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### MANOVA - Reporting (Type 2)

Prior to conducting the MANOVA, a series of Pearson correlations were performed between all of the dependent variables in order to test the MANOVA assumption that the dependent variables would be correlated with each other in the moderate range (Meyer, Gampst, & Guarino, 2006). As can be seen in Table 1, a meaningful pattern of correlations was observed amongst most of the dependent variables, suggesting the appropriateness of a MANOVA. Additionally, the Box’s M value of 127.71 was associated with a p value of .009, which was interpreted as non-significant based on Huberty and Petoskey’s (2000) guideline (i.e., p < .005). Thus, the covariance matrices between the groups were assumed to be equal for the purposes of the MANOVA.

A one-way multivariate analysis of variance (MANOVA) was conducted to test the hypothesis that there would be one or more mean differences between education levels (undergraduate, masters, PhD) and intelligence test scores. A statistically significant MANOVA effect was obtained, Pillais’ Trace = .30, F(18, 1218) = 11.94, p < .001. The multivariate effect size was estimated at .150, which implies that 15.0% of the variance in the canonically derived dependent variable was accounted for by educational level.

As the independent variable was associated with three levels, two eigenvalues and canonical correlations were extracted by the MANOVA. The first eigenvalue was equal to .41 and accounted for nearly all (98.0%) of the model variance. The canonical correlation associated with the first eigenvalue was equal to .54, which implies that 29.2% of the variance in the discriminant function derived scores was accounted for by education level. By contrast, the second eigenvalue was equal to .008 and a corresponding canonical correlation of .09, which was not found to be statistically significant (Wilks Λ = .99, F(8, 609), p = .757)

To help interpret the statistically significant MANOVA effect, the standardized discriminant function coefficients were consulted. As can be seen in Table 2, the standardized discriminant function coefficents suggested that the three education levels were maximally differentiated by a canonical variate with greater weightings from the verbal2 (.33), verbal3 (.40) and spatial1 (.35) subscales. However, it will be noted that although the standardized coefficients suggested relatively minimal unique contributions to the MANOVA effect from several subscales, the correlations between the subscale scores and the canonically derived scores were all positive and appreciable in magnitude (range = .44 to .72).

The estimate the group centroids (i.e., canonically derived group means) for the three education levels, the participant subscale raw scores were multiplied by the corresponding subscale unstandardized discriminant function coefficients and then summed across all participants. The PhD group was associated with the largest group centroid (M = 6.47, SD = 1.11), the MA group was associated with the next largest group centroid (M = 5.48, SD = .99), and, finally, the the undergraduate group was associated with the smallest group centroid (M = 4.91, SD = .90). In accordance with Enders (2003), an ANOVA was performed on the canonically derived data. However, because the sampling distribution associated with multivariate derived data is known to be different to that which is univariate (Neufeld & Gardner, 1990), a conservative approach to statistical significance testing was applied. Specifically, an alpha level of .001 was specified for the ANOVA. An ANOVA with three levels in the independent variable (undergraduate, masters and PhD) was performed on the canonically derived intelligence dependent variable, which yielded F(2, 616), = 126.88, p < .001, and η2 = .292, which implies that 29.2% of the variance in the canonically derived intelligence scores was accounted for by education level (note, this value corresponds to the canonical correlation associated with the first eigenvalue reported above, i.e., .542 = 29.2%).

Bonferroni adjusted (.001 / 4 = .00025) post-hoc tests were performed to specifically contrast the education levels on the canonically derived intelligence variable. All contrasts were found to be statistically significant (p < .00025). The Cohen’s d values were as follows: PhD vs. Undergrad = 1.56; PhD vs. MA = .94; and MA vs. Undergrad = .60. These values are suggestive of a large effect size, according to Cohen (1992).

References

Cohen, J. (1992). A power primer. Psychological Bulletin, 112, 155-159.

Enders, C.K. (2003). Performing multivariate group comparisons following a statistically significant MANOVA. Measurement and Evaluation in Counseling and Development, 36, 40-56.

Huberty, C. J., & Petoskey, M. D. (2000). Multivariate analysis of variance and covariance. In H. Tinsley and S. Brown (Eds.) Handbook of applied multivariate statistics and mathematical modeling. New York: Academic Press.

Meyers, L.S., Gamst, G., & Guarino, A. (2006). Applied multivariate research: Design and interpretation. Thousand Oaks, CA: Sage Publishers.

Neufeld, R. W. J., & Gardner, R. C. (1990). Data aggregation in evaluating psychological constructs: Multivariate and logical deductive considerations. Journal of Mathematical Psychology, 34, 276-296.