psych (version 2.4.1)

factor.fit: How well does the factor model fit a correlation matrix. Part of the VSS package

Description

The basic factor or principal components model is that a correlation or covariance matrix may be reproduced by the product of a factor loading matrix times its transpose: F'F or P'P. One simple index of fit is the 1 - sum squared residuals/sum squared original correlations. This fit index is used by VSS, ICLUST, etc.

Usage

factor.fit(r, f)

Value

fit

Arguments

r

a correlation matrix

f

A factor matrix of loadings.

Author

William Revelle

Details

There are probably as many fit indices as there are psychometricians. This fit is a plausible estimate of the amount of reduction in a correlation matrix given a factor model. Note that it is sensitive to the size of the original correlations. That is, if the residuals are small but the original correlations are small, that is a bad fit.

Let $$R* = R - FF'$$ $$fit = 1 - \frac{ \sum(R*^2)}{\sum(R^2)}$$.

The sums are taken for the off diagonal elements.

See Also

VSS, ICLUST

Examples

if (FALSE) {
#compare the fit of 4 to 3 factors for the Harman 24 variables
fa4 <- factanal(x,4,covmat=Harman74.cor$cov)
round(factor.fit(Harman74.cor$cov,fa4$loading),2)
#[1] 0.9
fa3 <- factanal(x,3,covmat=Harman74.cor$cov)
round(factor.fit(Harman74.cor$cov,fa3$loading),2)
#[1] 0.88

}