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Minimum Average Partial Correlation for Number of PCsminapvarlist[ifexp] [inrange]

minap,corr(corr matrix)

DescriptionVelicer(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.

minapcalculates 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 ofvarlistorcorrelation matrixmust be provided.

RemarksMany 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

minapcriteria is useful when principal components is being used as an approximation to factor analysis, as with the Statapcfoption to thefactorcommand. Gorsuch also points out that,whileminapwas 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.

minapwould 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

minapand 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.

minapreturns 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)

ReferencesGorsuch, 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.

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