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Minimum Average Partial Correlation for Number of PCs

minap varlist [if exp] [in range]

minap, corr(corr matrix)

Description

Velicer(1976) proposed that, when conducting principal components analysis as a
version of factor analysis, the number of components one should extract is that
at which the average partial correlation of the variables, after partialling
out m principal components, would be a minimum. minap calculates this minimum
average partial correlation.  It can take as input either a variable list or a
correlation matrix.

Options

corr(corr matrix) will estimate number of components from provided correlation
matrix. The correlation matrix must already exist.  One and only one of varlist
or correlation matrix must be provided.

Remarks

Many criteria for estimating the number of components in principal components
analysis, or of factors in factor analysis, have been proposed (Gorsuch, 1983).
One relatively little used of these criteria is the minimum average partial
correlation proposed by Velicer (1976). The minap criteria is useful when
principal components is being used as an approximation to factor analysis, as
with the Stata pcf option to the factor command. Gorsuch also points out
that,while minap was developed for pricipal components analysis, it may also be
usefu for common factor analysis.

This criterion has performed well in simulation studies with data with a
relatively clear factor structure (Zwick & Velicer, 1986).  Gorsuch (1976),
however, warns that minimum average partial correlation may not perform well
and may suggest underextraction when there are components or factors with only
a few loadings.  Similarly, in many applications of principal components
analysis, one may be interested in components on which only one or two
variables load.  minap would be inappropriate in those cases.

For comparison purposes, the number of eigenvalues greter than one, claimed by
Kaiser (1960) to be a good estimator of the number of components to extract, is
also provided. In most cases, this rule will recommend the extraction of more
components than will minap and Zwick and Velicer (1986) claim that it leads to
overextraction.

It should be noted that no criterion can be counted on by itself to determine
the number of components or factors to extract with real data.  Considerations
of interpretability are also important. In general, determining the precise
number of components to retain matters more when the component (or factor)
solution will be rotated.

minap returns the minimum average partial correlation and the Kaiser
(eigenvalue > 1) recommended number of components, as well as the eigenvector,
eigenvalue, and correlation matrices. The eigenvalue matrix is scaled so the
sqares of the collumn entries sum to the eigenvalues.

Examples

minap v1-v30

minap , corr(Harmon)

References

Gorsuch, R.L. (1983). Factor analysis (second edition).  Hillsdale, NJ:
Lawrence Erlbaum Associates.

Kaiser, h.F. (1960). The application of electronic computers to factor
analysis. Educational and Psychological Measurement, 20, 141-151.

Velicer, W.F. (1976). Determining the number of components from the matrix
of partial correlations. Psychometrka, 41, 321-327.

Author

Stephen Soldz
Boston Graduate School of Psychoanalysis
1581 Beacon St.
Brookline, MA 02446
Tel: (617) 277-3915 x27
ssoldz@bgsp.edu

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