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The case that the factor model does not account for all the covariances ofthe observed variables is considered. This is a quite realistic conditionbecause some model error as well as some sampling error should usually occurwith empirical data. It is shown that principal components representingcovariances not accounted for by the factors of the model can have a non-zerocorrelation with the common factors of the factor model. Non-zero correlationsof components representing variance not accounted for by the factor model withcommon factors were also found in a simulation study. Based on these results itshould be concluded that common factors can be correlated with variancecomponents representing model error as well as sampling error. In consequence,even when researchers decide not to represent some small or trivial variance bymeans of a common factor, these excluded variances can still be part of themodel.

The factor paradox: Common factors can be correlated with the variance not accounted for by the common factors!