In accordance with Cramer and Bock (1966), a MANOVA was first performed on the means to help protect against inflating the Type 1 error rate in the follow-up ANOVAs and post-hoc comparisons. However, 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 (i.e., .20 - .60; Meyers, 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.
Prior to conducting a series of follow-up ANOVAs, the homogeneity of variance assumption was tested for all nine intelligence subscales. Based on a series of Levene’s F tests, the homogeneity of variance assumption was considered satisfied, even though two of the nine Levene’s F tests were statistically significant (p > .05). Specifically, although the Levene’s F test suggested that the variances associated with the memory3 and spatial1 subscales were not homogenous, an examination of the standard deviations (see Table 2) revealed that none of the largest standard deviations were more than four times the size of the corresponding smallest, suggesting that the ANOVA would be robust in this case (Howell, 2009). A series of one-way ANOVA’s on each of the nine dependent variables was conducted as a follow-up tests to the MANOVA. As can be seen in Table 2, all of the ANOVA’s were statistically significant, with effect sizes (partial η2) ranging from a low of .073 (spatial2) to a high of .178 (verbal2).
Finally, a series of post-hoc analyses (Fisher’s LSD) were performed to examine individual mean difference comparisons across all three levels of education and all nine intelligence subscales. The results revealed that all post-hoc mean comparisons were statistically significant (p < .05). In all cases, the trend of the effect was linear. That is, on average, MA students were more intelligent than undergraduates and PhD students were, on average, more intelligent than MA students. The effect sizes as estimated by Cohen’s d are reported in Table 3. It can be observed that the largest effects tended to be associated with the verbal subscales with average Cohen’s d values equal to .74 to .75, which is a larger effect according to Cohen’s (1990) guidelines.
Cohen, J. (1992). A power primer. Psychological Bulletin, 112, 155-159.
Cramer, E. M., & Bock, R. D. (1966). Multivariate analysis. Review of Educational Research, 36, 604-617.
Howell, D. C. (2007). Statistical methods for psychology.
CA: Thompson . Wadsworth
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.
Academic Press. New York
Meyers, L.S., Gamst, G., & Guarino, A. (2006). Applied multivariate research: Design and interpretation.
Publishers. Thousand Oaks, CA