The Problem of Size and Shape
Before we enter into the discussion of methods of Multivariate Data Analysis we have to make an important distinction. It appears that there are two types (or flavors) of basic factor analysis, depending of what our interest is in the data.
Suppose we have a table that describes various sorts of animals (among which cats, dogs, tigers and wolves) by means of a number of morphometric measurements (such as weight, height, width, length, etcetera).
On the one hand, we may want to do an analysis that preserves the size of the animals. This means that in any graphic display of the factor analysis we expect to see the big animals (tigers and wolves) close together, the smaller ones (cats and dogs) clustered together, and the two groups neatly separated in accordance with their overall difference in size.
On the other hand, we may be interested primarily in the shape (profile or spectrum) of the animals, irrespective of their size. This analysis is expected to show felines together, canines clustered, and the two groups neatly separated by their phylogenetic differences.
The result of both types of analysis is usually represented in the form of a Cartesian diagram, with two (or, when using perspective viewing, at most three) dimensions. In this diagram each of the two (or three) coordinate axes represents a factor that has been extracted from the data table. The entities represented by the rows (objects, subjects or categories of a particular variable) or columns (measurements, observations or categories of another variable) are displayed as points in the diagram.
If the points in this diagram are related to the rows of the data table, then we speak of a “scores plot”.
If the points are associated with the columns of the table, then we deal with a “loadings plot”.
If both row- and column-entities of the data table are represented as points in the factor diagram then we obtain a “biplot”, which was invented by K. Ruben Gabriel in 1971. The biplot is a beautiful graphic device that, once its function is understood, offers remarkable synthetic and analytic possibilities. It is an important step forward in the art and science of Multivariate Data Analysis after the invention of the bivariate diagram by Descartes.