Factor Analysis
Cronbach's alpha also has a theoretical relation with factor analysis. As shown by Zinbarg, Revelle, Yovel and Li (2005), alpha may be expressed as a function of the parameters of the hierarchical factor analysis model which allows for a general factor that is common to all of the items of a measure in addition to group factors that are common to some but not all of the items of a measure. Alpha may be seen to be quite complexly determined from this perspective. That is, alpha is sensitive not only to general factor saturation in a scale but also to group factor saturation and even to variance in the scale scores arising from variability in the factor loadings. Coefficient omega_hierarchical (McDonald, 1999; Zinbarg, Revelle, Yovel & Li, 2005) has a much more straightforward interpretation as the proportion of observed variance in the scale scores that is due to the general factor common to all of the items comprising the scale.
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