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Resource type: this resource contains a tutorial
or tutorial notes. 

Completion status: this resource is considered to
be complete. 
 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
Student version of SPSS.

What is
MANOVA?
 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
 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)
 DVs
Several measures of cognitive functioning:
 Test of reading comprehension and retention
 Memory for names and faces
 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
 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. #* Use the SPSS Regression menus to calculate MD,
which will provide a score for each case which can be assessed
according to a χ^{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 χ^{2} value
(where the number of IVs equals is the χ^{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 /
alpha 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
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' /
lambda
is recommended unless there are problems with small N,
unequal ns, violations of assumptions, etc. in which case
Pillai's trace is more robust (Tabachnick & Fidell):
 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
 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
Also use effect
sizes to evaluate strength of the effects (particularly for
significant effects):
 Multivariate ANOVA:
 Wilks' λ  multivariate η:
Wilks' λ reflects the ratio of
withingroup variance across all discriminant functions to total
variance across all discriminant functions.
 Univariate ANOVA:
 η^{2} gives the proportion
of variance in the DV that is attributable to different levels of
an IV.
Pros and
cons
 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
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, indicating that the homogeneity of
variancecovariance matrix assumption was not 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’ λ = .42, F (4,52) = 7.03,
p < .001. The multivariate Wilks' λ 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, η^{2} =
.56, while the ANOVA based on the application scores was
nonsignificant, F(2,27)=4.20, p = .026, η^{2} =.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

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

Note: This table should also include skewness,
kurtosis, and descriptives for marginals.
Exercises
 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
 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
 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)
 French, A., Poulsen, J., & Yu, A. (2002). Multivariate Analysis of
Variance (MANOVA).
 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)
 Tabachnick, B. G., & Fidell, L. S. (1983). Using
multivariate statistics. New York: Harper & Row. (Chapter
9; more recent editions are available)
See also
External
links