MANOVA

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MANOVA

• LDF MANOVA
• Geometric example of MANOVA multivariate power
• MANOVA dimensionality
• Follow-up analyses if k gt 2
• Factorial MANOVA

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• 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

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• 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

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How MANOVA works -- two groups and 2 vars
Var 2
Var 1
Plot each participants position in this
2-space, keeping track of group membership.
Mark each groups centroid
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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
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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
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• 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 _____

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• 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 - -
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• 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.

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• There are two major types of follow-ups when k gt
  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 doesnt

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• 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!