











English
or languish  Probing the ramifications
of Hong Kong's language policy 



Multivariate
Analysis of Variance and Covariance
(MANOVA and MANOCOVA) 


project
index  statistical modelling (diagnostics) 


Key Features
 Variables
 Dependent variable  metric (interval or ratio)  2 or more
 Independent variables  nonmetric (categorical)  1 or more
 Covariate independent variables  metric (0, 1, or more)
 Objective  Detemine the effectiveness of various
treatments and/or factors on specific characteristics of a known
population through either random sampling or experimentation.
 Statistical procedure  Compare within group and between
group variances of multivariate groups in order to determine
whether the vector spaces between the groups' centroids are statistically
significant when measured as a whole. This is achieved through
partitioning of the sumsofsquares and crossproducts matrix
(SSCP).
MANOVA and MANOCOVA are multivariate extensions of ANOVA and
ANOCOVA, respectively.
Further testing is required in order to determine statistical
significance between individual pairs of group centroids.
 Null hypothesis  The equality
of the dependent mean vectors (centroids) for each of two or
more independent groups or treatments. See understanding
the nullhypothesis.



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Key Terms
 Analysis of variance (ANOVA)  A statistical technique
to determine if samples come from populations with equal means.
Simply conceived MANOVA is ANOVA with multiple dependent variables.
 Centroid  The mean vector of the dependent variables
for each treatment group.
 Covariate analysis  ANOVA and MANOVA statistical
experiments that include nuisance variables or covariates.
 Criterion variables  Another name for dependent variables.
With MANOVA and MANOCOVA experiements there is always more than
one.
 Effects
 Main (principal) effect  the effect of a single,
independent, nonmetric variable (factor) on the dependent variable
(ANOVA) or variables (MANOVA)
 Interaction (joint) effect  the joint effect of two
or more independent variables on the dependent variable (ANOVA)
or variables (MANOVA).
Variation in the dependent variabe(s) caused by main and interaction
effects are measured separately. It is possible, for example,
to have statistically significant interaction effects with no
significant main effects.
 Extraneous (nuisance) variables  Other names for
metric independent variables that are included as part of the
experimental design to remove noise when measuring the main and
interaction effects of the independent, nonmetric, categorical
variables on the metric dependent variables.
 Factor (a treatment or experimental variable)  A
nonmetric categorical variable manipulated by the researcher
to effect changes in the dependent variable.
In the case of oneway experimental design there is only one
factor. Twoway and threeway experimental designs have two and
three factors, respectively.
 Multivariate normal distribution  A generalization
of univariate normal distribution involving two or more dependent
variables.
 SSCP Matrix  Sum of squares and cross products matrix.
This matrix contains the various types of variation required
to calculate the effects and their statistical significance.
 Treatment level (treatment group)  A single factor
may consist of more than one treatment level or group. Each observation
of an experiment is subject to only one treatment per factor.
In effect, each observation of a given group is manipulated (treated)
in a way that observations of other groups of the same or different
factors are not.
 Wilk's Lambda Statistic  A multivariate extension
of the Ftest for ANOVA experiments. It is used to test for the
significance of individual main effects and interaction joint
effects of independent variable(s) on dependent variables.



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Assumptions
 Multivariate normality  the centroid for each treatment
group is assumed to be multivariate normal. This makes normal
distribution of the error terms possible.
 Homogeneity of variance across treatment groups 
the variancecovariance matrice for each treatment group is identical
to every other. In other words, the dispersion within each treatment
group is assumed identical for all groups.



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Experimental design
MANOVA and MANOCOVA experiments are conceptually speaking
expanded versions of ANOVA and ANOCOVA that employ two or more
dependent variables. In brief, MANOVA and MANOCOVA measure the
principal and joint effects of one or more nonmetric dependent
variables on a vector of dependent variables
A few of the more common experimental designs using these statistical
methods include
 Oneway MANOVA Problem
 A single nonmetric independent variable
 Two or more metric dependent variables
 Oneway MANOCOVA Problem
 A single nonmetric independent variable
 Two or more metric dependent variables
 One of more metric covariates (nuisance variables)
 Twoway MANOVA Problem
 Two nonmetric independent variables
 Two or more metric dependent variables
 Twoway MANOCOVA Problem
 Two nonmetric independent variable
 Two or more metric dependent variables
 One of more metric covariates (nuisance variables)
 Fixed, random, and mixed effects
 Fixed effect  The individual observations of a particular
treatment group are randomly selected from a larger population
before the experiment has taken place.
Assignment to individual treatment groups is usually performed
randomly, so as to more accurately measure the effect of the
treatment on the dependent variable.
 Random effect  The individual observations of a particular
treatment group are randomly selected from a larger population
after the experiment has already taken place.
 Mixed effect  A combination of fixed and random effect
experiments.



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Testing the
NullHypothesis
 Commonly employed statistics for two treatment groups
 Commonly employed statistics for three or more treatment
groups
The Wilks' Λ statistic for MANOVA corresponds to the
Fstatistic in ANOVA.
Smaller values of Λ indicate greater statistical significance
as the variance within groups becomes increasingly smaller
relative to the variance among groups.
The distribution of the Wilks' Λ statistic corresponds
well with that of the F  statistic distribution in the following
two cases:
 Case 1:
 Any number of dependent (criterion) variables, and
 Up to three independent variables (factors and covariates)
 Case 2:
 Two dependent variables, and
 Any number of independent variables
Among the available software statistics associated with MANOVA
and MANOCOVA Wilks' Λ is the most common.
 Bartlett's V Statistic
 Rao's Ra function



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Post Hoc Tests  Three or more treatment
groups
In cases where statistical significance for rejecting the
nullhypothesis has been found the researcher may wish to probe
more deeply in an effort to determine which between group differences
are significant.
Although one may be tempted to perform univariate Ftests
for each of the independent variables, this approach is discouraged,
as it ignores possible or even likely correlations among the
dependent variables.
There are two standard procedures to distinguish between informative
and noninformative variables  i.e., variables that do and do
not provide information about the population under study. These
procedures are:
 Protected F  and t  tests
 Canonical representation
Protected F  and t  tests
 The Protect F test
 α$$_{0} (expeimentwise
type 1 error)  If the overall multivariate test statistic (eg.
Wilks' Λ statistic) is not statistically significant,
then no further analysis is performed.
 α$$_{1} (variablewise
type 1 error)  If overall statistical significance is found,
then only those variables for which a univariate F  statistic
demonstrates statistical significanct are tested. A standard
rule of thumb is to set α$$_{1}
= α / p, where α is the desired error rate for the
entire set of comparisons and p is the number of independent
variables to be compared.
 The Protected Hotelling T$$2
and Protected t  tests
 α$$_{2} (comparisonwise
type 1 error)  Pairwise tests are performed on those variables
which were significant at the α$$_{1}
level. The standard rule of thumb for setting α$$_{2} is found by setting α$$_{2} = α$$_{3}
· q , where q is the number of variables subject to pairwise
comparison.
 α$$_{3} (comparisonwise
and variablewise type 1 error)  Only those variables pairs
for which statistical signifiance at the α$$_{3} is found are interpreted.
Canonical representation
The standard statistical technique for performing canonical
representation is discriminant analysis.
Canonical representation has as its objective the reduction
of dimensionality through the use of optimal artificial variates.
All of the original variates are retained but only a limited
number of linear combinations of them (the canonical variates)
are interpreted.
Briefly, canonical variates may be used to represent the centroids
of n groups in an n  1 dimensional subspace of the original
p (number of variates used in the MANOVA or MANOCOVA experiment)
such that p ≥ n  1. The interpretation of differences between
the populations is usually made by assigning some substantive
meaning to each of the canonical variates through an interpretation
of the coefficients of the linear combination.



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Reference
List
Hair, Joseph F., Jr., Rolph E. Anderson, Ronald L. Tatham,
and Bernie J. Grablowsky. 1984. Multivariate Data Analysis with
Readings. New York: Macmillan Publishing Co.
Hughes, Adele. 1984. Class notes to graduate coursework in
Applied Stastistical Methods.



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