6.5.3 Interpretation of correspondence analysis

The interpretation of the results of correspondence analysis comprises the interpretation of numerical results and factor graphics, yielded by CA. The former implies selection of significant axes and significant points.

Selection of Significant Axes

How many axes are significant and should be retained for further analysis or interpretation? Here significant means ‘necessary to study in detail’ – not in terms of statistical significance tests. Two types of factor axes are considered: First order factor axes and Second order factor axes. First order factor axes are considered on the basis of contributions to the total variance (or inertia), whereas the second order factor axes are considered on the basis of contributions to the eccentricity, that is. COS2 j .

Correspondence analysis issues eigenvalues for the min[(I, J)-1] factor axes; the eigenvalues are ranked in the decreasing order of magnitude.

First order factor axes

The number of (significant) axes, M, can be determined by any of the following rules:

  1. Sum of the inertia explained by the first M axes exceeds a certain threshold, typically 80% of the total inertia.
  2. Choose all the axes whose eigenvalues exceed

Second order factor axes

After having selected the first order factor axes, the second order factor axes are selected as follows:

Let M/ be the rank of a factor axis for which a point i of N (I) and or j of N (J) exists, such that

COS2 j (i) k

or

COS2 j (j) k

where k is typically = 0.25.

Thus, the number of axes chosen for interpretation = M + M/.

Rules for interpreting factorial axes by individual points

Explicative points

An explicative is a point whose absolute contribution CTRa (i) (for i I ) or CTRa (j) (for j J) are distinctly higher than the contributions of other points. The points i I whose contributions are higher than the average of the whole contribution are considered as explicative. The explicative points can be selected according to any of the following criteria:

Explained points

The points explained by an a -axis are the variable points i of N (I ) [ or j of N ( J )],. whose contributions to the eccentricity are greater than a certain threshold. The contributions to the eccentricity are similar to a squared coefficient of correlation (COS2 j ). Usually a threshold of 0.25 is used.

A point j can be an explained point (by an a -axis) without being an explicative point. Suppose that point i has an absolute contribution 40% and a squared correlation of 0.15 to an axis. This means that it contributes strongly to the creation of the axis, but it probably participates in the building of many other axes.

Thus, two sets of coefficients are calculated for each axis. These coefficients apply equally to the rows and columns of the data matrix.

Absolute contributions, which indicate the proportion of variance (i.e., inertia) explained by each variable in relation to each principal axis. This proportion is calculated with respect to the entire set of variables.

The squared correlations, which indicate the part of the variance of a variable explained by a principal axis.

The interpretation of absolute contributions is opposite to that of the relative contributions (COS2 j ). The latter indicate the extent to which each row category and each column category is described by the axis. The contribution to inertia, on the other hand, indicates the extent to which the geometric orientation of the axis is determined by the single variable categories