REPEATED Statement
 REPEATED factorspecification < / options >
;
When values of the dependent variables in the MODEL statement
represent repeated measurements on the same experimental unit,
the REPEATED statement enables you to test hypotheses about
the measurement factors (often called withinsubject
factors) as well as the interactions of withinsubject
factors with independent variables in the MODEL statement
(often called betweensubject factors).
The REPEATED statement provides multivariate and
univariate tests as well as hypothesis tests for
a variety of singledegreeoffreedom contrasts.
There is no limit to the number of
withinsubject factors that can be specified.
The REPEATED statement is typically used for handling repeated measures
designs with one repeated response variable.
Usually, the variables on the lefthand
side of the equation in the MODEL statement
represent one repeated response variable.
This does not mean that only one factor
can be listed in the REPEATED statement.
For example, one repeated response variable (hemoglobin count)
might be measured 12 times (implying variables Y1 to Y12 on
the lefthand side of the equal sign in the MODEL statement),
with the associated withinsubject factors treatment and
time (implying two factors listed in the REPEATED statement).
See the "Examples" section for an example of how PROC GLM handles this case.
Designs with two or more repeated response variables can,
however, be handled with the
IDENTITY transformation; see
Example 30.9 for an example of analyzing a
doublymultivariate repeated measures design.
When a REPEATED statement appears, the GLM procedure
enters a multivariate mode of handling missing values.
If any values for variables corresponding to each
combination of the withinsubject factors are missing,
the observation is excluded from the analysis.
If you use a CONTRAST or TEST statement with a
REPEATED statement, you must enter the CONTRAST
or TEST statement before the REPEATED statement.
The simplest form of the REPEATED statement
requires only a factorname.
With two repeated factors, you must specify the factorname
and number of levels (levels) for each factor.
Optionally, you can specify the actual values for the
levels (levelvalues), a transformation
that defines singledegreeof freedom contrasts, and
options for additional analyses and output.
When you specify more than one withinsubject factor,
the factornames (and associated
level and transformation information) must be
separated by a comma in the REPEATED statement.
These terms are described in the following section,
"Syntax Details."
Syntax Details
You can specify the following terms in the REPEATED statement.
 factorspecification

The factorspecification for the REPEATED statement can include
any number of individual factor specifications, separated by commas,
of the following form:
factorname levels < (levelvalues) >
< transformation >
where
 factorname
 names a factor to be associated with the dependent variables.
The name should not be the same as any variable name that
already exists in the data set being analyzed and should
conform to the usual conventions of SAS variable names.
When specifying more than one factor, list the dependent
variables in the MODEL statement so that the withinsubject
factors defined in the REPEATED statement are nested; that is,
the first factor defined in the REPEATED statement should be
the one with values that change least frequently.
 levels
 gives the number of levels associated
with the factor being defined.
When there is only one withinsubject factor, the number
of levels is equal to the number of dependent variables.
In this case, levels is optional.
When more than one withinsubject factor is defined,
however, levels is required, and the product of
the number of levels of all the factors must equal the number
of dependent variables in the MODEL statement.
 (levelvalues)
 gives values that correspond to
levels of a repeatedmeasures factor.
These values are used to label output and as spacings
for constructing orthogonal polynomial contrasts if you specify a
POLYNOMIAL transformation.
The number of values specified must correspond to the
number of levels for that factor in the REPEATED statement.
Enclose the levelvalues in parentheses.
define singledegreeoffreedom contrasts for
factors specified in the REPEATED statement.
Since the number of contrasts generated is always
one less than the number of levels of the factor,
you have some control over which contrast is omitted
from the analysis by which transformation you select.
The only exception is the IDENTITY transformation; this
transformation is not composed of contrasts and has the
same degrees of freedom as the factor has levels.
By default, the procedure uses the CONTRAST transformation.

CONTRAST < (ordinalreferencelevel ) >

generates contrasts between levels
of the factor and a reference level.
By default, the procedure uses the last level as the reference
level;
you can optionally specify a reference level
in parentheses after the keyword CONTRAST.
The reference level corresponds to the ordinal value
of the level rather than the level value specified.
For example, to generate contrasts between the
first level of a factor and the other levels, use
contrast(1)
 HELMERT

generates contrasts between each level of
the factor and the mean of subsequent levels.

IDENTITY

generates an identity transformation corresponding to the associated
factor. This transformation is not composed of contrasts; it has
n degrees of freedom for an nlevel factor, instead of n1.
This can be used for doublymultivariate repeated measures.

MEAN < (ordinalreferencelevel ) >

generates contrasts between levels of the factor
and the mean of all other levels of the factor.
Specifying a reference level eliminates the
contrast between that level and the mean.
Without a reference level, the contrast
involving the last level is omitted.
See the CONTRAST transformation for an example.

POLYNOMIAL

generates orthogonal polynomial contrasts.
Level values, if provided, are used as spacings
in the construction of the polynomials;
otherwise, equal spacing is assumed.

PROFILE

generates contrasts between adjacent levels of the factor.
You can specify the following options in
the REPEATED statement after a slash.
 CANONICAL

performs a canonical analysis of the H and
E matrices corresponding to the transformed
variables specified in the REPEATED statement.
 htype=n

specifies the type of the H matrix
used in the multivariate tests and the type of
sums of squares used in the univariate tests.
See the HTYPE= option in the specifications
for the MANOVA statement for further details.
 MEAN

generates the overall arithmetic means of the withinsubject variables.
 NOM

displays only the results of the univariate analyses.
 NOU

displays only the results of the multivariate analyses.
 PRINTE

displays the E matrix for each combination of
withinsubject factors, as well as partial correlation
matrices for both the original dependent variables and
the variables defined by the transformations specified
in the REPEATED statement.
In addition, the PRINTE option provides sphericity
tests for each set of transformed variables.
If the requested transformations are not orthogonal,
the PRINTE option also provides a sphericity test
for a set of orthogonal contrasts.
 PRINTH

displays the H (SSCP) matrix
associated with each multivariate test.
 PRINTM

displays the transformation matrices that
define the contrasts in the analysis.
PROC GLM always displays the M matrix so that the
transformed variables are defined by the rows, not
the columns, of the displayed M matrix.
In other words, PROC GLM actually displays M'.
 PRINTRV

displays the characteristic roots and
vectors for each multivariate test.
 SUMMARY

produces analysisofvariance tables for each
contrast defined by the withinsubject factors.
Along with tests for the effects of the independent variables
specified in the MODEL statement, a term labeled MEAN tests
the hypothesis that the overall mean of the contrast is zero.
When specifying more than one factor, list the dependent
variables in the MODEL statement so that the withinsubject
factors defined in the REPEATED statement are nested; that is,
the first factor defined in the REPEATED statement should be
the one with values that change least frequently.
For example, assume that three treatments are administered
at each of four times, for a total of twelve
dependent variables on each experimental unit.
If the variables are listed in the MODEL statement
as Y1 through Y12, then the following REPEATED statement
proc glm;
classes group;
model Y1Y12=group / nouni;
repeated trt 3, time 4;
run;
implies the following structure:
 Dependent Variables 
 Y1  Y2  Y3  Y4  Y5  Y6  Y7  Y8  Y9  Y10  Y11  Y12 
Value of trt  1  1  1  1  2  2  2  2  3  3  3  3 
Value of time  1  2  3  4  1  2  3  4  1  2  3  4 
The REPEATED statement always
produces a table like the preceding one.
For more information, see the section "Repeated Measures Analysis of Variance".
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.