Uses of the MANOVA ProcedureTop of PageStatMan Toolbar

MANOVA procedures are multivariate, significance test analogues of various univariate ANOVA experimental designs. MANOVA, as with its univariate counterparts typically involve random assignment of participants to levels of one or more nominal independent variables; however, all participants are measured on several continuous dependent variables.There are three basic variations of MANOVA:Hotelling's T: This is the MANOVA analogue of the two group T-test situation; in other words, one dichotomous independent variable, and multiple dependent variables.

One-Way MANOVA: This is the MANOVA analogue of the one-way F situation; in other words, one multi-level nominal independent variable, and multiple dependent variables.

Factorial MANOVA: This is the MANOVA analogue of the factorial ANOVA design; in other words, multiple nominal independent variables, and multiple dependent variables.

While all of the above MANOVA variations are used in somewhat different applications, they all have one feature in common: they form linear combinations of the dependent variables which best discriminate among the groups in the particular experimental design. In other words, MANOVA is a test of the significance of group differences in some m-dimensional space where each dimension is defined by linear combinations of the original set of dependent variables. This relationship will be represented for each design in the following sections.Why does MANOVA have to be so complicated? Why can't you just do separate ANOVA's for each dependent variables rather than form linear combinations of all dependent variables? There are several primary concerns to consider. First, very often and most likely, the dependent variables will be correlated with each other; thus, your findings from separate ANOVA's will be redundant and difficult to integrate. Second, the family-wise error rate becomes high; the odds of finding something significant simply because of chance rises with repeated use of the same sample of data.

Two Group Discrimination: Hotelling's TTop of PageStatMan Toolbar

The first diagram below represents the situation of two groups on a univariate dimension (y) where population differences are "real." The second diagram represents "real" population differences between group centroids--points defined by the y1 and y2 means for each group--of the two groups on y1 and y2.

Now the task becomes the formation of linear combinations of y1 and y2 in order to (a) correct for redundancy and (b) maximally discriminate amongst the groups. The discriminant weights of this composite (d1 and d2) are mathematically derived such that if we were to calculate a composite score for each participant, then the weights yield composite scores that will give the maximum F Ratio value. Essentially, this minimizes within-group variance and maximizes between-group variance. The new composite dimension (a linear combination) can be plotted against y1 and y2 to examine the univariate frequency distributions of the groups were overlap is minimized:With Hotelling's T we can construct only one dimension, and the test is basically checking whether the differences between the centroids are significantly different on the composite dimension. The statistic is distributed as an F, with q and n1+n2-q-1 degrees of freedom, where q is the number of dependent variables.

The One-Way MANOVATop of PageStatMan Toolbar

This MANOVA application is simply an extention of Hotelling's T to more than two groups. We have a single independent variable with more than two levels, and each participant is measured on several dependent variables. Rather than being limited to constructing a single dimension we can, the maximum number of dimensions that can be constructed in always the smaller of the following two values: (a) the number of groups minus one, or (b) the number of continuous dependent variables. The test statistic in this situation (e.g. Wilk's Lambda) determines whether the whole set of composites can significantly discriminate the groups.If more than one composite is constructed by the MANOVA procedure, all composite dimensions are uncorrelated. Therefore, information offered by the first composite is independent of that offered by the second, and so on. This occurs because the construction of each composite is built by using the residual of the previous composite.

Factorial MANOVATop of PageStatMan Toolbar

This MANOVA application is truly a super-factorial design. Participants are nested under multiple treatment combinations (multiple independent variables) and are measured using several continuous dependent variables. It maintains the advantages that factorial designs have over simple one-way designs (i.e. interaction information) plus it can form composite dimensions specific to each effect in the design which optimally separate the groups being evaluated for each effect. In other words, factorial MANOVA involves the calculation of several sets of composite variables and each set is specific to a particular effect.For factorial MANOVA, we can form a total number of uncorrelated linear combinations of all dependent variables for each effect that is equal to the smaller of the number of degrees of freedom for that effect (Number of cells minus one) or the number of dependent variables. For example, in a 3 x 4 design with 20 dependent variables we can examine the following:The A Effect: As in any factorial design, we examine the A main effect by collapsing across all other variables. Since we have three levels of A (dfA = 2), we can form a maximum of 2 independent linear composite dimensions (like those in Hotelling's T). Then calculate Lambda to evaluate whether these two dimensions as a set significantly discriminate the A groups.

The B Effect: By the same logic as above, we can form a total of 3 linear combinations. Again, calculate Lambda to determine whether the B groups differ significantly on all three dimensions.

The AB Interaction: Since the degrees of freedom for the interaction equals six, we can calculate six independent composite dimensions on which we can calculate Lambda to determine if we have a significant interaction.

Solving the RMD Problem: The Poor TechniqueTop of PageStatMan Toolbar

The MANOVA procedure does have another application. The Repeated Measures design and the Mixed design both assume compound symmetry (i.e. the variances are equal among groups and that the covariances among levels are equal). When this is violated, each independent variable can be treated as a separate dependent variable (using dummy coding). A MANOVA is then performed in order to look for effects, and a significant linear combination of variables would indicate a significant difference between the original levels. This can also be extended to the violation of compound symmetry for several dependent variables; in this case, the procedure is known as a doubly multivariate analysis.

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