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.
ApostolT. (2013). Calculus. New York, NY: Springer.
2.
HowardW. J.RhemtullaM.LittleT. D. (2012). Using principle components as auxiliary variables in missing data estimation. Multivariate Behavioral Research, 50, 285-299.
3.
JohnsonR. A.WichernD. W. (2002). Applied multivariate statistical analysis. Upper Saddle River, NJ: Prentice Hall.
4.
TimmN. H. (2002). Applied multivariate analysis. New York, NY: Springer.
5.
RaykovT.MarcoulidesG. A. (2008). An introduction to applied multivariate analysis. New York, NY: Taylor & Francis.
6.
RaykovT.MarcoulidesG. A. (2011). Introduction to psychometric theory. New York, NY: Taylor & Francis.
7.
RaykovT.MarcoulidesG. A.LiT. (2017). On the fallibility of principal components in research. Educational and Psychological Measurement, 77, 165-178.
