![]() |
![]() |
The GLMPOWER Procedure |
The univariate linear model has the form
![]() |
where is the
vector of responses,
is the
design matrix,
is the
vector of model parameters corresponding to the columns of
, and
is an
vector of errors with
![]() |
In PROC GLMPOWER, the model parameters are not specified directly, but rather indirectly as
, which represents either conjectured response means or typical response values for each design profile. The
values are manifested as the dependent variable in the MODEL statement. The vector
is obtained from
according to the least squares equation,
![]() |
Note that, in general, there is not a 1-to-1 mapping between and
. Many different scenarios for
might lead to the same
. If you specify
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 specified in the MODEL statement. For example, if
exhibits an interaction effect but the corresponding interaction term is left out of the model, then the cell means (
) derived from
differ from
. In particular, the cell means thus derived are the projection of
onto the model space.
It is convenient in power analysis to parameterize the design matrix in three parts,
, defined as follows:
The essence design matrix
is the collection of unique rows of
. Its rows are sometimes referred to as "design profiles." Here,
is defined simply as the number of unique rows of
.
The weight vector
reveals the relative proportions of design profiles. Row
of
is to be included in the design
times for every
times row
is included. The weights are assumed to be standardized (that is, sum up to 1).
The total sample size is . This is the number of rows in
. If you gather
copies of the
row of
, for
, then you end up with
.
It is useful to express the crossproduct matrix in terms of these three parts,
![]() |
since this factors out the portion () depending on sample size and the portion (
) depending only on the design structure.
A general linear hypothesis for the univariate model has the form
![]() |
![]() |
|||
![]() |
![]() |
where is an
contrast matrix (assumed to be full rank) and
is the null value (usually just a vector of zeros). Note that effect tests are just contrasts that use special forms of
. Thus, this scheme covers both effect tests and custom contrasts.
The test statistic is
![]() |
where
![]() |
![]() |
|||
![]() |
![]() |
|||
![]() |
![]() |
where . Note that
if
has full rank.
Under ,
. Under
,
is distributed as
with noncentrality
![]() |
Muller and Peterson (1984) give the exact power of the test as
![]() |
Sample size is computed by inverting the power equation.
See Muller et al. (1992) and O’Brien and Shieh (1992) for additional discussion.
![]() |
![]() |
Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.