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
