Archive:MANOVA

File:Wikademia.logo.png Resource type: this resource contains a tutorial or tutorial notes. Completion status: this resource is ~25% complete. The purpose of this tutorial is to teach use of multivariate analysis of variance (MANOVA), with practical exercises based on using SPSS.

What is MANOVA?

• A multivariate extension of univariate ANOVA
• If you have two or more dependent variables (ANOVA analyses only a single DV at a time)

Example

Effects of chemotherapy and memory enhancement training on cognitive functioning in Alzheimer's patients

IVs (factors)

1. Chemotherapy (drug vs no-drug)
2. Memory training (training vs no-training)

DVs

Several measures of cognitive functioning:

1. Test of reading comprehension and retention
2. Memory for names and faces
3. Ratings provided by family members

Usage

• Alternatively, could use a series of univariate ANOVAs - one for each dependent variable - MANOVA does all these univariate tests simultaneously.
• Most commonly used in laboratory research (where we experimentally manipulate factors), but can be used in any design where we want to answer questions about whether different levels of the IVs affect a combination of DVs.
• "Because of the increase in complexity and ambiguity of results with MANOVA, one of the best overall recommendations is: Avoid it if you can." Tabachnick and Fidell (1983, p.230)
• In other words - be sure it is really the best approach to use (e.g., sometimes a mixed ANOVA could be a better approach).

Assumptions

1. Sample size - sample size rule of thumb:the sample in each cell must be greater that the number of dependent variables
2. Univariate and multivariate normality - (when cell size > 30 this is less important)
3. Linearity - linear relationships among all pairs of dependent variables
4. Homogeneity of regression - covariates must have a homogeneity of regression effect (must have equal effects on the dependent variable across the groups)
5. Homogeneity of variance-covariance matrix (Box's M) – the F test from Box’s M statistics should be interpreted cautiously in that a significant result may be due to violation of the multivariate normality assumption and a nonsignificant result may be due to small sample size and lack of power– fairly robust if equal sample sizes
6. Multicollinearity and singularity - when there is strong multicollinearity, you have redundant dependent measures and this decreases statistical efficiency
7. Outliers - MANOVA is very sensitive to the effect of outliers because they impact on the Type I error (use Mahalanobis distance to check for multivariate outliers)

How does it work?

MANOVA combines concepts from factorial ANOVA and discriminant analysis:

• It examines the effect of several independent variables (main effects and interaction effects), as does univariate ANOVA
• These IV effects are examined on several DVs that are combined to form one or more linear composites, as in discriminant analysis.
• Factor A main effect - evaluated by combining the original DVs to form one or more orthogonal discriminant functions (roots) which provide the greatest possible separation of the groups representing the levels of Factor A.
• Factor B main effect - evaluated by combining the original DVs to form one or more orthogonal discriminant functions (roots) which provide the greatest possible separation of the groups representing the levels of Factor B.
• A X B Interaction - assessed by forming one or more discriminant functions that maximise the separation of cells of the factorial data matrix.
• For each effect (A, B, and A x B) the discriminant functions will differ (so the composite DV being examined can change)

Tests of significance across groups

These criteria can be used to assess differences across "dimensions" of the DVs

1. Roy's greatest characteristic root
   • tests for differences on only the first discriminant function
   • most appropriate when DVs are strongly interrelated on a single dimension
   • highly sensitive to violation of assumptions - most powerful when all assumptions are met
2. Wilk's lambda (U)
   • most commonly used statistic for overall significance
   • considers differences over all the characteristic roots
   • the smaller the value of Wilk's lambda, the larger the between-groups dispersion
3. Hotelling's trace
   • considers differences over all the characteristic roots
4. Pillai's criterion
   • considers differences over all the characteristic roots
   • more robust than Wilk's, should be used when sample size decreases, unequal cell sizes or homogeneity of covariances is violated

Effect sizes

Also use effect sizes to evaluate strength of the effects (particularly for significant effects):

• Univariate ANOVA: eta-square gives the proportion of variance in the DV that is attributed to the different levels of the significant IV
• Multivariate ANOVA: Wilks' Lambda - multivariate eta-square: Wilks' Lambda reflects the ratio of within-group variance across all discriminant functions to total variance across all discriminant functions

Stepdown analysis

• Used to assess individually the differences of the DVs
• Procedure involves computing a univariate F statistic for a DV after eliminating the effects of other DVs preceding it in the analysis.
• Previous DVs are treated as covariates
• Somewhat similar to hierarchical multiple linear regression
• Researcher determines the order in which the DVs are entered, based on some theoretical conceptualisation
• Often used when have correlated dependent variables

Pros and cons

Advantages of MANOVA

• Tests the effects of several IVs and several outcome (DVs) within a single analysis
• Has the power of convergence (no single operationally defined DV is likely to capture perfectly the conceptual variable of interest)
• IVs of interest are likely to affect a number of different conceptual variables - for example: an organisation's non-smoking policy will affect satisfaction, production, absenteeism, health insurance claims, etc
• Can provide a more powerful test of significance than available when using univariate tests
• Reduced error rate compared with performing a series of univariate tests
• Interpretive advantages over a series of univariate ANOVAs

Disadvantages of MANOVA

• Discriminant functions are not always easy to interpret - they are designed to separate groups, not to make conceptual sense. In MANOVA, each effect evaluated for significance uses different discriminant functions (Factor A may be found to influence a combination of DVs totally different from the combination most affected by Factor B or the interaction between Factors A and B).
• Like discriminant analysis, the assumptions on which it is based are numerous and difficult to assess and meet

References

• Hair et al. (1998) Chapter 6
• Tabachnick & Fidell (1996) Chapter 9 (more recent editions are available)

See also

• MANOVA

External links

• MANOVA (ucspace)
  • MANOVA Notes (Handout)