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MANOVA. LDF & MANOVA Geometric example of MANVOA & multivariate power MANOVA dimensionality Follow-up analyses if k > 2 Factorial MANOVA. ldf & MANOVA 1 grouping variable and multiple “others” (quantitative or binary) Naming conventions :

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MANOVA

  • LDF & MANOVA

  • Geometric example of MANVOA & multivariate power

  • MANOVA dimensionality

  • Follow-up analyses if k > 2

  • Factorial MANOVA


  • 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”


  • 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


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”


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


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


  • 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 = _____


  • “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 - -


  • 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.


  • 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


  • 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!


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