# Summary Results for Within Effects in GLM and ANOVA

Within-Subject (Repeated Measures) Designs

The Within effects results are available on the GLM and ANOVA Results - Summary tab.

Within effects. The options in the Within effects group box allow you to review, as appropriate for the given design, various results statistics for the within-subject (repeated measures) design. If the current design does not include within-subject (repeated measures) factors, these options are not displayed on this tab.

Multiv. tests. Click the Multiv. tests button to display a spreadsheet with the multivariate results for the within-subject (repeated measures) factors. If there are no within-subject (repeated measures) factors with more than two levels, then this option is dimmed. The multivariate tests that are computed can be selected in the Multiv. tests group box (see GLM and ANOVA Results - Summary tab).

Note: univariate and multivariate tests for repeated measures. The way in which within-subject (repeated measures) factors are handled in the general linear model is described in the Within-subject (repeated measures) designs topic (see also the Introductory Overview to the ANOVA/MANOVA module); a detailed discussion of the reasons for computing multivariate tests for repeated measures effects with more than 1 degree of freedom is presented in the ANOVA/MANOVA section on sphericity and compound symmetry). In short, when a repeated measures effect has more than a single degree of freedom (e.g., a main effect Time with three levels, which would have 2 degrees of freedom), the univariate F statistic reported in the ANOVA table is based on the assumption that the hypotheses associated with each single degree of freedom are orthogonal (e.g., that the changes from Time 1 to Time 2 and 3, and from Time 2 to Time 3 are uncorrelated). This assumption is often not tenable (you can review the Error Correlations (see below) to get an idea of how the M-transformed dependent variables are correlated), and in that case such a violation of the so-called sphericity and compound symmetry assumption may invalidate the univariate F-test (see, for example, Winer, Brown, and Michels, 1991, Section 4.4, for details). One way to remedy this problem is to use various adjustment factors to arrive at a corrected F statistic; for example, the Greenhouse-Geisser and Huynh-Feldt statistics (see below) will compute reduced degrees of freedom (to reflect the fact that the within-subject hypotheses are correlated, and not independent) for the univariate F statistic. The preferred approach, however, is to treat the simultaneous within-subject hypotheses (M-transformed dependent variables) as multivariate dependent variables, and then to compute the standard multivariate results. You can review the hypothesis and error matrices for the within-subject model via the Effect SSCPs and Error SSCPs options described below.

Univ. tests. Click the Univ. tests button to display a spreadsheet with the univariate ANOVA results for the within-subject (repeated measures) effects. For effects with more than a single degree of freedom, these tests should be interpreted with caution, because they are only valid under the assumptions of sphericity and compound symmetry regarding sums of squares error matrices; see also the discussion of the Multivariate designs for additional detail.

G-G and H-F. Click the G-G and H-F button to display a spreadsheet with the adjusted univariate ANOVA tests for the within-subject (repeated measures) effects. These tests are discussed in the ANOVA/MANOVA section on sphericity and compound symmetry. In short, they aim at adjusting the degrees of freedom for the respective tests, based on the intercorrelations among the respective within-subject (repeated measures) hypotheses (which are assumed to be orthogonal for the simple univariate tests). These methods are also known as the Greenhouse-Geisser and Huynh-Feldt adjustments (see Greenhouse & Geisser, 1959; Huynh & Feldt, 1970). STATISTICA will report results for both of these adjustments, in the columns labeled G-G (Greenhouse & Geisser) and H-F (Huynh & Feldt); additional columns will report the results based on the lower bound estimate for the adjustment factor (Lowr. Bnd).

Effect SSCPs. Click the Effect SSCPs button to display the sums of squares and cross-product matrices (SSCP matrices) for the within-subject (repeated) measures effects (i.e., the respective M-transformed SSCP matrices for the between effects). Note that the sum of the diagonal elements for each matrix (and for each dependent measure) is equal to the sums of squares for the respective Univariate tests (see above). See also the discussion of the Multivariate designs in the Introductory Overview for details.

Sphericity. Click the Sphericity button to display a spreadsheet with the results for Mauchly's test of sphericity. This test is only applicable to designs with within-subject (repeated measures) factors with more than two levels. The sphericity assumption is related to the compound symmetry assumption discussed in the context of the Note: univariate and multivariate tests for repeated measures described above; both are concerned with the conditions under which the univariate approach to repeated measures ANOVA yields valid statistical significance tests.

The Mauchly test of sphericity evaluates the hypothesis that the sphericity assumption holds (null hypothesis); if the test is significant, then we reject the hypothesis of sphericity and declare the assumption violated. However, Monte Carlo studies (Rogan, Keselman & Mendoza, 1979; Keselman, Rogan, Mendoza & Breen, 1980) have shown the Mauchly test to be very sensitive to departures from multivariate normality; yet, even minor (non-significant) violations of the sphericity assumption can lead to erroneous conclusions in the ANOVA (see Bock, 1975; Box, 1954; Kirk, 1982). Therefore, we advise you to always compute the multivariate statistics for repeated measures, just in case.

Error SSCPs. Click the Error SSCPs button to display the error sums of squares and cross-product matrices (SSCP matrices) for the within-subject (repeated) measures effects (i.e., the respective M-transformed SSCP error matrices). Note that the sum of the diagonal elements for each matrix (and for each dependent measure) is equal to the sums of squares error for the respective Univariate tests (see above). See also the discussion of the Multivariate designs in the Introductory Overview for details.

Error Corrs. Click the Error Corrs button to display the same Error SSCP matrix (see previous option), with the entries converted to correlations (instead of sums of squares and cross-products). This matrix is useful for evaluating the sphericity and compound symmetry assumptions, and whether or not univariate tests for the within-subject (repeated measures) effects are valid. See the discussion of the Multivariate designs in the Introductory Overview for details; see also the ANOVA/MANOVA section on sphericity and compound symmetry.