The Practitioner’s Guide to Multivariate Techniques Training Course

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.

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.

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

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.

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.