ABSTRACT
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
= +, (1)
where is a p1 random vector of observed random variables, is a pm matrix of
factor pattern coefficients (which indicate how much a unit change of a common factor
effects a change in a corresponding observed variable), is an m1 random vector of
latent (hypothetical) common factor variables, is a pp diagonal matrix of unique
factor pattern loadings, and is a p1 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( )]}=I and
var()=I, where var( ) denotes the pp variance-covariance matrix of the common factor
variables. A fundamental set of assumptions of the common factor model, which
introduce prior constraints into the model, is that cov( , )=0 and var()=I, which implies
that the common factor variables are uncorrelated with the unique factor variables and
further that the unique factor variables are also mutually uncorrelated. From the model
equation, the additional fact that we assume that is a pp diagonal matrix, and these
covariance constraints one can readily derive the fundamental theorem of common factor
analysis given by the equation
=+2, (2)
where =var() is the pp variance-covariance matrix for the observed variables, is
the pm matrix of factor pattern coefficients (as before), =var( ) is the mm
variance/covariance matrix for the common factors, and 2= var()=I is a
diagonal matrix of unique variances, representing in its principal diagonal the respective
variance within each observed variable due to its corresponding unique factor variable.
