6.4.3 Factor Graphics

As mentioned earlier, the objective of principal components analysis is to represent the rows and columns of a data matrix in a subspace of reduced dimensionality and to highlight the relations unseen by computations. Therefore the points of N ( i ) and N ( j ) in a p-dimensional subspace, generated by the first factorial axes, are plotted in a series of two-dimensional plots.

Variable points

Since the a -coordinates of the variable points, j of N( j ), are similar to correlation coefficient (or Cosine squared), these points can be projected on to a circle (called a correlation circle), whose axes are pairs of factorial axes. The variables are represented in a correlation circle.

Individual points

The points i of N ( I ) are projected on to a 2-dimensional factor space.

Joint Graphics

The points of clouds N ( I ) and N ( J ) can be projected on to the same graphic, though the meaning of both sets is not the same. The proximity between one point i of N ( I ) and point j of N ( J ) cannot be interpreted, because the points of N ( I ) are centered at G, the center of gravity of N ( I ), whereas the points of N ( J ) are centered at O (the origin of the row variables).

Graphics for Supplementary Elements:

The supplementary points can be plotted in the factor space generated by explicative factor axes. To avoid confusion, it is suggested that the symbols for supplementary and principal points should not be the same.

Interpretation of Graphics

Distances between individual points are interpreted in terms of similar patterns of response to the variables Distances between variables are interpreted (when using normalized principal components analysis) in terms of correlation. However, a great deal of caution is needed in interpreting the distance between a variable point and an individual point, because these two points do not belong to the same space. However, we can legitimately compare relative positions of two individuals with respect to the entire set of variables or two variables with respect to the entire set of individuals