The program issues the following statistics:
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) =Ga 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 CorN( j ) or CorN( 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 eccentricity: An element j is selected as significant when its contribution to eccentricity of a factorial axis, CORa.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: