How well does the factor model fit a correlation matrix. Part of the VSS package
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.
factor.fit(r, f)
r |
a correlation matrix |
f |
A factor matrix of loadings. |
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 - sum(R*^2)/sum(R^2)
.
The sums are taken for the off diagonal elements.
fit
William Revelle
## Not run: #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 ## End(Not run)
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