The Assumption of Homogeneity of Variance

The assumption of homogeneity of variance is an assumption of the ANOVA that assumes that all groups have the same or similar variance.  The ANOVA utilizes the F statistic, which is robust to the assumption, as long as group sizes are equal.  Equal group sizes may be defined by the ratio of the largest to smallest group being less than 1.5.  If group sizes are vastly unequal and homogeneity of variance is violated, then the F statistic is considered liberal when large sample variances are associated with small group sizes.  When this occurs, the alpha value is greater than the level of significance.  This indicates that the null hypothesis is being falsely rejected.  On the other hand, the F statistic is considered too conservative if large variances are associated with large group sizes.  This would mean that the actual alpha value is less than the level of significance.  This does not cause the same problems as falsely rejecting the null hypothesis, however, it can cause a decrease in the power of the study.  

To test for homogeneity of variance there are several statistical tests that can be used; these tests include: Hartley’s F_max, Cochran’s, Levene’s and Barlett’s test.  Several of these assessments have been found to be too sensitive to non-normality and are not frequently used.  Of these tests, a more common assessment for homogeneity of variance is Levene’s test.  The test statistic for Levene’s test is calculated by diverging the data for each group from the group mean, and then comparing the absolute values.  Levene’s test is presented with the F statistic, as an ANOVA is conducted to compare the absolute values.  A p value less than .05 indicates a violation of the assumption.  If a violation occurs, it is likely that conducting the non-parametric equivalent of the analysis is more appropriate. 

ANOVAs and MANOVAs

An ANOVAs is used to assess differences on time and/or group for one continuous variable and a MANOVA is used to assess differences on time and/or group for multiple continuous variables, but what other factors go into the decision to conduct multiple ANOVAs or a single MANOVA?  MANOVAs are best conducted when the dependent variables used in the analysis are highly negatively correlated and are also acceptable if the dependent variables are found to be correlated around .60, either positive or negative.  The use of MANOVA is discouraged when the dependent variables are not related or highly positively correlated.  

MANOVA is discouraged with highly positively correlated variables because, although the overall multivariate analysis works well, once the highest priority dependent variables has been assessed, the tests conducted and results presented on the remaining dependent variables will be vague.  The reason for this is because once the highest priority dependent variable becomes a covariate, the variance that remains for the lower priority dependent variables is not enough to be significantly related to the main effects or interactions.  Additionally, the univariate ANOVA results are misleading.  

MANOVA is also discouraged when the dependent variables are not significantly related.  A multivariate analysis has lower power than univariate analyses, therefore the difference between univariate and step-down analysis is small.  In this instance the only benefit to conducting a MANOVA over univariate ANOVAs is a reduction in the likelihood of Type I error.  If multiple ANOVAs are the more appropriate analysis, Type I error can be controlled for with the use of the Bonferroni correction, α = 1 - (1 - α₁)(1 - α₂)…(1 - α_n).

In the case where some the dependent variables are correlated in different sets, it may be more appropriate to run two separate MANVOAs; one with each set of correlated variables.