{smcl}
{.-}
help for {cmd:minap}
{.-}

{title:Minimum Average Partial Correlation for Number of PCs}
 
{p 8 27}
{cmd:minap}
{it:varlist}
[{cmd:if} {it:exp}]
[{cmd:in} {it:range}]

{p 8 27}
{cmd:minap,}
{cmd:corr(}{it:corr matrix}{cmd:)}


{title:Description}

{p 0 0}
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. {cmd:minap} calculates this minimum average 
partial correlation.  It can take as input either a variable list or a 
correlation matrix.

{title:Options}

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

{title:Remarks}

{p}
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 {cmd:minap} criteria is useful when principal components
is being used as an approximation to factor analysis, as with the Stata
{cmd:pcf} option to the {cmd:factor} command. Gorsuch also points out that,while
{cmd:minap} was developed for pricipal components analysis, it may also be usefu
for common factor analysis.

{p}
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.  {cmd:minap} would be inappropriate in those cases.

{p}
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 {cmd:minap} and Zwick and Velicer (1986) claim that it leads to
overextraction.

{p}
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.

{p}
{cmd: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.

{title:Examples}

	{cmd:minap v1-v30}

	{cmd:minap , corr(Harmon)}

{title:References}

{p 4 4}
Gorsuch, R.L. (1983). {it:Factor analysis (second edition)}. 
Hillsdale, NJ: Lawrence Erlbaum Associates.

{p 4 4}
Kaiser, h.F. (1960). The application of electronic computers to factor 
analysis. {it:Educational and Psychological Measurement, 20}, 141-151.
	
{p 4 4}
Velicer, W.F. (1976). Determining the number of components
from the matrix of partial correlations. {it:Psychometrka, 41}, 321-327.

{title:Author}

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