Correspondence Analysis

Abstract

Correspondence analysis is defined in some instances as a way of interpreting contingency tables that may be defined through principal components analysis (Mardia et al. 1979). In correspondence analysis used in this book, a factor is represented by the eigenvector of the normalized covariance or correlation matrix (Usunoff and Guzman-Guzman 1989). It can be further viewed as a simultaneous linear regression scheme with dual scaling, which allows the interpretation of both sample sites and variables in the same factor space. The points, i (sample) and the points j (variables) can be simultaneously reported on the planes associated with the factor axes. The proximity of a point j to a group of points i is taken as an indication that the variable actually characterizes this group of samples. The contribution, CR of the points i or j, in the variability accounted for by an axis (a) can be computed and this aids interpretation. The CR of whole points i or j amounts to one, according to calculations (Razack and Dazy 1990):

$$\sum\limits_i {C{R_\alpha }\left( i \right) = \sum\limits_j {C{R_\alpha }\left( j \right) = 1} }$$

((14.1))

for a given axis (a), and the quality (QT) which expresses the quality of representation of points on the axis (a) is given by:

$$\sum\limits_\alpha {Q{T_{\left( i \right)}} = \sum\limits_{_\alpha } {Q{T_{\left( j \right)}} = 1} }$$

((14.2))

for a given point i or j. (14.2)