# Repeated measures ANOVA

The Repeated measures ANOVA procedure analyzes groups of related dependent variables that represent different measurements of the same attribute. Note that the order in which you specify within-subjects factors is important. Each factor constitutes a level within the previous factor. 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 Repeated measures ANOVA procedure provides 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. These matrices are called SSCP (sums-of-squares and cross-products) matrices. In addition to testing hypotheses, Repeated measures ANOVA produces estimates of parameters.

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.

- Example
- In a weight-loss study, suppose the weights of several people are measured
each week for five weeks. In the data file, each person is a subject or case. The weights for the
weeks are recorded in the variables
*weight1*,*weight2*, and so on. The gender of each person is recorded in another variable. The weights, measured for each subject repeatedly, can be grouped by defining a within-subjects factor. The factor could be called*week*, defined to have five levels. The variables*weight1*, ...,*weight5*are used to assign the five levels of*week*. The variable in the data file that groups males and females (*gender*) can be specified as a between-subjects factor to study the differences between males and females. - Measures
- If subjects were tested on more than one measure at each time, define the measures. For example, the pulse and respiration rate could be measured on each subject every day for a week. These measures do not exist as variables in the data file but are defined here. A model with more than one measure is sometimes called a doubly multivariate repeated measures model.
- 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).

## Data considerations

- Data
- The within-subjects factor variables should be quantitative.
- Assumptions
- The multivariate approach considers the measurements on a subject to be a sample from a multivariate normal distribution.
- 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 Mixed between-within ANOVA. 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 a Repeated measures ANOVA

This feature requires Custom Tables and Advanced Statistics.

- From the menus choose:
- Enter a Factor name under the
Within-subject factors section. The factor names represent the independent
variables that constitute the different time points or conditions at which the dependent variables
are measured. You can use the Number of levels control to specify the number
of levels for the associated factor name.
You can optionally click Add a factor to enter additional factor names.

- Click Select variables under the
Measures section to define dependent variables that were repeatedly measured
across the within-subjects factor levels. The Select variables dialog allows to specify a variable
for each factor level that is specified in the Within-subject factors
section. Click OK when done.
You can optionally click Add to create additional measures.

- Optionally, you can select the following options from the
Additional settings menu:
- Click Contrasts to test for differences among the factor variables.
- Click Statistics to select which statistics to include in the procedure.
- Click EM means to select the factors and interactions for which you want estimates of the population marginal means in the cells.
- Click Plots to enable the display of charts in the output and to select the charting settings.
- Click Options to specify null hypothesis settings and control of the treatment of missing data.
- Click Save to dataset to add values predicted by the model, residuals, and related measures to the dataset as new variables.

- Click Run analysis.

This procedure pastes GLM: Repeated Measures command syntax.