Contemporary Psychometrics: A Festschrift for Roderick P. McDonald

By Albert Maydeu-Olivares | Go to book overview

7
Looking Back on
the Indeterminacy
Controversies in
Factor Analysis

Stanley A. Mulaik
Georgia Institute of Technology


ROTATIONAL INDETERMINACY

The common factor analysis model has two forms of indeterminacy: (a) rotational indeterminacy and (b) factor indeterminacy. To discuss these, let us first specify the model equation of the common factor analysis model and the fundamental theorem derived from it. The model equation of common factor analysis is

where η is a p × 1 random vector of observed random variables, Λ is a p × m matrix of factor pattern coefficients (which indicate how much a unit change of a common factor effects a change in a corresponding observed variable), ξ is an m × 1 random vector of latent (hypothetical) common factor variables, Ψ is a p × p diagonal matrix of unique factor pattern loadings, and ε is a p × 1 random vector of unique factor variances. We will assume without loss of generality that all variables have zero means and that the common factor and unique factor variables have unit variances so that {diag[var(ξ)]| = I and var(ε) = I, where var(ξ) denotes the p × p variance–covariance matrix of the common factor variables. A fundamental set of assumptions of the common factor model, which introduce prior constraints

-173-

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