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# Looking Back on the Indeterminacy Controversies in Factor Analysis

DOI link for Looking Back on the Indeterminacy Controversies in Factor Analysis

Looking Back on the Indeterminacy Controversies in Factor Analysis book

# Looking Back on the Indeterminacy Controversies in Factor Analysis

DOI link for Looking Back on the Indeterminacy Controversies in Factor Analysis

Looking Back on the Indeterminacy Controversies in Factor Analysis book

## 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.