Joreskog (1971): Play Along with Example 1 in AMOS

Joreskog's paper Statistical Analysis of Sets of Congeneric Tests was, it is probably fair to say, a watershed moment in psychometrics.

In it, Joreskog reported on how he had been able to employ the "basic equations of the factor analytic model with one common factor" and combine it with the "maximum likelihood method" of estimation, to allow for a "statistical test of the assumption that the tests [indicators] are congeneric" (p. 112), based on "the testing of goodness of fit by the likelihood ratio technique" (p. 109).

Forget about that strange word "congeneric" for a moment. What is important to know is that this is the moment that factor analysis meets statistical significance testing. And if there were two things that psychologists liked at the time it was factor analysis and significance testing. It was always going to be a winning combination, and it went on to be called confirmatory factor analysis - you've probably heard of it.

Anyway. Joreskog goes on to apply his technique to some data pertaining to marks for four essays. The following table appears on page 114:

The first table (1a) is the covariance-variance matrix for the scores, the second table (1b) is the model fit for the parallel, tau-equivalent and congeneric models. The third I'm not too worried about.

The congeneric model is just the common factor model. The tau-equivalent model is the common factor model plus item factor loadings constrained to be equal. The parallel model is the common factor model plus item factor loadings and item variances constrained to be equal.

Looking at it from the perspective of the covariance-variance matrix, the tau-equivalent model dictates that covariances are equal (p. 113), and the parallel model says that covariances and variances are equal (p.113). These are strict demands. The congeneric model involves neither of these strict demands.

I have converted the covariance-variance matrix from Table 1a to a correlation matrix, so that you can run the analysis in AMOS. The correlation matrix looks like this:

And download the .sav file here from my dropbox.

What you need to do in AMOS is to create 3 models: (1) the normal single factor model (test of congeneric-ness), (2) a single factor model where factor loadings are constrained to be equal (tests of tau-equivalence), and (3) a single factor model where factor loading and variance are constrained to be equal (test of parallel-ness).

The first model is no problem - just draw a single-factor model how you usually do. For the second model (tau-equivalence), you first need to name parameters and then go to Analyze -> Manage Models. Set factor loadings to be equal like so:

For the third model (parallel), you need to set factor loadings and variances to be equal like so:

Now run the analysis with a normal Maximum likelihood estimation procedure. You will get the same (almost the same) results as Joreskog did back in the 70s. I imagine figures differ subtly due to rounding in the original paper. We can now also can get fit indexes (e.g. CFI, RMSEA) that have been created since Joreskog's paper was published.