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##

1 Exploratory and Confirmatory Factor Models

There are two general classes of factor models: exploratory and
confirmatory. In exploratory factor analysis one does not postulate
an *a priori* factor structure; that is, the number of latent
factors, correlations among them, and the *factor loading pattern*
(the pattern of relative weights of
the observed variables on the latent factors) is calculated from the
data in some manner which maximizes the amount of variance/covariance
explained by the latent factors. More formally, in exploratory factor
analysis:
- There are no hypotheses about factor loadings (all variables
load on all factors, and factor loadings cannot be constrained to be
equal to other loadings)
- There are no hypotheses about interfactor correlations (either
all correlations are zero -- orthogonal factors, or all may
correlate -- oblique factors)
- Only one group is analyzed
- Unique factors (those that relate only to one variable) are
uncorrelated,
- All observed variables need to have specific variances.

These models often are fitted using a statistical package such as SPSS
or SAS, in which one may *explore* the relationships among
observed variables in a latent variable framework.
In contrast, confirmatory factor analysis requires one to formulate a
hypothesis about the number of latent factors, the relationships
between the observed and latent factors (the factor pattern), and the
correlations among the factors. Thus, a possible model of the data is
formulated in advance as a *factor structure,* and the factor loadings and correlations are
estimated from the data^{}. As usual, this model-fitting process
allows one to *test* the ability of the hypothesized factor
structure to account for the observed covariances by examining the
overall fit of the model. Typically the model involves certain
constraints, such as equalities among certain factor loadings or
equalities of some of the factor correlations. If the model fails
then we may relax certain constraints or add more factors, test for
significant improvement in fit using the chi-squared difference test,
and examine the overall goodness of fit to see if the new model
adequately accounts for the observed covariation. Likewise, some or
all of the correlations between latent factors may be set to zero or
estimated. Then we can test if these constraints are consistent with
the data. Confirmatory factor models are the type we are concerned
with using Mx.

** Next:** 2 Building a Phenotypic
** Up:** 2 Phenotypic Factor Analysis
** Previous:** 2 Phenotypic Factor Analysis
** Index**
Jeff Lessem
2002-03-21