Archive:MANOVA

File:Wikademia.logo.png Resource type: this resource contains a tutorial or tutorial notes. The purpose of this tutorial is to teach use of multivariate analysis of variance (MANOVA), with practical exercises based on using SPSS. Note that the MANOVA procedure is not available with the Studentware version of SPSS.

What is MANOVA?

• An extension of univariate ANOVA procedures: To the situation in which there are two or more dependent variables (ANOVA analyses only a single DV at a time).
• Covers situations where there are two or more (correlated) DVs and where these DVs cannot simply be combined.
• The MANOVA procedure identifies (inferentially) whether there are:
  • Difference levels of the IVs have a significant effect on the DVs
  • Interactions between the IVs and a linear combination of the DVs.

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

• MANOVA is appropriate when we have several DVs which all measure different aspects of some cohesive theme, e.g., several different types of academic achievement (e.g., Maths, English, Science).
• MANOVA works well in situations where there are moderate correlations between DVs. For very high or very low correlation in DVs, it is not suitable: if DVs are too correlated, there isn’t enough variance left over after the first DV is fit, and if DVs are uncorrelated, the multivariate test will lack power (so why sacrifice degrees of freedom?) (French et al., 2002)
• Alternatively, consider use a series of univariate ANOVAs (one for each DV) or possibly Mixed ANOVA.
• "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 & Fidell, 1983, p.230). In other words - be sure it is really the best approach to use.
• Covariates can also be included → MANCOVA

How does it work?

Simple explanation

• The MANOVA procedure creates a new DV which is a linear combination of the multiple DVs. This particular combination of DVs is chosen to maximise the difference between the IV groups. (Francis, 2007)
• The MANOVA procedure then assesses whether this new DV differs significantly between the IV groups. (Francis, 2007)

More complex explanation

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)

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)

Tests of significance across groups

Choose from these test statistics 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

Alternatives to MANOVA

• Combine or eliminate DVs so that only one DV need be analysed.
• Use factor analysis to find orthogonal factors that make up the DVs, then use univariate ANOVAs on each factor (because the factors are orthogonal each univariate analysis should be unrelated)

Example writeup

A one-way multivariate analysis of variance (MANOVA) was conducted to determine the effect of the three types of study strategy (thinking, writing and talking) on the two dependent variables (recall and application test scores). A nonsignificant Box’s M indicated that the homogeneity of variance-covariance matrix assumption was not violated. No outliers were evident and MANOVA was considered to be an appropriate analysis technique. Significant differences were found among the three study strategies on the dependent measures, Wilks’ <math>\lambda</math> = .42, F (4,52) = 7.03, p < .001. The multivariate Wilks' <math>\lambda</math> was quite strong at .35. Table 1 presents the means and standard deviations of the dependent variables for the three strategies.

Analyses of variance (ANOVA) for each dependent variable were conducted as follow-up tests to the MANOVA. Using the Bonferroni method of adjusting Type I error for multiple comparisons, each ANOVA was tested at the .025 level. The ANOVA of the recall scores was significant, F (2,27) = 17.11, p <.001, <math>\eta^2</math> =.56, while the ANOVA based on the application scores was nonsignificant, F(2,27)=4.20, p = .026, <math>\eta^2</math> =.24.

Post hoc analysis for the recall scores consisted of conducting pairwise comparisons to determine which study strategy affected performance most strongly. Each pairwise comparison was tested at the .025/3, or .008, significance level. These revealed that the writing group produced significantly superior performance on the recall questions in comparison with either of the other two groups, and the thinking and talking groups did not differ significantly from each other.

Table 1 Means and Standard Deviations for each Dependent Variable by Strategy

	Recall		Application
Strategy	M	SD	M	SD
Thinking	3.30	0.68	3.20	1.23
Writing	5.80	1.03	5.00	1.76
Talking	4.20	1.14	4.40	1.17

Exercises

• Data: SCHL8.sav (Francis 5.3; p. 132 (5th ed.))
• DVs (Academic achievement):
  • Maths (mathsach)
  • English (engach)
• IVs:
  • Socio-economic status (SES; Low, Moderate, High)

SPSS Steps

• Analyze - General Linear Model - Multivariate (add IV(s) (fixed factors) and DVs)
• Graphs - could use any of:
  • Clustered Bar Chart (Summaries of separate variables) or
  • Clustered Error-bar (Summaries of separate variables) or
  • Multiple Line Graph (Summaries of separate variables)

[[../References|References]]

• Francis, G. (2007). Introduction to SPSS for Windows: v. 15.0 and 14.0 with Notes for Studentware (5th ed.). Sydney: Pearson Education.
• French, A., Poulsen, J., & Yu, A. (2002). Multivariate Analysis of Variance (MANOVA).
• Hair et al. (1998) Chapter 6
• Tabachnick, B. G., & Fidell, L. S. (1983). Using multivariate statistics. New York: Harper & Row. (see Chapter 9; more recent editions are available)

See also

• MANOVA

External links

• MANOVA (ucspace)
  • MANOVA Notes (Handout)