# GLM Repeated Measures

The GLM Repeated Measures procedure provides analysis of variance when the same measurement is made several times on each subject or case. If between-subjects factors are specified, they divide the population into groups. Using this general linear model procedure, you can test null hypotheses about the effects of both the between-subjects factors and the within-subjects factors. You can investigate interactions between factors as well as the effects of individual factors. In addition, the effects of constant covariates and covariate interactions with the between-subjects factors can be included.

In a doubly multivariate repeated measures design, the dependent variables represent measurements of more than one variable for the different levels of the within-subjects factors. For example, you could have measured both pulse and respiration at three different times on each subject.

The GLM Repeated Measures procedure provides both univariate and multivariate analyses for the repeated measures data. Both balanced and unbalanced models can be tested. A design is balanced if each cell in the model contains the same number of cases. In a multivariate model, the sums of squares due to the effects in the model and error sums of squares are in matrix form rather than the scalar form found in univariate analysis. These matrices are called SSCP (sums-of-squares and cross-products) matrices. In addition to testing hypotheses, GLM Repeated Measures produces estimates of parameters.

Commonly used *a priori* contrasts are available to perform
hypothesis testing on between-subjects factors. Additionally, after
an overall *F* test has shown significance, you can use post
hoc tests to evaluate differences among specific means. Estimated
marginal means give estimates of predicted mean values for the cells
in the model, and profile plots (interaction plots) of these means
allow you to visualize some of the relationships easily.

Residuals, predicted values, Cook's distance, and leverage values can be saved as new variables in your data file for checking assumptions. Also available are a residual SSCP matrix, which is a square matrix of sums of squares and cross-products of residuals, a residual covariance matrix, which is the residual SSCP matrix divided by the degrees of freedom of the residuals, and the residual correlation matrix, which is the standardized form of the residual covariance matrix.

WLS Weight allows you to specify a variable used to give observations different weights for a weighted least-squares (WLS) analysis, perhaps to compensate for different precision of measurement.

**Example.** Twelve students are assigned to a high- or low-anxiety
group based on their scores on an anxiety-rating test. The anxiety
rating is called a between-subjects factor because it divides the
subjects into groups. The students are each given four trials on a
learning task, and the number of errors for each trial is recorded.
The errors for each trial are recorded in separate variables, and
a within-subjects factor (trial) is defined with four levels for the
four trials. The trial effect is found to be significant, while the
trial-by-anxiety interaction is not significant.

**Methods.** Type I, Type II, Type III, and Type IV sums of
squares can be used to evaluate different hypotheses. Type III is
the default.

**Statistics.** Post hoc range tests and multiple comparisons
(for between-subjects factors): least significant difference, Bonferroni,
Sidak, Scheffé, Ryan-Einot-Gabriel-Welsch multiple *F*, Ryan-Einot-Gabriel-Welsch
multiple range, Student-Newman-Keuls, Tukey's honestly significant
difference, Tukey's *b*, Duncan, Hochberg's GT2, Gabriel, Waller
Duncan* t *test, Dunnett (one-sided and two-sided), Tamhane's
T2, Dunnett's T3, Games-Howell, and Dunnett's *C*. Descriptive
statistics: observed means, standard deviations, and counts for all
of the dependent variables in all cells; the Levene test for homogeneity
of variance; Box's *M*; and Mauchly's test of sphericity.

**Plots.** Spread-versus-level, residual, and profile (interaction).

GLM Repeated Measures Data Considerations

**Data.** The dependent variables should be quantitative. Between-subjects
factors divide the sample into discrete subgroups, such as male and
female. These factors are categorical and can have numeric values
or string values. Within-subjects factors are defined in the Repeated
Measures Define Factor(s) dialog box. Covariates are quantitative
variables that are related to the dependent variable. For a repeated
measures analysis, these should remain constant at each level of a
within-subjects variable.

The data file should contain a set of variables for each group
of measurements on the subjects. The set has one variable for each
repetition of the measurement within the group. A within-subjects
factor is defined for the group with the number of levels equal to
the number of repetitions. For example, measurements of weight could
be taken on different days. If measurements of the same property were
taken on five days, the within-subjects factor could be specified
as *day* with five levels.

For multiple within-subjects factors, the number of measurements
for each subject is equal to the product of the number of levels of
each factor. For example, if measurements were taken at three different
times each day for four days, the total number of measurements is
12 for each subject. The within-subjects factors could be specified
as *day(4)* and *time(3)*.

**Assumptions.** A repeated measures analysis can be approached
in two ways, univariate and multivariate.

The univariate approach (also known as the split-plot or mixed-model
approach) considers the dependent variables as responses to the levels
of within-subjects factors. The measurements on a subject should be
a sample from a multivariate normal distribution, and the variance-covariance
matrices are the same across the cells formed by the between-subjects
effects. Certain assumptions are made on the variance-covariance matrix
of the dependent variables. The validity of the *F* statistic
used in the univariate approach can be assured if the variance-covariance
matrix is circular in form (Huynh and Mandeville, 1979).

To test this assumption, Mauchly's test of sphericity can be used,
which performs a test of sphericity on the variance-covariance matrix
of an orthonormalized transformed dependent variable. Mauchly's test
is automatically displayed for a repeated measures analysis. For small
sample sizes, this test is not very powerful. For large sample sizes,
the test may be significant even when the impact of the departure
on the results is small. If the significance of the test is large,
the hypothesis of sphericity can be assumed. However, if the significance
is small and the sphericity assumption appears to be violated, an
adjustment to the numerator and denominator degrees of freedom can
be made in order to validate the univariate *F* statistic. Three
estimates of this adjustment, which is called **epsilon**, are
available in the GLM Repeated Measures procedure. Both the numerator
and denominator degrees of freedom must be multiplied by epsilon,
and the significance of the *F* ratio must be evaluated with
the new degrees of freedom.

The multivariate approach considers the measurements on a subject
to be a sample from a multivariate normal distribution, and the variance-covariance
matrices are the same across the cells formed by the between-subjects
effects. To test whether the variance-covariance matrices across the
cells are the same, Box's *M* test can be used.

**Related procedures.** Use the Explore procedure to examine
the data before doing an analysis of variance. If there are *not* repeated
measurements on each subject, use GLM Univariate or GLM Multivariate.
If there are only two measurements for each subject (for example,
pre-test and post-test) and there are no between-subjects factors,
you can use the Paired-Samples T Test procedure.

Obtaining GLM Repeated Measures

This feature requires SPSS® Statistics Standard Edition or the Advanced Statistics Option.

- From the menus choose:
- Type a within-subject factor name and its number of levels.
- Click Add.
- Repeat these steps for each within-subjects factor.
To define measure factors for a doubly multivariate repeated measures design:

- Type the measure name.
- Click Add.
After defining all of your factors and measures:

- Click Define.
- Select a dependent variable that corresponds to each combination of within-subjects factors (and optionally, measures) on the list.

To change positions of the variables, use the up and down arrows.

To make changes to the within-subjects factors, you can reopen the Repeated Measures Define Factor(s) dialog box without closing the main dialog box. Optionally, you can specify between-subjects factor(s) and covariates.

This procedure pastes GLM: Repeated Measures command syntax.