REPEATED Statement
- REPEATED factor-specification < / 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 within-subject factors),
as well as the interactions of within-subject factors with
independent variables in the MODEL statement (often called
between-subject factors).
The REPEATED statement provides multivariate and
univariate tests as well as hypothesis tests for a
variety of single-degree-of-freedom contrasts.
There is no limit to the number of within-subject
factors that can be specified.
For more details, see the "Repeated Measures Analysis of Variance" section in Chapter 30, "The GLM Procedure."
The REPEATED statement is typically used for handling repeated measures
designs with one repeated response variable.
Usually, the variables on the left-hand
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
left-hand side of the equal sign in the MODEL statement), with
the associated within-subject factors treatment and time
(implying two factors listed in the REPEATED statement).
See the "Examples" section for an
example of how PROC ANOVA handles this case.
Designs with two or more repeated response variables can,
however, be handled with the IDENTITY transformation;
see Example 30.9 in Chapter 30, "The GLM Procedure," for an example
of analyzing a doubly-multivariate repeated measures design.
When a REPEATED statement appears, the ANOVA procedure
enters a multivariate mode of handling missing values.
If any values for variables corresponding to each
combination of the within-subject factors are missing,
the observation is excluded from the analysis.
The simplest form of the REPEATED statement
requires only a factor-name.
With two repeated factors, you must specify the factor-name
and number of levels (levels) for each factor.
Optionally, you can specify the actual values for the levels
(level-values), a transformation that defines
single-degree-of freedom contrasts, and options for
additional analyses and output.
When more than one within-subject factor is
specified, factor-names (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.
- factor-specification
-
The factor-specification for the REPEATED statement can include
any number of individual factor specifications, separated by commas,
of the following form:
factor-name levels < (level-values) >
< transformation >
where
- factor-name
- 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 within-subject
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
- specifies the number of levels associated
with the factor being defined.
When there is only one within-subject factor, the number
of levels is equal to the number of dependent variables.
In this case, levels is optional.
When more than one within-subject 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.
- (level-values)
-
specifies values that correspond to
levels of a repeated-measures factor.
These values are used to label output; they are also used as spacings
for constructing orthogonal polynomial contrasts if you specify a
POLYNOMIAL transformation.
The number of level values specified must correspond to the
number of levels for that factor in the REPEATED statement.
Enclose the level-values in parentheses.
The following transformation keywords define
single-degree-of-freedom 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 it has the
same degrees of freedom as the factor has levels.
By default, the procedure uses the CONTRAST transformation.
-
CONTRAST < (ordinal-reference-level ) >
-
generates contrasts between levels
of the factor and a reference level.
By default, the procedure uses the last 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 n-level factor, instead of n-1.
This can be used for doubly-multivariate repeated measures.
-
MEAN < (ordinal-reference-level ) >
-
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.
For examples of the transformation matrices generated by these
contrast transformations,
see the section "Repeated Measures Analysis of Variance"
in Chapter 30, "The GLM Procedure."
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.
- 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 within-subject
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 ANOVA 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 ANOVA actually displays M'.
- PRINTRV
-
produces the characteristic roots and
vectors for each multivariate test.
- SUMMARY
-
produces analysis-of-variance tables for each
contrast defined by the within-subjects 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 within-subject
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
repeated trt 3, time 4;
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 on repeated measures
analysis and on using the REPEATED statement,
see the section "Repeated Measures Analysis of Variance" in Chapter 30, "The GLM Procedure."
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
