MANOVA

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MANOVA - PowerPoint PPT Presentation

MANOVA. LDF &amp; MANOVA Geometric example of MANVOA &amp; multivariate power MANOVA dimensionality Follow-up analyses if k &gt; 2 Factorial MANOVA. ldf &amp; 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!