factor.residuals.RdThe 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. Find the residuals of the original minus the reproduced matrix. Used by factor.fit, VSS, ICLUST, etc.
factor.residuals(r, f)The basic factor equation is \(_nR_n \approx _{n}F_{kk}F_n'+ U^2\). Residuals are just R* = R - F'F. The residuals should be (but in practice probably rarely are) examined to understand the adequacy of the factor analysis. When doing Factor analysis or Principal Components analysis, one usually continues to extract factors/components until the residuals do not differ from those expected from a random matrix.
rstar is the residual correlation matrix.
fa2 <- fa(Harman74.cor$cov,2,rotate=TRUE)
#> Specified rotation not found, rotate='none' used
fa2resid <- factor.residuals(Harman74.cor$cov,fa2)
fa2resid[1:4,1:4] #residuals with two factors extracted
#> VisualPerception Cubes PaperFormBoard Flags
#> VisualPerception 0.6533350 0.10174203 0.1656341 0.19205836
#> Cubes 0.1017420 0.86326270 0.1638187 0.05229164
#> PaperFormBoard 0.1656341 0.16381874 0.8232249 0.10052094
#> Flags 0.1920584 0.05229164 0.1005209 0.76340745
fa4 <- fa(Harman74.cor$cov,4,rotate=TRUE)
#> Specified rotation not found, rotate='none' used
#> Warning: NaNs produced
fa4resid <- factor.residuals(Harman74.cor$cov,fa4)
fa4resid[1:4,1:4] #residuals with 4 factors extracted
#> VisualPerception Cubes PaperFormBoard Flags
#> VisualPerception 0.44982015 -0.03531402 -0.01016952 0.04145577
#> Cubes -0.03531402 0.77015627 0.04339997 -0.04969772
#> PaperFormBoard -0.01016952 0.04339997 0.66152929 -0.02945683
#> Flags 0.04145577 -0.04969772 -0.02945683 0.65020432