Why is error variance important in CFA?

Question

I am reading the book related to SEM (Byrne, 1998) and it is stated that regression of the observed variables on the factor, and the variances of both the errors of measurement and the factor, as well the error covariance, are of primary interest. Why is error variance important in CFA, and particularly error covariance? Thanks.

Answer 1

Basic SEM attempts to explain variance in measured variables through predictive relationships with a latent variable. In a simple, single common factor model, one latent factor is estimated as a potential explanation for all common variance in a set of related measurements. In a typical case of strongly correlated measurements, each measurement will correlate with the sum total of the measurements. A latent factor is estimated as a linearly weighted composite of the measurements rather than a simple sum. Weights are based on the amount of common variance in each measurement. Factor loadings are the correlations between the measurements and the latent factor.

Smaller factor loadings result in larger error variances. Error variances are the portions of variance in each measurement that do not covary with the latent factor. These are interesting in as much as they can indicate "good" and "bad" measures of a latent factor. I hesitate to make such evaluative judgments myself, as it seems to me that one might sometimes want a measure of a latent factor that correlates somewhat weakly with other measures to improve coverage of the latent factor's total variance. Nonetheless, latent factor modelers often set out to establish a unidimensional set of indicators, which is to say they want all indicators to load heavily on one common factor and have small, uncorrelated error variances. Thus if a given indicator has a low loading and lots of error variance, this is "interesting" because it means that indicator is "bad" (e.g., unreliable, largely explained by other latent factors, or otherwise very different from other indicators in the set).

AFAIK, error covariance refers to error variances that covary. E.g., if a set of ten indicators all purport to reflect the influence of one latent causal factor, say eight of them do, and two of them also share a second common influence that is uncorrelated with the primary factor of interest. In this case, those two misfit indicators will have low loadings on the primary factor, a lot of error variance, and their respective error variances will covary. "Nuisance factors" such as the second one in this example are often useful to model as well, because they leave less variance unexplained by a SEM, thus improving fit statistics and the reliability of path coefficents. Look into bifactor analysis if you have nuisance factors to estimate and control.

Answer 2

This is covered quite well by Timothy A. Brown in Confirmatory Factor Analysis for Applied Research (the chapter Model Revision and Comparison). He explains how error covariance will actually inflate factor loadings of the involved items (i.e. the items displaying error covariance), while loadings of remaining items in the cluster will be deflated.

From my own experience, if you have two competing factor sollutions, addressing error correlations (by allowing the involved items to correlate freely) may change the relative fit of the models. If you are not aware of this, you could potentially claim that your analysis, based on global fit indices alone, supports model A over model B, while in fact, this is an artifact of substantive error correlations. Now, whether error correlations are substantive or not, is another matter. It should be supported by both a content analysis (i.e. is there high overlap between the content of the two items or is one a reversed version of the other?) and EPC values.