MANOVA

1. MANOVA • LDF & MANOVA • Geometric example of MANOVA & multivariate power • MANOVA dimensionality • Follow-up analyses if k > 2 • Factorial MANOVA

2. ldf & MANOVA • 1 grouping variable and multiple “others” (quantitative or binary) • Naming conventions : • LDF -- if the groups are “naturally occurring” • bio-taxonomy to diagnostic categories & measurement • grouping variable is called the “criterion” • others called the “discriminator” or “predictor” variables • MANOVA -- if the groups are the “result of IV manipulation” • multivariate assessment of agricultural “programs” • grouping variable is called the “IV” • others called the “DVs”

3. Ways of thinking about the “new variable” in MANOVA • (like regression) involves constructing a “new” quantitative variate from a weighted combination of quantitative, binary, or coded predictors, discriminators or DVs • The “new” variable is constructed so that when it is used as the DV in an ANOVA, the F-value will be as large as possible (simultaneously maximizing between groups variation and minimizing within-groups variation) • the “new” variable is called • MANOVA variate -- a “variate” is constructed from variables • linear discriminant function -- a linear function of the original variables constructed to maximally discriminate among the “groups” • canonical variate -- alludes to canonical correlation as the general model within which all corr and ANOVA models fit

4. How MANOVA works -- two groups and 2 vars Var #2 Var #1 Plot each participant’s position in this “2-space”, keeping track of group membership. Mark each groups “centroid”

5. Look at the group difference on each variable, separately. Var #2 Var #1 The dash/dot lines show the mean difference on each variable -- which are small relative to within-group differences, so small Fs

6. The MANOVA variate “positioned” to maximize resulting F Var #2 Var #1 In this way, two variables with non-significant ANOVA Fs can combine to produce a significant MANOVA F

7. Like ANOVA, ldf can be applied to two or more groups. • When we have multiple groups there may be an advantage to using multiple discriminant functions to maximally discriminate between the groups. • That is, we must decide whether the multiple groups “line up” on a single dimension (called a concentrated structure), or whether they are best described by their position in a multidimensional “space” (called a diffuse structure). • Maximum # dimensions for a given analysis: • the smaller of# groups - 1 • # predictor variables • e.g., 4 groups with 6 predictor variables ? Max # ldfs = _____

8. “Anticipating” the number of dimensions (MANOVAs) • By inspecting the “group profiles,” (means of each group on each of the predictor variables) you can often anticipate whether there will be more than one ldf … • if the groups have similar patterns of differences (similar profiles) for each predictor variable (for which there are differences), then you would expect a single discriminant function. • If the groups have different profiles for different predictor variables, then you would expect more than one ldf Group Var1 Var2 Var3 Var4 Group Var1 Var2 Var3 Var4 1 10 12 6 8 1 10 12 6 14 2 18 12 10 2 2 18 6 6 14 3 18 12 10 2 3 18 6 2 7 Concentrated + 0 + - Diffuse 1st + - 0 0 2nd 0 0 - -

9. Determining the number of dimensions (variates) • Like other “determinations”, there is a significance test involved • Each variate is tested as to whether it “contributes to the model” using one of the available F-tests of the -value. • The first variate will always account for the most between-group variation (have the largest F and Rc) -- subsequent variates are “orthogonal” (providing independent information), and will account for successively less between group variation. • If there is a single variate, then the model is said to have a concentrated structure • if there are 2 or more variates then the model has a diffuse structure • the distinction between a concentrated and a diffuse structure is considered the “fundamental multivariate question” in a multiple group analysis.

10. There are two major types of follow-ups when k > 2 • Univariate follow-ups -- abandoning the multivariate analysis, simply describe the results of the ANOVA (with pairwise comparisons) for each of the predictors (DVs) • MANOVA variate follow-ups -- use the ldf(s) as DVs in ANOVA (with pairwise comparisons) to explicate what which ldfs discriminate between what groups • this nicely augments the spatial & re-classification depictions • if you have a concentrated structure, it tells you exactly what groups can be significantly discriminated • if you have a diffuse structure, it tells you whether the second variate provides discriminatory power the 1st doesn’t

11. Factorial MANOVA • A factorial MANOVA is applied with you have . . . • a factorial design • multiple DVs • A factorial MANOVA analysis is (essentially) a separate MANOVA performed for each of the factorial effects, in a 2-way factorial . . . • Interaction effect • one main effect • other main effect • It is likely that the MANOVA variates for the effects will not be the same. Said differently, different MANOVA main and interaction effects are likely to be produced by different DV combinations & weightings. So, each variate for each effect must be carefully examined and interpreted!