On the Unlikely Case of an Error-Free Principal Component From a Set of Fallible Measures

This note extends the results in the 2016 article by Raykov, Marcoulides, and Li to the case of correlated errors in a set of observed measures subjected to principal component analysis. It is shown that when at least two measures are fallible, the probability is zero for any principal component-and in particular for the first principal component-to be error-free. In conjunction with the findings in Raykov et al., it is concluded that in practice no principal component can be perfectly reliable for a set of observed variables that are not all free of measurement error, whether or not their error terms correlate, and hence no principal component can practically be error-free.