Advanced ANOVA/MANOVA

What is MANOVA?[edit]
 Developed as a theoretical construct by Samual S. Wilks in 1932 (Biometrika).
 An extension of univariate ANOVA procedures to situations in which there are two or more related dependent variables (ANOVA analyses only a single DV at a time). DVs should be correlated (but not overly so; otherwise they should be combined) or conceptually related.
 The MANOVA procedure identifies (inferentially) whether:
 Different levels of the IVs have a significant effect on a linear combination of each of the DVs
 There are interactions between the IVs and a linear combination of the DVs.
 There are significant univariate effects for each of the DVs separately.
Example[edit]
 Effects of chemotherapy and memory enhancement training on cognitive functioning in Alzheimer's patients
 IVs (factors)
 Chemotherapy (drug vs nodrug)
 Memory training (training vs notraining)
Several measures of cognitive functioning:
 Test of reading comprehension and retention
 Memory for names and faces
 Ratings provided by family members
Usage[edit]
 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?)^{[1]}
 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?[edit]
 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.^{[2]}
 The MANOVA procedure then assesses whether this new DV differs significantly between the IV groups.^{[2]}
 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[edit]
 Sample size
 Rule of thumb: the n in each cell > the number of DVs
 Larger samples make the procedure more robust to violation of assumptions
 Normality:
 MANOVA sig. tests assume multivariate normality, however when cell size > ~20 to 30 the procedure is robust violating this assumption
 Note that univariate normality is not a guarantee of multivariate normality, but it does help.
 Check univariate normality via histograms, normal probability plots, skewness, kurtosis, etc. and check multivariate normality using Mahalanobis' distance. These procedures will also help to check for possible outliers.
 Outliers:
 MANOVA is sensitive to the effect of outliers (they impact on the Type I error rate); first check for univariate outliers, then use Mahalanobis' distance to check for multivariate outliers (MVOs).
 MVOs are cases with an unusual combination of scores for the DVs of interest.
 The SPSS Regression menus can be used to calculate Mahalanobis' Distance, which will provide a score for each case which can be assessed according to a ^{2} distribution
 Analyze  Regression  Linear  Dependent (add a unique identifier e.g., ID)  Independent (add all the MANOVA DVs)  Save  MD  Paste/OK.
 Cases which can be considered MVOs are those with MD values above the critical ^{2} value (where the number of IVs equals is the ^{2} df).
 MANOVA can tolerate a few outliers, particularly if their scores are not too extreme and there is a reasonable N. If there are too many outliers, or very extreme scores, consider deleting these cases or transforming the variables involved (see Tabachnick & Fidell).
 Linearity
 Linear relationships among all pairs of DVs
 Assess via scatterplots and bivariate correlations (check for each level of the IV(s) i.e., cells  use Split File)
 Homogeneity of regression
 This assumption is only important if using stepdown analysis, i.e., there is reason for ordering the DVs.
 Covariates must have a homogeneity of regression effect (must have equal effects on the DV across the groups)
 Multicollinearity and singularity
 MANOVA works best when the DVs are only moderately correlated.
 When correlations are low, consider running separate ANOVAs
 When there is strong multicollinearity, there are redundant DVs (singularity) which decreases statistical efficiency.
 Correlations above .7, and particularly above .8 or .9 are reason for concern.
 Consider removing one of the strongly correlated pairs or combining them to form a single measure.
 Homogeneity of variancecovariance matrix (Box's M)
 The F test from Box’s M statistics should be interpreted cautiously because it is a highly sensitive test of the violation of the multivariate normality assumption, particularly with large sample sizes.
 MANOVA is fairly robust to this assumption where there are equal sample sizes for each cell.
 Homogeneity of error variances (Levene's test)
 If this assumption is violated, use a more conservative critical level for determining significance for that variable in the univariate Ftest. Tabachnick and Fidell suggest .025 or .01 rather than the conventional .05 level.
Multivariate test statistics[edit]
Choose from among these multivariate test statistics to assess whether there are statistically significant differences across the levels of the IV(s) for a linear combination of DVs. In general Wilks' is recommended unless there are problems with small N, unequal ns, violations of assumptions, etc. in which case Pillai's trace is more robust^{[3]}:
 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
 Wilks' lambda (λ)
 Most commonly used statistic for overall significance
 Considers differences over all the characteristic roots
 The smaller the value of Wilks' lambda, the larger the betweengroups dispersion
 Hotelling's trace
 Considers differences over all the characteristic roots
 Pillai's criterion
 Considers differences over all the characteristic roots
 More robust than Wilks'; should be used when sample size decreases, unequal cell sizes or homogeneity of covariances is violated
Tests of betweensubject effects[edit]
 What should be done once it is found that an overall F for MANOVA is significant?
 If there is a significant multivariate effect, examine the Tests of BetweenSubjects Effects for each of the DVs.
 Since there are multiple tests, control for the Type I errorrate (e.g., use a Bonferroni adjusment  divide the original alpha level by the number of tests).
 However, note that the DVs are usually correlated, therefore this approach would result in confounded results.
 Stepdown F ratios provide a similar approach, without the counfounded results. In this approach, all DVs are prioritised (by the researcher) from most to least important. The most important variable is considered first without correcting for the lower priority variables. All subsequent variables are tested after removing the effects of the higher priority variables (by specifying the higher priority variables as covariates). Thus, stepdown analysis:
 Is used to assess IV effects on individual DVs
 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
 Is most appropriate when the DVs are correlated.
 See also: Analyses Following a Significant MANOVA (uwsp.edu)
Effect sizes[edit]
Also use effect sizes to evaluate strength of the effects (particularly for significant effects):
 Multivariate ANOVA:
 Wilks'  multivariate : Wilks' reflects the ratio of withingroup variance across all discriminant functions to total variance across all discriminant functions.
 Univariate ANOVA:
 ^{2} gives the proportion of variance in the DV that is attributable to different levels of an IV.
Pros and cons[edit]
 Advantages
 Tests the effects of several IVs and several outcome (DVs) within a single analysis.
 Uses 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  e.g., an organisation's nonsmoking policy may affect employee satisfaction, production, absenteeism, health insurance claims, etc.
 Can provide a more powerful test of significance than available when via univariate tests.
 Reduced Type I error rate compared with performing a series of univariate tests.
 Interpretive advantages over a series of separate univariate ANOVAs.
 Disadvantages
 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
 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[edit]
A oneway multivariate analysis of variance (MANOVA) was conducted to determine the effect of the three types of study strategies (thinking, writing and talking) on two dependent variables (recall and application test scores). A nonsignificant Box’s M, indicated a lack of evidence that the homogeneity of variancecovariance matrix assumption was violated. No univariate or multivariate 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’ = .42, F (4,52) = 7.03, p < .001. The multivariate Wilks' was quite strong at .35. Table 1 presents the means and standard deviations of the dependent variables for the three strategies.
Univariate analyses of variance (ANOVAs) for each dependent variable were conducted as followup tests to the MANOVA. Using the Bonferroni method for controlling Type I error rates 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, = .56, while the ANOVA based on the application scores was nonsignificant, F(2,27)=4.20, p = .026, =.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. The writing group produced significantly superior performance on the recall questions in comparison with either of the other two groups. The thinking and talking groups did not differ significantly from each other.
Table 1 Means and Standard Deviations for each Dependent Variable by Strategy























Note: This table should also include skewness, kurtosis, and descriptives for marginals.
Exercises[edit]
 Data
 Data: SCHL8.sav (Francis 5.3; p. 132 (5th ed.))
 1st MANOVA
 DVs (Academic achievement):
 Maths (mathsach)
 English (engach)
 IVs:
 Socioeconomic status (SES; Low, Moderate, High)
 2nd MANOVA
 DVs (Classroom behaviour):
 Attentiveness in Year 8 (attent)
 Settledness in Year 8 (settle)
 Sociability in Year 8 (sociab)
 IVs:
 Gender (Sex; Male, Female)
 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 Errorbar (Summaries of separate variables) or
 Multiple Line Graph (Summaries of separate variables)
 3rd MANOVA (withinsubjects)
 DVs
 Year level or Time
 Year 7 and 8 (same participants over Time)
 Classroom behaviour
 Attentiveness
 Settledness
 Sociability
 Year level or Time
 4th MANOVA (withinsubjects)
 DVs
 Students' perceptions of maths and english teachers
 Maths and English teachers (same students assessing these)
 Student ratings of teacher qualities
 Responsiveness
 Expectations
 Enjoyable class
 Students' perceptions of maths and english teachers
References[edit]
 ↑ French, A., Poulsen, J., & Yu, A. (2002). Multivariate Analysis of Variance (MANOVA).
 ↑ ^{2.0} ^{2.1} Francis, G. (2007). Introduction to SPSS for Windows: v. 15.0 and 14.0 with Notes for Studentware (5th ed.). Sydney: Pearson Education. (Section 5.3)
 ↑ Tabachnick, B. G., & Fidell, L. S. (1983). Using multivariate statistics. New York: Harper & Row. (Chapter 9; more recent editions are available
 Hair, J. F., Anderson, R. E., Tatham, R. L., & Black, W. C. (1998). Multivariate data analysis (5th ed.). New York: Macmillan (Chapter 6).
 Pallant, J. (2005). SPSS survival manual: A step by step guide to data analysis using SPSS for Windows (Versions 1214). Crows Nest, NSW, Australia: Allen & Unwin. (Chapter 21)
See also[edit]
External links[edit]
 MANOVA (ucspace)
 MANOVA Notes (Handout)