INTERPRETING THE REPEATED-MEASURES ANOVAUSING THE SPSS GENERAL LINEAR MODEL PROGRAMRM ANOVAIn this scenario (based on a RM ANOVA example from Leech, Barrett, and Morgan, 2005) – each of 12 participants has evaluated fourproducts (e.g., four brands of DVD players) on 1-7 (1 = very low quality to 7 = very high quality) Likert-type scales. There is oneIndependent Variable with four (4) test conditions (P1, P2, P3, and P4) for this scenario, where P1-4 represents the four different products.The Dependent Variable is a rating of preference by consumers about the products. Each participant rated each of the four products – and assuch, the analysis to determine if a difference in preference exists is the RM ANOVA.RM ANOVA SyntaxGLM p1 p2 p3 p4 /WSFACTOR = product 4 Polynomial /METHOD = SSTYPE(3) /PLOT = PROFILE( product ) /PRINT = DESCRIPTIVE /CRITERIA = ALPHA(.05) /WSDESIGN = product .General Linear ModelThis first table identifies the four levels of the within subjects repeated measures independent variable, product. For each level (P1 to P4), there is arating of 1-7, which is the dependent variable.Within-Subjects FactorsMeasure: MEASURE_1p1p2p3p4product1234DependentVariableThis next table provides the descriptive statistics (Means, Standard Deviations, and Ns) for the average rating for each of the four levels…Descriptive Statistics4.67 1.923 123.58 1.929 123.83 1.642 123.00 1.651 12p1 Product 1p2 Product 2p3 Product 3p4 Product 4Mean Std. Deviation NThe next table shows four similar multivariate tests of the within subjects effect. These are actually a form of MANOVA (Multivariate Analysis ofVariance). In this case, all four tests have the same Fs and are significant. If the sphericity assumption is violated, a multivariate test could be used(such as one of the procedures shown below), which corrects the degrees of freedom. Typically, we would report the Wilk’s Lambda (Λ) line ofinformation – which indicates significance among the four test conditions (levels).This table presents four similar multivariate tests of the within-subjects effect (i.e., whether the four products are rated equally). Wilk’sLambda is a commonly used multivariate test. Notice that in this case, the Fs, dfs, and significance levels are the same: F(3, 9) = 19.065, p <0.001. The significant F means that there is a difference somewhere in how the products are rated. The multivariate tests can be used whetheror not sphericity is violated. However, if epsilons are high, indicating that one is close to achieving sphericity, the multivariate tests may beless powerful (less likely to indicate statistical significance) than the corrected univariate repeated-measures ANOVA.Multivariate Testsb.864 19.065a3.000 9.000 .000.136 19.065a3.000 9.000 .0006.355 19.065a3.000 9.000 .0006.355 19.065a3.000 9.000 .000Pillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootEffectproductValue F Hypothesis df Error df Sig.Exact statistica. Design: Intercept Within Subjects Design: productb. RM ANOVA EXAMPLEPAGE 2This next table shows the test of an assumption of the univariate approach to repeated-measures ANOVA known as sphericity. As is commonly thecase, the Mauchly statistic is significant and, thus the assumption is violated. This is shown by the Sig. (p) value of .001 is less than the a priori alphalevel of significance (.05). The epsilons, which are measures of degree of sphericity, are less than 1.0, indicating that the sphericity assumption isviolated. The "lower-bound" indicates the lowest value that epsilon could be. The highest epsilon possible is always 1.0. When sphericity is violated,you can either use the multivariate results or use epsilon values to adjust the numerator and denominator degrees of freedom. Typically, whenepsilons are less than .75, use the Greenhouse-Geisser epsilon, but use Huynh-Feldt if epsilon > .75.This table shows that the Mauchly’s Test of Sphericity is significant, which indicates that these data violate the sphericity assumption of theunivariate approach to repeated-measures ANOVA. Thus, we should either use the multivariate approach, use the appropriate non-parametrictest (Friedman), or correct the univariate approach with the Greenhouse-Geisser adjustment or other similar correction.Mauchly's Test of SphericitybMeasure: MEASURE_1.101 22.253 5.001 .544.626 .333Within Subjects EffectproductMauchly's WApprox.Chi-Square df Sig.Greenhouse-GeisserHuynh-Feldt Lower-boundEpsilonaTests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional toan identity matrix.May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in theTests of Within-Subjects Effects table.a. Design: Intercept Within Subjects Design: productb. RM ANOVA EXAMPLEPAGE 3In the next table, note that 3 and 33 would be the dfs to use if sphericity were not violated. Because the sphericity assumption is violated, we will usethe Greenhouse-Geisser correction, which multiplies 3 and 33 by epsilon, which in this case is .544, yielding dfs of 1.632 and 17.953.You can see in the Tests of Within-Subjects Effects table that these corrections reduce the degrees of freedom by multiplying them by Epsilon.In this case, 3  .544 = 1.632 and 33  .544 = 17.953. Even with this adjustment, the Within-Subjects Effects (of Product) is significant,F(1.632, 17.952) = 23.629, p < 0.001, as were the multivariate tests. This means that the ratings of the four products are significantlydifferent. However, the overall (Product) F does not tell you which pairs of products have significantly different means.Tests of Within-Subjects EffectsMeasure: MEASURE_117.229 3 5.743 23.629 .00017.229 1.632 10.556 23.629 .00017.229 1.877 9.178 23.629 .00017.229 1.000 17.229 23.629 .0018.021 33 .2438.021 17.953 .4478.021 20.649 .3888.021 11.000 .729Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSourceproductError(product)Type III Sumof Squares df Mean Square F Sig.RM ANOVA EXAMPLEPAGE 4FYI: SPSS has several tests of within-subjects contrasts – such as the example shown below… These contrasts can serve as a viable method ofinterpreting the pairwise contrasts of the repeated-measures main effect.Tests of Within-Subjects ContrastsMeasure: MEASURE_113.537 1 13.537 26.532 .000.188 1 .188 3.667 .0823.504 1 3.504 20.883 .0015.613