Summary Results for Within Effects in GLM and ANOVA

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.

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).

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.

See also GLM - Index.