Equivalencies among Canonical Factor Analysis, Canonical Reliability Analysis, and Principal Components Analysis: Implications for Data Reduction of Fallible Measures

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Abstract
If variables are scaled so that they have equal errors of measurement, then the maximally reliable composites found in a canonical reliability analysis have highly desirable properties. Under this scaling transformation, canonical factor analysis on all nonerror variance, principal components analysis and canonical reliability analysis all yield equivalent results. Thus, the resulting components are successively maximally reliable and simultaneously provide a least squares fit to both the covariance matrix of rescaled observed scores and of rescaled true scores. In addition, the proportion of true to observed score variance retained in an s-dimensional least squares fit is equal to the ratio of the s-dimensional canonical reliability to the canonical reliability of the entire battery. Based on the above, it is argued that data reduction (when there is no a priori reason to keep the original units of measurement) should proceed by doing a components analysis on the covariance matrix rescaled so as to yield equal errors of measurement.

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