plus bold-italic epsilon

where bold y is the N times 1 vector of responses, bold upper X is the N times k design matrix, bold-italic beta is the k times 1 vector of model parameters that correspond to the columns of bold upper X, and bold-italic epsilon is an N times 1 vector of errors with

epsilon 1 comma ellipsis comma epsilon Subscript upper N Baseline tilde normal upper N left-parenthesis 0 comma sigma squared right-parenthesis left-parenthesis normal i normal i normal d right-parenthesis

In PROC GLMPOWER, the model parameters bold-italic beta are not specified directly, but rather indirectly as bold y Superscript star, which represents either conjectured response means or typical response values for each design profile. The bold y Superscript star values are manifested as the dependent variable in the MODEL statement. The vector bold-italic beta is obtained from bold y Superscript star according to the least squares equation,

bold-italic beta equals left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper X prime bold y Superscript star

Note that, in general, there is not a one-to-one mapping between bold y Superscript star and bold-italic beta. Many different scenarios for bold y Superscript star might lead to the same bold-italic beta. If you specify bold y Superscript star with the intention of representing cell means, keep in mind that PROC GLMPOWER allows scenarios that are not valid cell means according to the model that is specified in the MODEL statement. For example, if bold y Superscript star exhibits an interaction effect but the corresponding interaction term is left out of the model, then the cell means (bold upper X bold-italic beta) that are derived from bold-italic beta differ from bold y Superscript star. In particular, the cell means that are derived in this way are the projection of bold y Superscript star onto the model space.

It is convenient in power analysis to parameterize the design matrix bold upper X in three parts, StartSet ModifyingAbove bold upper X With two-dots comma bold w comma upper N EndSet, defined as follows:

  1. The q times k essence design matrix ModifyingAbove bold upper X With two-dots is the collection of unique rows of bold upper X. Its rows are sometimes referred to as "design profiles." Here, q less-than-or-equal-to N is defined simply as the number of unique rows of bold upper X.

  2. The q times 1 weight vector bold w reveals the relative proportions of design profiles, and bold upper W equals normal d normal i normal a normal g left-parenthesis bold w right-parenthesis. Row i of ModifyingAbove bold upper X With two-dots is to be included in the design w Subscript i times for every w Subscript j times that row j is included. The weights are assumed to be standardized (that is, they sum up to 1).

  3. The total sample size is N. This is the number of rows in bold upper X. If you gather upper N w Subscript i Baseline equals n Subscript i copies of the ith row of ModifyingAbove bold upper X With two-dots, for i equals 1 comma ellipsis comma q, then you end up with bold upper X.

The preceding quantities are derived from PROC GLMPOWER syntax as follows:

  • Values for ModifyingAbove bold upper X With two-dots, bold y Superscript star, and bold w are specified in the exemplary data set (from using the DATA= option in the PROC GLMPOWER statement), and the corresponding variables are identified in the CLASS, MODEL, and WEIGHT statements.

  • N is specified in the NTOTAL= option in the POWER statement.

It is useful to express the crossproduct matrix bold upper X prime bold upper X in terms of these three parts,

bold upper X prime bold upper X equals upper N ModifyingAbove bold upper X With two-dots prime bold upper W ModifyingAbove bold upper X With two-dots

because this expression factors out the portion (N) that depends on sample size and the portion (ModifyingAbove bold upper X With two-dots prime bold upper W ModifyingAbove bold upper X With two-dots) that depends only on the design structure.

A general linear hypothesis for the univariate model has the form

StartLayout 1st Row 1st Column upper H 0 colon 2nd Column bold upper L bold-italic beta equals bold-italic theta 0 2nd Row 1st Column upper H Subscript upper A Baseline colon 2nd Column bold upper L bold-italic beta not-equals bold-italic theta 0 EndLayout

where bold upper L is an l times k contrast matrix with rank r Subscript upper L and bold-italic theta 0 is the null value (usually just a vector of zeros).

Note that model effect tests are just contrasts that use special forms of bold upper L. Thus, this scheme covers both effect tests (which are specified in the MODEL statement and the EFFECTS= option in the POWER statement) and custom contrasts (which are specified in the CONTRAST statement).

The model degrees of freedom normal upper D normal upper F Subscript normal upper M are equal to the rank of bold upper X, denoted r Subscript upper X. The error degrees of freedom normal upper D normal upper F Subscript normal upper E are equal to N – r Subscript upper X. The sample size N must be at least normal upper D normal upper F Subscript normal upper M plus the number of covariates.

The test statistic is

upper F equals StartStartFraction left-parenthesis StartFraction normal upper S normal upper S Subscript normal upper H Baseline Over r Subscript upper L Baseline EndFraction right-parenthesis OverOver ModifyingAbove sigma With caret squared EndEndFraction

where

StartLayout 1st Row 1st Column normal upper S normal upper S Subscript normal upper H 2nd Column equals StartFraction 1 Over upper N EndFraction left-parenthesis bold upper L ModifyingAbove bold-italic beta With caret minus bold-italic theta 0 right-parenthesis prime left-parenthesis bold upper L left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper L prime right-parenthesis Superscript negative 1 Baseline left-parenthesis bold upper L ModifyingAbove bold-italic beta With caret minus bold-italic theta 0 right-parenthesis 2nd Row 1st Column ModifyingAbove bold-italic beta With caret 2nd Column equals left-parenthesis bold upper X prime bold upper X right-parenthesis Superscript minus Baseline bold upper X prime bold y 3rd Row 1st Column ModifyingAbove sigma With caret squared 2nd Column equals StartFraction 1 Over normal upper D normal upper F Subscript normal upper E Baseline EndFraction left-parenthesis bold y minus bold upper X ModifyingAbove bold-italic beta With caret right-parenthesis prime left-parenthesis bold y minus bold upper X ModifyingAbove bold-italic beta With caret right-parenthesis EndLayout

Under upper H 0, upper F tilde upper F left-parenthesis r Subscript upper L Baseline comma normal upper D normal upper F Subscript normal upper E Baseline right-parenthesis. Under upper H Subscript upper A, F is distributed as upper F left-parenthesis r Subscript upper L Baseline comma normal upper D normal upper F Subscript normal upper E Baseline comma lamda right-parenthesis with noncentrality

lamda equals upper N left-parenthesis bold upper L bold-italic beta minus bold-italic theta 0 right-parenthesis prime left-parenthesis bold upper L left-parenthesis ModifyingAbove bold upper X With two-dots prime bold upper W ModifyingAbove bold upper X With two-dots right-parenthesis Superscript negative 1 Baseline bold upper L prime right-parenthesis Superscript negative 1 Baseline left-parenthesis bold upper L bold-italic beta minus bold-italic theta 0 right-parenthesis sigma Superscript negative 2

The value of sigma is specified in the STDDEV= option in the POWER statement.

Muller and Peterson (1984) give the exact power of the test as

normal p normal o normal w normal e normal r equals upper P left-parenthesis upper F left-parenthesis r Subscript upper L Baseline comma normal upper D normal upper F Subscript normal upper E Baseline comma lamda right-parenthesis greater-than-or-equal-to upper F Subscript 1 minus alpha Baseline left-parenthesis r Subscript upper L Baseline comma normal upper D normal upper F Subscript normal upper E Baseline right-parenthesis right-parenthesis

The value of alpha is specified in the ALPHA= option in the POWER statement.

Sample size is computed by inverting the power equation.

See Muller and Benignus (1992) and O’Brien and Shieh (1992) for additional discussion.

Last updated: December 09, 2022