# MANOVA – Post-Hoc Tests Using Univariate ANOVAs or Discriminant Analysis

bonferronidiscriminant analysismanovapost-hocstatistical-power

I am using a MANOVA test to compare nine different dependent variables (from neuropsychological and neuropsychiatric assessment) between three groups. The output shows a significant influence from GROUP on my variables ($p < .001$).

Ofcourse, I am interested in how the three groups influence every dependent variable. I have studied Field's "Discovering Statistics Using IBM SPSS Statistics" chapter 16, and he states that the preferred post-hoc analysis is a discriminant analysis, because of the linear combination in which the dependent variables are related to group membership in a MANOVA. Discriminant analysis could account for this linear combination, so Field states.

Otherwise, I read some literature, on basic statistical sites, where is stated that I can use multiple univariate ANOVA's with Bonferroni correction, and use a post-hoc on these univariate ANOVA's when they are significant.

Which of these method's is better? That is, which will make my chance of a Type I or II error the least?

#### Best Answer

"Significant influence from group" means that $H_0: {\mu_1}=\mu_2=\mu_3$ has been rejected, where $\mu_i$ is the mean vector of the dependent variables in group $i$. This can happen if $\mu_1=\mu_2\neq \mu_3$. In this case, discriminant analysis between group 1 and 2 would fail. You would have first to decompose the overall hypothesis into $\mu_1 = \mu_2$, $\mu_2 = \mu_3$ and $\mu_1 = \mu_3$. There, of course, multiplicity adjustments (e.g. Bonferroni) are again necessary.

Even if it does not fail, discriminant analysis gives you rather estimates of the effects, not test results. If you are in fact interested in such a tool (e.g. in order to diagnose to which group a new patient would belong), discriminant analysis will of course still be necessary.

Multiplicity adjustments of the hypotheses $H_0 ^ {j}:\;\mu_1^{j}=\mu_2^{j}=\mu_3^{j}$, where $j$ denotes a dependent variable, can be done with Bonferroni-method. The interpretation of a significant result would be that in the dependent variable you identified not all groups have equal means. Usually you would want to decompose this result as well into pairwise comparisons as above. Also you have to keep in mind that it may happen that you can reject the global hypothesis but fail with the post-hoc analyses.

Your last question: As Bonferroni is quite conservative, you may consider using different methods, e.g. like in the SimComp R-package. This would estimate the unknown dependency between the variables. Said information would lead to a less conservative adjustment, thus, better power.