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Why-o-why-do-we-need-multivariate-statistics!

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Why-o-why-do-we-need-multivariate-statistics!

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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
       of your IV
     • 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
      • choose your covariates wisely
         – 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
     MANOVA answers
      • 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)

    – select your DVs carefully
      • there is a danger in include too many DVs
      • one “bad” variable can drastically alter your
        results
    – sample size
      • MANOVA does require larger sample sizes
        than ANOVA
      • have to factor in number of DVs
      • Scott’s rules of thumb
         – 10 participants per DV, and
         – minimum of 10 in each condition/group combination

								
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