MANOVA
Steps to Conduct a MANOVA in SPSS 1. Click Analyze, then General Linear Model, then Multivariate 2. Move your dependent variables to the “Dependent Variables” box on the right 3. Move your between-subject IVs to the “Fixed Factor(s)” box on the right 4. Click EM MEANS a. Click on your IVs and (Overall) and move to “Display Means for:” box on right b. If any of your IVs have more than 3 levels, you can choose a post-hoc test by checking “Compare man effects” and selecting an adjustment: LSD, Bonferroni, Sidak. 5. Click Options a. Click Descriptive Statistics, Estimates of Effect Size, Homogeneity Tests b. Click Continue 6. Either click OK to run or click Paste to paste syntax into a syntax window. Stepdown Analysis: To conduct a MANOVA with a Roy-Bargmann Stepdown Analysis as a follow-up to a significant MANOVA, you will need to run the following SPSS syntax. Note that you need to replace the red text with the variables names from your data and you need to order the DVs in the order you want them tested in the stepdown analysis (see Tabachnick & Fidell p. 266): manova dv1, dv2, dv3 by iv (min, max) /print=signif(stepdown), /method=sequential /design iv. EXAMPLE A researcher was interested in was interested in comparing the effectiveness of two types of instructional methods to teach clarinet to students. One set of twelve elementary-aged clarinet students used a programmed instruction technique (experimental group) to learn clarinet and another group of eleven elementary-aged clarinet students received instruction via traditional classroom instruction (control group). At the end of the semester, two judges rated each clarinet student on the following three performance aspects: interpretation (INT), tone (TONE), and rhythm (RHY). The data, representing the average of two judges’ ratings, are listed in the table below, with GP = 1 referring to the experimental group and GP = 2 referring to the control group. GP INT TONE RHY 1 4.2 4.1 3.2 1 4.1 4.1 3.7 1 4.9 4.7 4.7 1 4.4 4.1 4.1 1 3.7 2.0 2.4 1 3.9 3.2 2.7 1 3.8 3.5 3.4 1 4.2 4.1 4.1 1 3.6 3.8 4.2 1 2.6 3.2 1.9 1 3.0 2.5 2.9 1 2.9 3.3 3.5 2 2.1 1.8 1.7 2 4.8 4.0 3.5 2 4.2 2.9 4.0 2 3.7 1.9 1.7 2 3.7 2.1 2.2 2 3.8 2.1 3.0 2 2.1 2.0 2.2 2 2.2 1.9 2.2 2 3.3 3.6 2.3 2 2.6 1.5 1.3 2 2.5 1.7 1.7 correlations INT TONE RHY. Correlations Correlations INT Interpretation TONE Tone RHY Rhythm INT Interpretation Pearson Correlation 1 .713 .739 Sig. (2-tailed) .000 .000 N 23 23 23 TONE Tone Pearson Correlation .713 1 .823 Sig. (2-tailed) .000 .000 N 23 23 23 RHY Rhythm Pearson Correlation .739 .823 1 Sig. (2-tailed) .000 .000 N 23 23 23 GLM INT TONE RHY BY GP /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /EMMEANS=TABLES(OVERALL) /EMMEANS=TABLES(GP) /PRINT=DESCRIPTIVE ETASQ HOMOGENEITY /CRITERIA=ALPHA(.05) /DESIGN= GP. General Linear Model Between-Subjects Factors Value Label N GP Instruction group 1 Experimental 12 2 Control 11 Descriptive Statistics GP Instruction group Mean Std. Deviation N INT Interpretation 1 Experimental 3.775 .6690 12 2 Control 3.182 .9325 11 Total 3.491 .8431 23 TONE Tone 1 Experimental 3.550 .7634 12 2 Control 2.318 .8171 11 Preliminary Analyses As a preliminary step, check the correlations among the DVs. We MANOVA works best with moderately correlated variables (around |.6|). Ours are a little on the high side, but we will proceed with the MANOVA anyways. MANOVA syntax using the GLM command Number of cases in each group. Descriptive Statistics Total 2.961 .9953 23 RHY Rhythm 1 Experimental 3.400 .8224 12 2 Control 2.345 .8311 11 Total 2.896 .9707 23 Box's Test of Equality of Covariance Matricesa Box's M 2.023 F .284 df1 6 df2 3119.084 Sig. .945 Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. a. Design: Intercept + GP Multivariate Testsa Effect Value F Hypothesis df Error df Sig. Partial Eta Squared Intercept Pillai's Trace .956 137.666b 3.000 19.000 .000 .956 Wilks' Lambda .044 137.666b 3.000 19.000 .000 .956 Hotelling's Trace 21.737 137.666b 3.000 19.000 .000 .956 Roy's Largest Root 21.737 137.666b 3.000 19.000 .000 .956 GP Pillai's Trace .431 4.794b 3.000 19.000 .012 .431 Wilks' Lambda .569 4.794b 3.000 19.000 .012 .431 Hotelling's Trace .757 4.794b 3.000 19.000 .012 .431 Roy's Largest Root .757 4.794b 3.000 19.000 .012 .431 a. Design: Intercept + GP b. Exact statistic Box’s Test of homogeneity of variance- covariance matrices. We want to fail to reject this test. In this particular case since we have approximately equal sample sizes in each group across each DV, violations of this assumption are not serious. If we had drastically different sample sizes in each group and we have evidence that we don’t meet this assumption, we should use the Pillai’s Trace criterion to evaluate the MANOVA. MANOVA Tests. These are our omnibus MANOVA tests. Since our IV only has two groups, each of the four ways to evaluate the MANOVA are identical. Generally, we report Wilk’s λ; I would use Pillai’s Trace if we violate the homogeneity of variance assumptions (see the MANOVA Handout for details on each of these criteria).The results from the MANOVA F(3, 19) = 4.79, p = .012, Wilk’s λ = .569, partial η2 = .43 indicate that the two groups differ on the combination of the three performance aspects. Partial η2 indicates that group differences are explaining around 43% of the variation in the linear combination of outcomes. Since our IV only has two groups, we can calculate a multivariate effect size similar to Cohen’s d in the univariate case; it is called Mahalanobis distance or D2. To calculate D2, we need the Hotelling’s trace value (Tr = .757) from the Multivariate Tests table. D2 = n1 + n2 n1n2 Tr(N − 2) = 12 + 11 12 ∗ 11 . 757(23 − 2) = 2.77, ?ℎ??ℎ ?? ? ???? ????? ?????? ???? Levene's Test of Equality of Error Variancesa Levene Statistic df1 df2 Sig. INT Interpretation Based on Mean 2.966 1 21 .100 Based on Median 2.494 1 21 .129 Based on Median and with adjusted df 2.494 1 20.902 .129 Based on trimmed mean 3.060 1 21 .095 TONE Tone Based on Mean .057 1 21 .813 Based on Median .071 1 21 .792 Based on Median and with adjusted df .071 1 18.085 .792 Based on trimmed mean .014 1 21 .907 RHY Rhythm Based on Mean .010 1 21 .921 Based on Median .096 1 21 .759 Based on Median and with adjusted df .096 1 20.008 .759 Based on trimmed mean .030 1 21 .865 Tests the null hypothesis that the error variance of the dependent variable is equal across groups. a. Design: Intercept + GP Tests of Between-Subjects Effects Source Dependent Variable Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model INT Interpretation 2.019a 1 2.019 3.114 .092 .129 TONE Tone 8.708b 1 8.708 13.975 .001 .400 RHY Rhythm 6.382c 1 6.382 9.342 .006 .308 Intercept INT Interpretation 277.759 1 277.759 428.298 .000 .953 TONE Tone 197.630 1 197.630 317.142 .000 .938 RHY Rhythm 189.450 1 189.450 277.297 .000 .930 Homogeneity of variance test across groups within each DV. From these results, we have evidence that we meet this assumption. However, we wouldn’t be too worried given the that we have approximately equal sample sizes in each group. If we did violate this assumption (and we have more than 2 groups), we would report Pillai’s Trace criteria. Follow-up Univariate ANOVAs. The following table contains results from univariate ANOVAs that compare the IV on each DV separately. GP INT Interpretation 2.019 1 2.019 3.114 .092 .129 TONE Tone 8.708 1 8.708 13.975 .001 .400 RHY Rhythm 6.382 1 6.382 9.342 .006 .308 Error INT Interpretation 13.619 21 .649 TONE Tone 13.086 21 .623 RHY Rhythm 14.347 21 .683 Total INT Interpretation 295.990 23 TONE Tone 223.430 23 RHY Rhythm 213.580 23 Corrected Total INT Interpretation 15.638 22 TONE Tone 21.795 22 RHY Rhythm 20.730 22 a. R Squared = .129 (Adjusted R Squared = .088) b. R Squared = .400 (Adjusted R Squared