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. COS^{2} 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:

- Sum of the inertia explained by the first
*M*axes exceeds a certain threshold, typically 80% of the total inertia. - 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

COS^{2} j
(*i*) ³ *k*

or

COS^{2} j
(*j*) ³ *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* Ì *I *) or CTRa (*j*)
(for *j* Ì *J*) are
distinctly higher than the contributions of other points. The points *i* Ì *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:

- CTRa (
*i*) ³ average CTRa of all points - The points
*i*Ì*I*are ordered by their contribution to CTRa (*i**i*) ³*p}*is truncated at the lowest value*i*_{0 }_{Ì}_{ }*I*such that the truncated sum is ³*p.*The set is the set of explicative points. The same procedure is followed for*J*.

*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 (COS^{2} 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 (COS^{2} 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