MANOVA

Question 1

What is MANOVA?

Answer 1

• MANOVA is a multivariate technique used to analyse multiple DVs
  
  • Creates a variate for each participant comprised of a combination of scores from DVs
  • Can assess multiple DVs as well as combinations and interactions between DVs
• When to use MANOVA
  
  • Particularly useful when the target construct is complex (eg happiness)
  • Best used when DVs are highly negatively correlated, or when DVs are moderately correlated in either direction

Question 2

What are the advantages of MANOVA?

Answer 2

1. Controls for Type 1 Error (especially compared to multiple anovas)
2. Considers the combined effects of multiple DVs (interactions etc)
3. When #DV < 5, MANOVA has the same power as multiple ANOVAS
4. Multivariate test statistics can be used to interpret results
5. More DVs can result in a more robust model due to increased power and reduced error variance

Question 3

What are the disadvantages of MANOVA?

Answer 3

• Complex technique; issues with interpretation
• Questionable ability to reduce Type 1 Error
  
  • Could simply adjust significance levels
• Selection of DVs requires strong theoretical foundation
• Generally less powerful than ANOVA
  
  • Especially when assessing DVs independently
• Issues of redundancy when DVs are highly correlated

Question 4

What are Wilks Lamda and Pillai’s Trace?

Answer 4

• Wilks’ Lamda (Λ); Measures amount of variance not accounted for by IVs.
  
  • When 2+ variates; Λ = product of unexplained variances
  • Most common, appropriate when no assumption violations
• Pillai-Bartlett Trace (V); proportion of variance accounted for by IVs
  
  • When 2+ variates; V = sum of explained variances
  • Robust to assumption violations (but issues with homogeneity of variance and unequal sample sizes)

Question 5

What are Hotellings Trace and Roys Largest Root?

Answer 5

• Both are related to the eigenvalues of the matrix
• Hotellings T²; Sum of the eigenvalues for each variate
  
  • sum of explained variance divided by unexplained variance for each variate
• Roy’s Largest Root; The eigenvalue for the first (largest) variate
  
  • Proportion of explained variance to unexplained variance for 1st variant
  • Powerful when there is only one variate and assumptions are met

Question 6

What is a matrix?

Answer 6

• A matrix is an array of numbers in columns and rows
  
  • Dimensions are expressed numerically as #rows x #columns (eg 2x3)
    
    • Square Matrix; when #rows = #columns
    • Identity Matrix; square matrix with diagonal =1, others =0
  • Values are referred to as components or elements
  • Columns and rows are referred to as vectors

Question 7

What is the Sum of Squares and Cross Products Matrix?

Answer 7

• The SSCP Matrix is how the test statistic is calculated in MANOVA
  
  • ‘On Diagonal’ : each cell represents Sum Squares for one DV
  • ‘Off Diagonal’ : represents combined effects of DVs
  • ​Hypothesis SSCP; Effect variance matrix
  • Error SSCP; Error variance matrix
  • Total SSCP; Total vairance matrix
• Test Statistic; compares ratio of systematic variance to unsystematic variance for multiple DVs through Partitioning of Variances
  
  • ​Divides H SSCP by E SSCP (multiply by inverse)
  • Output requires interpretation via test statistics

Question 8

What is Discriminant Function Analysis?

Answer 8

• DFA is conducted as a follow up to a significant MANOVA analysis. It allows assessment of how the DVs discriminate the groups.
• The group variables are predicted by a linear combination of the outcome variables (like logistic regression)
  
  • essentially the inverse of the MANOVA
• DFA produces multiple functions (or variates) which each represent a different linear combination of the DVs

Question 9

What assumptions are required for MANOVA?

Answer 9

• Homogeneity of covariance matrices;
  
  • Variances for each DV are roughly equal AND
  • The correlation between any two DVs is similar in all groups
  • Tested using Box’s M (p < .001 result = violation). Don’t bother when sample sizes are equal, notoriously oversensitive.
• Multivariate Normality: Are DVs and linear combinations of DVs normally distributed?
  
  • Can be assumed if cell size > 30
  • No direct test, use judgment and combinations of other tests
• Multicollinearity/singularity: Correlations higher thatn .90

Question 10

How do you interpret results of DFA?

Answer 10

• Eigenvalue Tables; Shows number of functions that can be extracted (will be lesser of #groups-1 or #predictors).
  
  • In order of variance explained.
• Wilks Lambda; Contribution of functions to group separation (1st is combination).
  
  • When reporting consider all functions together.
• Canonical DF Coefficients; Provides regression coefficients for formula of each function
• Structure Matrix; Values indicate how well and in what direction each predictor correlates with each function.
  
  • Loading > .3 used in interpretation
• Group Centroids; Shows which groups are differentiated best by that function
  
  • Group with value of opposite sign best discriminated