# Why-o-why-do-we-need-multivariate-statistics! by akgame

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```									         Multivariate Analysis of Variance
(MANOVA)
• Let's revisit ANOVA to set the conceptual background for
MANOVA
– mean differences between groups of a factor are tested via
the F-test
• F = MSbg / MSwg
– MSbg = SSbg / dfbg
– MSwg = SSwg / dfwg
• if no treatment effect F approximately equals
1
• if there is a treatment effect F is large
ANOVA Basics continued

• let’s do a little example with the following data
set
Experimental       Control
Group           Group
2              6
3              7
1              5
Treatment Mean          2              6
Sums of Squares         2              2
Overall Mean                   4
Total SS                      28
ANOVA Basics continued
• Calculating SSwg and SSbg
– SSwg is the pooled within-group variability
• so simply add the SS for each group (=4)
– SSbg is the difference the mean of each
group and the grand mean
• this quantity is then multiplied by n (=24)
• Calculating MSwg and MSbg
– divide the SS values by degrees of freedom
• MSwg = 4 / 4 = 1
• MSbg = 24 / 1 = 24
• Calculate F
– F = 24 / 1 = 24
• translate this into a p-value (=.008)
Multivariate Analysis of Variance (MANOVA)
• Obviously MANOVA is an extension of
ANOVA
• 2 or more correlated DVs analyzed together
• DVs form a “set”
• this is yet another form of the general linear
model
– Y1 + Y2 + Y3 + ... + Yn = X1 + X2 + X3 + ... + Xn
• each individual gets a new composite DV
– where the Y-values are continuous variables
– and the X-values are discrete variables
• or dummy-coded variables if we have multiple
groups/conditions for specific discrete variables
Multivariate Analysis of Variance (MANOVA)
• Why not do separate ANOVAs?
– MANOVA is sometimes powerful because
we have taken this correlation among DVs
into account
• i.e., better deals with overlapping variance
– partial redundancy among DVs
• does it by creating a linear composite of the
individual DVs
• helps protect against Type I error
– however, we could also do the same
analyses using ANOVA if
– used a more conservative alpha level
– DVs are not “too highly correlated”
Multivariate Analysis of Variance (MANOVA)
• Examples
– Clinical
• clients randomly assigned to psychodynamic,
CBT, or no therapy
• we have DV difference scores for depression,
life satisfaction, and physical health
– Educational
• effect of parent support on child achievement
• achievement measured through grades &
standardized tests
Multivariate Analysis of Variance (MANOVA)
– Developmental
• gender differences in social coping during
transition to junior high
• coping assessed with a variety of measures
– # friends in social network, # after-school activities,
self-ratings of being liked, etc.
– in all examples, we have more than 1 DV
– similarities/differences to repeated
measures ANOVA
• similar in that we expect correlations between
these DVs
• different in that we have different measures for
the DV
– although this is not absolutely necessary
Multivariate Analysis of Variance (MANOVA)
• MANOVA and null hypothesis significance
testing (NHST)
– t-test
• 1 =  2
– ANOVA
• 1 =  2 =  3 =  n
– MANOVA
• 1 =  2 =  3 =  n for DV 1 AND
• 1 =  2 =  3 =  n for DV 2 AND
• 1 =  2 =  3 =  n for DV 3 AND
• ....
– the alternative hypothesis is that there is at least 1
differences (across groups) in at least 1 of the DVs
or in the DV composite
Multivariate Analysis of Variance (MANOVA)
• Testing the null
– In ANOVA, variance is partitioned into:
• SS total = SS between + SS within
• so if SS between is much larger than SS within the
null is probably not correct
– Similar approach in MANOVA
• however, SS (which are scalars) are replaced
by
– sums of squares and cross-product (SSCP) matrices
• because we need to take correlations
(covariances) of the DVs into account
– we use determinants to get a summary index of
variance in these matrices
• like mean square in ANOVA
Multivariate Analysis of Variance (MANOVA)
• Omnibus tests
– Wilk’s 
•   most popular test
•   ratio of determinanterror to determinanttotal
•   represents percent variance not explained
•   1 -  is an index of variance explained
– 2, a measure of effect size
– analogous to R2 in regression
• want  to be small, and this value is transformed
into an approximate F for hypothesis-testing
Multivariate Analysis of Variance (MANOVA)
• Other tests include
– Pillai’s Trace
– Hotelling’s Trace
– Roy’s Largest (or Characteristic) Root
• Statistical power
– Pillai’s > Wilks > Hotelling’s > Roy’s
• Robustness to violations of assumptions
– Pillai’s most robust
• In general, however, all 4 omnibus tests will
agree
Multivariate Analysis of Variance (MANOVA)
• Follow-up analyses
– for which DVs are there differences?
• results from univariate ANOVAs are provided
– like doing an ANOVA for each DV
• you should use a more conservative alpha level
• individual test  = experimenterwise  /
number of DVs
– ANCOVA, controlling for all other DVs
– specific contrasts
• simple, polynomial
– posthoc comparison procedures
• Tukey’s, Bonferroni, etc.
– conduct a discriminant function analysis
(DFA)
Multivariate Analysis of Variance (MANOVA)
• Assumptions and practical issues of MANOVA
– participants’ scores on each DV are
statistically independent from other
participants’ scores on each DV
• i.e., the errors are statistically independent
– evaluate intraclass correlation coefficient
– multivariate normality
• each DV has a normal distribution
• all linear combinations of DVs have normal
distributions
• all “groups” must have normal distributions
• and, after saying all of this, MANOVA is fairly
robust to violations from normality
Multivariate Analysis of Variance (MANOVA)
– homogeneity of variance/covariance
• variance for each DV is the same across
groups/conditions
• and, covariance between each pair of DVs is the
same across groups/conditions (pooled for the error
term)
• statistically evaluate this using Box’s M test and/or
Bartlett’s chi-square and/or Levene’s test
– problem: all are sensitive to small violations
• if assumption is not satisfied...
– continue on and don’t worry about it, or
– use posthoc methods such as Games-Howell, Tamhane’s
T2, Dunnett’s C, etc.
– these correct for violation of this assumption
• use a conservative alpha (.001) to determine this
Multivariate Analysis of Variance (MANOVA)
– linearity among DVs
• linear combinations for the DVs are used to
maximize the separation (i.e., differences)
between groups
– homogeneity of regression
• specific to MANCOVA
• the relationship between the DVs and the
covariate(s) must be the same across
groups/conditions
• if not, this implies that the DV-covariate
relationship is not consistent across the groups
• you have an IV-covariate interaction: yikes!!!
Multivariate Analysis of Variance (MANOVA)
• Kinds of research questions
– main effects for the IVs
– interactions among the IVs
– importance of DVs part I
• a different linear combination of DVs is created
for each “effect” (i.e., for each main effect and
interaction)
• therefore, different DVs may be differentially
influenced by type of effect
– importance of DVs part II
• analogous to hierarchical or sequential
regression
• DVs are organized in terms of their importance
• Roy-Bargman stepdown analysis
Multivariate Analysis of Variance (MANOVA)
– effect sizes
• 2, can exceed 1 because DVs are correlated
• partial 2, which adjusts for the overestimation
– incorporated with more sophisticated
designs
• inclusion of covariates for the MANCOVA
– correlations with DVs not with the IVs
– including a time element
• Repeated-measures MANOVA
• having multiple DVs measured on multiple
occasions
Multivariate Analysis of Variance (MANOVA)

• Some final thoughts
– Back to the type of research questions that
• multiple univariate questions
– when DVs have low correlations
• intrinsically multivariate questions
– when DVs have “high” correlations ( .60 )
– used when the principal concern is how the DVs
differ/change with respect to the “new” variable
Multivariate Analysis of Variance (MANOVA)