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Multivariate Analysis of Variance (Manova)
description | simple example | MAIA example | how it works | caveatsDescription: Manova creates a linear combination of the dependent variables (DV's) and then tests for differences in the new variable using methods similar to Anova. The independent variable (IV) used to group the cases is categorical. Manova tests whether the categorical variable explains a significant amount of variability in the new dependent variable.
{2 or more DV's} = f (1 or more categorical IV's).
Simple example: Suppose you want to test whether stream size and shape differ across ecoregions. The independent categorical variable would be ecoregion and the set of dependent variables might include {width, depth, flow, and gradient}. The dependent variables are correlated which is appropriate for Manova. Manova constructs new variables from {width, depth, flow, and gradient} and tests whether the new composite variables differs across ecoregions.
MAIA example: For the Maryland fish IBI, Roth et al. (1999) first grouped stream sites using cluster analysis of fish species data. The cluster analysis computed the distance between each set of species for every pair of sites and yielded a dendogram that grouped sites by cluster. Cluster analysis does not have statistical testing associated with it, but the authors were interested in determining which clusters were significantly different. To test for significance, they used a Manova model.
For the Manova model, the relative abundances of the different fish species were the dependent variables; and they used cluster assignment as the independent variable. They tested each branching point successively down the cluster tree for statistical significance.
Figure
Figure: Schematic representation of the Manova analysis for the Maryland fish index. (The complete cluster tree was too complicated to illustrate here.) In the sketch above, the first branch of the cluster analysis was significant, A and B included significantly different fish assemblages. At the next level, AA and AB were not significantly different but BA and BB were. Successive branches are not pictured, but the same algorithm was applied down each branch. Thus, this diagram illustrates three significant clusters, sites grouped as A, BA and BB.
How the method works: A new variable is created that combines all the dependent variables on the left hand side of the equation such that the differences between group means are maximized. (The f-statistic from Anova is maximized, that is, the ratio of explained variance to error variance). The simplest significance test treats the first, new variable just like a single dependent variable in Anova, and uses the tests as in Anova. Additional, multivariate tests can also be computed that involve multiple new variables derived from the initial set of dependent variables.
Assumptions/limitations: Dependent variables can be correlated or independent of each other. Like Anova, Manova isn't too bothered by slight departures from normality, but extreme outliers can be more of a problem.
Manova can require rather large sample sizes for complicated models because the number of cases in each category must be larger than the number of dependent variables. Manova also prefers that the groups have a similar number of cases in each group. In addition, Manova expects that the variance of dependent variables and the correlation between them are similar within groups.