On Extended Guttman Condition in High Dimensional Factor Analysis
It is well-known that factor analysis and principal component analysis often yield similar estimated loading matrices. Guttman (Psychometrika 21:273–285, 1956) identified a condition under which the two matrices are close to each other at the population level. We discuss the matrix version of the Guttman condition for closeness between the two methods. It can be considered as an extension of the original Guttman condition in the sense that the matrix version involves not only the diagonal elements but also the off-diagonal elements of the inverse matrices of variance-covariances and unique variances. We also discuss some implications of the extended Guttman condition, including how to obtain approximate estimates of the inverse of covariance matrix under high dimensions.
KeywordsHigh dimensions Principal components Unique variances
The authors are thankful to Dr. Dylan Molenaar’s comments. Ke-Hai Yuan’s work was supported by the National Science Foundation under Grant No. SES-1461355.
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