INTERPRETING THE ONE-WAY MANOVAAs a means of checking multicollinearity, the circled correlation (between the dependentvariables) should be low to moderate. If the correlation were .60 (some argue .80) or above, wewould consider either making a composite variable (in which the highly correlated variableswere summed or averaged) or eliminating one of the dependent variables.Correlationsa1 .439** .413**. .000 .000.439** 1 .449**.000 . .000.413** .449** 1.000 .000 .Pearson CorrelationSig. (2-tailed)Pearson CorrelationSig. (2-tailed)Pearson CorrelationSig. (2-tailed)Grades in High SchoolMath Achievement TestFather's Education LevelGrades inHigh SchoolMathAchievementTestFather'sEducationLevelCorrelation is significant at the 0.01 level (2-tailed).**. Listwise N=75a. Ideally, we would like to see a significant relationship between the independent variable(s) and the dependent variables.To meet the assumptions – it is best to have approximately equal cell sizes. That meaning, thelargest cell size (N) is not more than 1.5 times larger than the smallest cell size (N). For thisexample, we do not have a concern.Descriptive Statistics5.16 1.375 255.20 1.555 256.76 1.332 255.71 1.592 758.4134 5.49859 2513.5733 5.21092 2515.7067 7.10659 2512.5645 6.67031 75Father's Education LevelHS grad or lessSome CollegeBS or MoreTotalHS grad or lessSome CollegeBS or MoreTotalGrades in High SchoolMath Achievement TestMean Std. Deviation NThe Box’s Test of Equality of Covariance Matrices checks the assumption of homogeneity ofcovariance across the groups using p < .001 as a criterion. For our example, we do not have aconcern – as Box’s M (4.572) was not significant, p (.624) > α (.001) – indicating that there areno significant differences between the covariance matrices. Therefore, the assumption is notviolated and Wilk’s Lambda is an appropriate test to use.Box's Test of Equality of Covariance Matricesa4.572.7316129201.2.624Box's MFdf1df2Sig.Tests the null hypothesis that the observed covariancematrices of the dependent variables are equal across groups.Design: Intercept+FathEduca. The following is the MANOVA using the Wilk’s Lambda test. Using an alpha level of .05, wesee that this test is significant, Wilk’s Λ = .662, F(4, 142) = 8.118, p < .001, multivariate η2 = .19.This significant F indicates that there are significant differences among the Fathers Education(FathEduc) groups on a linear combination of the three dependent variables. If we had violatedthe assumption of homogeneity of variance-covariance, one could use the Pillai’s Trace test.Multivariate Testsc.945 613.377a2.000 71.000 .000 .945.055 613.377a2.000 71.000 .000 .94517.278 613.377a2.000 71.000 .000 .94517.278 613.377a2.000 71.000 .000 .945.364 8.012 4.000 144.000 .000 .182.662 8.118a4.000 142.000 .000 .186.470 8.219 4.000 140.000 .000 .190.358 12.883b2.000 72.000 .000 .264Pillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootPillai's TraceWilks' LambdaHotelling's TraceRoy's Largest RootEffectInterceptFathEducValue F Hypothesis df Error df Sig.Partial EtaSquaredExact statistica. The statistic is an upper bound on F that yields a lower bound on the significance level.b. Design: Intercept+FathEducc. The multivariate η2 = .186 indicates that approximately 19% of multivariate variance of thedependent variables is associated with the group factor.INTERPRETING MANOVAPAGE 2The Levene’s Test of Equality of Error Variances tests the assumption of MANOVA and ANOVAthat the variances of each variable are equal across the groups. If the Levene’s test is significant,this means that the assumption has been violated – and data should be viewed with caution – orthe data could be transformed so as to equalize the variances. As we see in this example, theassumption is met for both dependent variables (Grades in High School, p > .05, and MathAchievement Test, p > .05).Levene's Test of Equality of Error Variancesa.818 2 72 .4453.105 2 72 .051Grades in High SchoolMath Achievement TestF df1 df2 Sig.Tests the null hypothesis that the error variance of the dependent variable isequal across groups.Design: Intercept+FathEduca. Because the MANOVA was significant, we will now examine the univariate ANOVA results.Note that these tests are identical to the three separate univariate one-way ANOVAs we wouldhave performed if we opted not to do the MANOVA – provided that there are no missing data.Because the Grades in High School and Math Achievement Test dependent variables arestatistically significant and there are three levels or values of Fathers Education, we would needto do post hoc multiple comparisons or contrasts to see which pairs of means are different. The pvalues for the ANOVAs on the MANOVA output do not take into account that multiple ANOVAshave been conducted. To protect against Type I error, we can use a traditional Bonferroniprocedure and test each ANOVA at the .025 level (.05 divided by the number of ANOVAsconducted). As can be seen, both ANOVAs are significant at the .025 adjusted alpha level (p < .001 for both).Tests of Between-Subjects Effects41.627a2 20.813 10.270 .000 .222703.079b2 351.539 9.775 .000 .2142442.453 1 2442.453 1205.158 .000 .94411839.912 1 11839.912 329.216 .000 .82141.627 2 20.813 10.270 .000 .222703.079 2 351.539 9.775 .000 .214145.920 72 2.0272589.402 72 35.9642630.000 7515132.393 75187.547 743292.481 74Dependent VariableGrades in High SchoolMath Achievement TestGrades in High SchoolMath Achievement TestGrades in High SchoolMath Achievement TestGrades in High SchoolMath Achievement TestGrades in High SchoolMath Achievement TestGrades in High SchoolMath Achievement TestSourceCorrected ModelInterceptFathEducErrorTotalCorrected TotalType III Sumof Squares df Mean Square F Sig.Partial EtaSquaredR Squared = .222 (Adjusted R Squared = .200)a. R Squared = .214 (Adjusted R Squared = .192)b. INTERPRETING MANOVAPAGE 3Follow-up univariate ANOVAs (shown above) indicated that both Grades in High School andMath Achievement Test were significantly different for children of fathers with different degreesof education, F(2, 72) = 10.270, p < .001, η2 = .222 and F(2, 72) = 9.775, p < .001, η2 = .214,respectively.The results of the pairwise comparisons are shown below. We had previously controlled for TypeI error across the two univariate ANOVAs by testing each at the .025 alpha level. To