Dual scaling

In the previous chapter we saw that canonical analysis can be applied to a sample of p-variate observations partitioned into g≥2 classes for the purpose of obtaining insight into relationships among the classes. Canonical variate analysis was introduced as a special case of canonical analysis in which the data matrix for one of the two sets of variables is specialized so as to account for the partition of a sample. Suppose now that we have a sample of individuals which are simultaneously classified with respect to two categorical variables. Dual scaling may be set up as a double canonical variate analysis in which both data matrices are specialized so that each corresponds to one of the two variables of classification for the analysis of data of this kind. Doubly-classified observations arise frequently. In palynology, for example, the data generally consist of counts of fossil pollen taxa at a number of stratigraphic levels. In a similar way, studies of natural communities often lead to estimates of the density of different animal species in a variety of plant communities. The data in such cases are conveniently organized in the form of a 2-way array or table. We may then enquire as to the nature of the relationships within and between the row and column categories of the table. This is the question addressed by dual scaling. Dual scaling is in fact applicable to an m-way classification of individuals (m≥2), though for simplicity we shall confine our attention to the case where m is strictly equal to two.