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For every complex problem there is an answer that is clear, simple, and wrong.
(H. L. Mencken)

DECEMBER 16, 2010

How many axes in a factor analysis?

A new entry on the CROP survey house blog explores the historic and practical reasons behind the main techniques used to determine the number of axes used in factor analysis. Let us remind the reader that factor analysis is used to reduce redundancy in a data set by identifying variables that appear to contribute to the same underlying dimension. The goal is to determine the true number of underlying dimensions in a given data set.

The author begins the exploration of the topic by explaining that the Kaiser-Guttman rule places axis rotation BEFORE the operation seeking to limit the number of factor axis based on the premise that a factor's eigenvalue must be equal or superior to 1 for this factor to be retained. This order in operations means that the complete final analytic space is not taken into consideration when determining the number of factor axes to be used in the analysis. The author also brings to light that axes reduction using scree plots suffers from the same shortcomings.

The author offers two explanations for this phenomenon. The first explanation is that exploratory factor analysis (the term "exploratory" differentiates factor analysis as used in market research from its "pure" form, which originates in psychology) is linked to artificial intelligence and data mining, where questions of performance and automation win over meaning.

The second explanation refers to history: when the Kaiser-Guttman rule was put forward, in the 50's and 60's, computerized data analysis was very expensive. Since axis rotation can require a large number of mathematical operations, the determination of an eigenvalue threshold BEFORE axis rotation was an economical solution. That is to say that the so-called "habitual" procedure used to reduce the number of factor axes before the axes are rotated is purely a matter of habit and practicality, rather than one with underlying statistical or methodological reasoning.

The author suggests a new rule, which he calls KG+. In this new paradigm, the Kaiser-Guttman rule is used to determine the MINIMUM number of axes; the actual determination of the number of significant axes takes place after the rotation, which, according to the author, should allow for a better interpretation of the axes.

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