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\title{Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. \emph{Psychometrika}, \emph{30}, 179--185. (3362 citations as of 4/1/2016)}

\author{Online comments by Jack McArdle}



\institute{There is no best ``number of factors'' rule, but there is a worst rule --- the number of eigenvalues greater than one. Bill Revelle}

\date{Psychometric Society, Asheville, NC, July, 2016}



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\frametitle{The Number of Factors Problem}

Henry Kaiser's approach to factor analysis (dating from 1960) was disdainfully labeled ``Little Jiffy'' by Chet Harris:

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``principal components with associated eigenvalues greater than one followed by normal varimax rotation''

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Horn (1965) introduced a different strategy for selecting the number of components (aka, ``factors''):

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select as many components that have eigenvalues greater than the values expected from randomly constructed data

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\frametitle{Parallel Analysis}
This comparison method in Horn (1965) has become known in the literature as ``parallel analysis''

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The word ``parallel'' refers to the construction of random data sets that have the same number of variables and number of subjects as the real correlation matrix being analyzed.

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The ordered eigenvalues are obtained for each correlation matrix constructed; these ordered vectors are then averaged to obtain the value expected for a particular (ordered) eigenvalue when generated from random data.


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\frametitle{The Guttman Lower Bound}

The Kaiser rule is based on some earlier work by Guttman on a lower bound for the number of factors:

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the number of eigenvalues greater than one for the (observed) correlation matrix

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Note the following: if the population correlation matrix is the identity, so all the variables have zero correlations, then there are \emph{no} latent roots of the population correlation matrix that are strictly greater than one.
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However, an observed correlation matrix based on data sampled from a population having such an identity correlation matrix, one would expect about one-half of the eigenvalues to be greater than 1.0 and about  one-half to be less than 1.0.

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This because the sum of all the eigenvalues (all of which must be nonnegative) equals the number of variables.

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\frametitle{The Illustration in Horn (1965)}

Horn (1965) gives an illustration involving 65 variables and 297 subjects --- this is his Table 1 that we show next
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Real data number of eigenvalues greater than 1.0: 16

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Real data number of eigenvalues greater than that found for random data: 9

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Horn also noted something of an inflection at this point:

given that Horn was Cattell's student, did he see the proverbial elbow in the scree plot?

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\frametitle{Further Work}

A lot of work has been stimulated by Horn (1965):

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using percentiles of the distribution for the random eigenvalues;

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using permutation distributions to generate the parallel random data;

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using other criteria, such as Velicer's minimum average partial (MAP) test.

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