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:

1. 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)
2. There are no hypotheses about interfactor correlations (either all correlations are zero -- orthogonal factors, or all may correlate -- oblique factors)
3. Only one group is analyzed
4. Unique factors (those that relate only to one variable) are uncorrelated,
5. 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.