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