Editing Factor analysis (section)

=== Variance versus covariance ===
Factor analysis takes into account the [[random error]] that is inherent in measurement, whereas PCA fails to do so. This point is exemplified by Brown (2009),<ref name=Brown>{{cite web|last=Brown|first=J. D.|title=Principal components analysis and exploratory factor analysis – Definitions, differences and choices.|date=January 2009|url=http://jalt.org/test/PDF/Brown29.pdf|publisher=Shiken: JALT Testing & Evaluation SIG Newsletter|access-date=16 April 2012}}</ref> who indicated that, in respect to the correlation matrices involved in the calculations:

{{Blockquote|"In PCA, 1.00s are put in the diagonal meaning that all of the variance in the matrix is to be accounted for (including variance unique to each variable, variance common among variables, and error variance). That would, therefore, by definition, include all of the variance in the variables. In contrast, in EFA, the communalities are put in the diagonal meaning that only the variance shared with other variables is to be accounted for (excluding variance unique to each variable and error variance). That would, therefore, by definition, include only variance that is common among the variables."|Brown (2009)|Principal components analysis and exploratory factor analysis – Definitions, differences and choices}}

For this reason, Brown (2009) recommends using factor analysis when theoretical ideas about relationships between variables exist, whereas PCA should be used if the goal of the researcher is to explore patterns in their data.