6.4.2 Interpretation of Principal Components Analysis

The program issues the following statistics:

1. Ranked listed of eigenvalues of X^TX for all the principal components
2. Factor coordinates of the variable points
3. Contribution of variable to the variance (inertia) of each factorial axis
4. Contribution of the variables to the eccentricity of the factorial axis.
5. Quality of representation of the variable in the subspace by the set of principal axes.

Relative contribution of an element j (variable or individual) to the inertia of the a - axis is given by

where Fa _j is the coordinate of the j ^th variable on a -axis.

Relative contribution of an element j (variable or individual) to the eccentricity of the a - axis is given by

Cor _a (j) =G_a ²

Rules for Selecting Significant Axes

In practice, the data set is usually so large that it is impossible to process the results of factor analysis, coordinate by coordinate, contribution by contribution or factor by factor. The following rules are suggested for the selection of significant elements that will help in the interpretation of data. Here, the word ‘significant’ should be understood in terms of relevance, not in terms of statistical significance.

First order significant axes

Let N be number of axes to be retained.

Rule 1: N is the number of factorial axes such that

Rule 2: N is the number of factorial axes such that, where t _a is the percentage of variance explained by the a ^th factor, and p is the percentage of variance explained (usually 80%).

Second order significant axes:

Rule 3: Let N be the rank of the a -axis to be retained. N is chosen so that at least one variable or one individual exists such that Cor_N( j ) or Cor_N( _I ) is greater that a given k (k is similar to COSine squared).

This rule allows us to retain as significant those axes, which highlight local effects.

Rules for Selecting Significant Elements of Factorial Axes

Significant elements are selected on the basis of their contribution to the inertia (CTR) or the eccentricity of a factorial axis.

• Contribution to inertia: Significant elements of a cloud of column or row points are those, which explain the dispersion of factorial axes l a . The explicative points are those which have the greatest value of CTRa ( i), for each l a .
• A subset of points i can be selected whose contributions in terms of CTRa ( i ) are greater than the average of the contributions for the whole set.
• Rank order CTRa ( i ) in the decreasing order and select the subset t for which
• å CTRa i ) ³ r

· Contribution to eccentricity: An element j is selected as significant when its contribution to eccentricity of a factorial axis, COR_a.³ k.. .

The selection procedure for interpretation is based on two given values fixed by the user: r is often taken equal to 80% and k is taken equal to 0.5.

Quality of Representation of the Factor space.

The quality of representation of the factor space is given by:

where s is the number of factors in the factor space

Quality of representation of points in the s-dimensional factor space.

The quality of representation of a point ( i or j ) in the factor space generated by the principal components analysis can be measured as follows: