The Practitioner’s Guide to Multivariate Techniques Training Course

Multivariate Analysis of Variance (MANOVA)

Whereas the Analysis of Variance technique (ANOVA) investigates possible systematic
differences between prescribes groups of individuals on a single variable, the technique of
Multivariate Analysis of Variance is simply an extension of that procedure to numerous
variates viewed collectively. These variates could be distinct in nature; for example
Height, Weight etc, or repeated measures of a single variate over time or over space. When
the variates are repeated measures over time or space, the analyses may often be reduced
to a succession of univariate analyses, with easier interpretation. This procedure is often
referred to as Repeated Measure Analysis.

Principal Component Analysis

If only two variates are recorded for a number of individuals, the data may conveniently be
represented on a two-dimensional plot. If there are ‘p’ variates then one could imagine a
plot of the data in ‘p’ dimensional space. The technique of Principal Component Analysis
corresponds to a rotation of the axes so that the maximum amounts of variation are
progressively represented along the new axes. It has been described as …….‘peering into
multidimensional space, from every conceivable angle, and selecting as the viewing angle
that which contains the maximum amount of variation’ The aim therefore is a reduction
of the dimensionality of multivariate data. If for example a very high percentage (say
90%) of the variability is contained in the first two principal components, a plot of these
components would be a virtually complete pictorial representation of the variability.

Discriminant Analysis

Suppose that several variates are observed on individuals from two identified groups. The
technique of discriminant analysis involves calculating that linear function of the variates
that best separates out the groups. The linear function may therefore be used to identify
group membership simply from the pattern of variates. Various methods are available to
estimate the success in general of this identification procedure.

Canonical Variate Analysis

Canonical Variate Analysis is in essence an extension of Discriminant Analysis to
accommodate the situation where there are more than two groups of individuals.

Cluster Analysis

Cluster Analysis as the name suggests involves identifying groupings (or clusters) of
individuals in multidimensional space. Since here there is no ‘a priori’ grouping of

individuals, the identification of so called clusters is a subjective process subject to various
assumptions. Most computer packages offer several clustering procedures that may often
give differing results. However the pictorial representation of the so called ‘clusters’, in
diagrams called dendrograms, provides a very useful diagnostic.

Factor Analysis

If ‘p’ variates are observed on each of ‘n’ individuals, the technique of factor analysis
attempts to identify say ‘r’ (< p) so called factors which determine to a large extent the
variate values. The implicit assumption here therefore is that the entire array of ‘p’ variates
is controlled by ‘r’ factors. For example the ‘p’ variates could represent the performance
of students in numerous examination subjects, and we wish to determine whether a few
attributes such as numerical ability, linguistic ability could account for much of the
variability. The difficulties here stem from the fact that the so-called factors are not
directly observable, and indeed may not really exist.

Factor analysis has been viewed very suspiciously over the years, because of the measure
of speculation involved in the identification of factors. One popular numerical procedure
starts with the rotation of axes using principal components (described above) followed by a
rotation of the factors identified.