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.
References
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.
Belmont , 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.
New York
Meyers, L.S., Gamst, G., & Guarino, A. (2006). Applied multivariate research: Design and interpretation. Thousand Oaks , CA
