1. Analysis of variance (ANOVA)-the General Linear Model (GLM)
Kazimieras Pukėnas

2. 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.

3. GLM Repeated Measures
For example, we have a independent groups with n subjects under test and the same participants take part in b conditions (repetitions) of an experiment, i.e., the within-subjects factor has b levels and between-subjects factor, which divide the population into groups, has a 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).
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.

4. GLM Repeated Measures Assumptions:
The measurements on a subject should be a sample from a 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 as responses to the levels of within-subjects factors. 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.

5. GLM Repeated Measures
The two‐way GLM Repeated Measures tests three hypotheses:
the main effects on repeated measures of between-subjects (grouping) factor A;
the main effects of within-subjects (conditions) factor B;
interaction effect between factors.
For interval scale dependent variables with unknown means , and variance ,
where a – the number of categories of factor A, b – the number of categories of factor B, we can test the hypotheses

6. GLM Repeated Measures
where null hypothesis H0 is that the grouping factor A has no influence on the response variable;
where null hypothesis H0 is that the conditions factor B has no influence on the response variable;
where

7. GLM Repeated Measures
The same decision rule is used for each hypothesis:
The null hypothesis H0 is rejected if ;
The null hypothesis H0 is not rejected if ;
where is the significance level;

8. GLM Repeated Measures Obtaining GLM Repeated Measures:
Open the file with the data analyzed. The data file should contain a set of variables equal to the number of repetitions.
From the menus choose: Analyze  General Linear Model  Repeated Measures...
Type a within-subject factor name (field Within- Subject Factor Name) and its number of levels (field Number of Levels) in Repeated Measures Define Factor(s) dialog box (Fig.1) and click Add.
For multiple within-subjects factors repeat these steps for each within-subjects factor. To aid identification which variables correspond to the particular combinations of within-subject factors, at first the largest temporal scale is defined.
If multivariate repeated measures analysis of variance is used, type successively the measure name in field Measure Name and click Add.

9. GLM Repeated Measures
Fig. 1. Repeated Measures Define Factor(s) dialog box

10. GLM Repeated Measures
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 in Repeated Measures dialog box (Fig. 2).
Note. 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.
In a multivariate repeated measures design (several correlated parameters are measured simultaneously), the number of measurements for each subject is equal to the product of number of parameters and the total number of levels of the within-subjects factors.
Specify between-subjects factor(s) (and optionally, covariates).

11. GLM Repeated Measures
Fig. 2 . Repeated Measures dialog box

12. GLM Repeated Measures
The functions of dialog boxes Model..., Contrasts..., Plots..., Post Hoc..., Save... are explained in Analysis of variance (ANOVA)-the General Linear Model (GLM), therefore we will focuses only on the Options… dialog box (Fig.3);
Select:
Estimates of effect size;
Observed power;
Homogeneity tests;
For more about this options, see Analysis of variance (ANOVA)-the General Linear Model (GLM);

13. GLM Repeated Measures
Fig. 3. Repeated Measures: Options dialog box

14. GLM Repeated Measures
By analogy to Post hoc multiple comparison tests (can determine which means across between-subjects factors differ) also pairwise comparisons among estimated marginal means for within-subjects factors can be provided (if the number of measurements is at least three);
Select the within-subjects factors in Factor(s) and Factor interactions (these means are adjusted for the covariates, if any) and move to the right-hand box Display Means for. Check Compare main effects and select least significant difference (LSD), Bonferroni, or Sidak adjustment to the confidence intervals and significance from Confidence interval adjustment drop-down list. Bonferroni adjustment is not applicable at the large number of measurements.

15. Example
Example. We will consider free-throw performance dynamics of young basketball players. There are two groups of players trained by two different training methods. All players were tested five times at regular time intervals. Each test was performed in two series of 50 shots and total score was used for this study (hypothetical data). The data file fragment is show in Fig. 4, who dependent variables test_01, ..., test_05 are levels of within-subjects factor and group is between-subjects factor.
The following basic tables are obtained from the GLM Repeated Measures output.

16. Example
Fig. 4. Data View

17. Example
Box's M tests the assumption of multivariate models that the dependent variables are drawn from a multivariate normal distribution and that there is homogeneity of the covariance matrices of the dependent variables across all level combinations of the between-subjects factors. A finding of nonsignificance upholds the multivariate tests. For this example, Box's M is nonsignificant (Fig. 5) so we can conclude that the assumptions of the multivariate model are not violated. Due to hight sensitivity of Box's M test, the null hypothesis that the covariance matrices are equal across design cells are rejected at ;

18. Example
Fig. 5. Box’s M test of the homogeneity of the covariance matrices
of the dependent variables;

19. Example
The Multivariate Tests table (Fig.6) provides F tests of the within-subjects factor and its interactions with between-subjects grouping factor. Four variants of the F test are provided (Pillai's trace, Wilks' lambda, Hotelling's trace, and Roy's largest root). Usually the same substantive conclusion emerges from any variant. For these data, we conclude that effect of the within-subjects factor are highly signifficant, p(Sig.) = 0.000, with a Partial Eta-Squared effect size of 0.941, and Observed Power 1.000, and its interaction with between-subjects grouping factor is non-significant.

20. Example
Fig. 6. Multivariate Tests table

21. Example
Mauchly's test of sphericity (Fig. 7) is used in relation to meeting the assumptions of univariate models (and tests of within-subject effects, illustrated below). A finding of non-significance corresponds to concluding that assumptions are met. For the example below, there is a finding of non-significance, meaning that there is no violation of sphericity.

22. Fig. 7. Mauchly’s Test of Sphericity table

23. Example
The Tests of Within-Subjects Effects table (Fig. 8) displays univariate tests of within-subjects effects. Of course, if Mauchly's test showed no violation of sphericity, this table would be interpreted in terms of the "Sphericity assumed" rows. If Mauchly's test shows violation of sphericity, this may be compensated by an epsilon adjustment: Greenhouse- Geisser, Huynh-Feldt, and lower bound. Greenhouse- Geisser is the most widely-used. As in Multivariate Tests case, we conclude that effect of the within-subjects factor are highly signifficant, p = 0.000, with a Partial Eta-Squared effect size of 0.840, and Observed Power 1.000, and its interaction with between-subjects grouping factor is non-significant due to small Partial Eta-Squared effect size and non-acceptable Observed Power (despite that p<0.05);

24. Example
Fig. 8. Tests of Within-Subjects Effects table

25. Example
Levene's test tests homogeneity of variance. In a well-fitting model, error variance of each repeated measures dependent variable should be the same across groups formed by the between-subjects (grouping) factors. If the Levene statistic is significant at the .05 level or better, the researcher rejects the null hypothesis that the groups have equal variances. In our example, resulting p-value of Levene's test is greater than significance level as are shown in table Levene’s Test of Equality of Error Variances (Fig.9). That is, assumptions are met. Note, that the Levene’s test is robust in the face of departures from normality.

26. Example
Fig. 9. Levene’s Test of Equality of Error Variances table

27. Example
The Tests of Between-Subjects Effects table (Fig. 10) displays tests of between-subjects (grouping) effects. In this example, we conclude that the between-subject effects of group is non-significant .
The Pairwise Comparisons table (Fig. 11) gives pairwise comparisons among estimated marginal means for levels of within-subject factor. By looking at the significance values we can see that significant differences are between all levels of within-subject factor, except 4 and 5 levels.

28. Example
Fig. 10. Tests of Between-Subject Effects table

29. Example
Fig. 11. Pairwise Comparisons table

30. Example
The profile plot Estimated Marginal Means (Fig. 12) shows the estimated marginal means for each level of the within-subject factor (for each of the fifth of measurements), and for each value of between-subject factor (group). As we can see, there is no-significant effect of interaction between within-subject and between-subject factors: parallel or roughly parallel lines indicate lack of interaction effects. Graphical analysis of interactions must be confirmed by Multivariate Tests and Tests of Within-Subjects Effects findings.

31. Example
Fig. 12. Estimated Marginal Means profile plots

32. References GLM UNIVARIATE, ANOVA, AND ANCOVA
By G. David Garson, Statistical Associates Publishing
MULTIVARIATE GLM, MANOVA, AND MANCOVA
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