# CFA items are averages: should we account for the variance?

Imagine a standard confirmatory factor analysis with one latent variable, L, causing the observed variables (items) q1, q2 and q3. It is my goal to rank all observations on their estimated scores on the latent variable L. So far this is straight-forward.

Yet in my case, the observed q1 to q3 each are averages. I also have access to the corresponding variances of q1 to q3 for each observation, let's call these v1, v2 and v3.

Since each observation can differ in both their mean and variance for each of the q variables, I believe I should somehow incorporate the v variables into the generation of my latent index that I will use as the basis of the ranking.

What do you think is the best way?

I can think of the following solutions:

1. Estimate L as before based on variables q1 to q3. Analogously estimate latent variable K based on variables v1 to v3. Then do a OLS such that L = K + e, in order to predict L given K.
2. Estimate the latent variables S, L and K, whereby L and K are estimated as above, but both are sub-scales of S.

There might be more ways I haven't thought of.

Item variances are rarely constant across items in factor analysis. Most typically, items are only congeneric, which means that their loadings on the common factor and their residual variances both may differ across items, so that their total variances may differ. So there is nothing unusual here.

Your observed variables are averages--does that mean equally weighted sums divided by the number of items in each sum? Then the weights are not subject themselves to random sampling error, so they don't contribute to uncertainty about the values of the sums.

Deriving a common factor from the variances doesn't make a lot of sense to me.

If you wanted to overcome the information loss due to the construction of averages, you might try modeling the original items in a second order factor analysis. However, my bet would be that the model would not fit well.

However, you have a further problem--factor indeterminacy (see, e.g., Grice 2001; Rigdon et al 2019). The value of the common factor for each case is indeterminate--an infinite number of different sets of scores for the factor will be equally consistent with the data and model. Regression-method "factor scores" ignore this indeterminacy, at the cost of not exactly reproducing the factor. "Plausible factor score" methods arbitrarily choose one set of scores from among this infinity of sets, concealing the indeterminacy. Whether or not you are comfortable with this may depend on what you mean to do with the scores.

Grice, J. W. (2001). Computing and evaluating factor scores. Psychological methods, 6(4), 430.

Rigdon, E. E., Becker, J. M., & Sarstedt, M. (2019). Factor indeterminacy as metrological uncertainty: Implications for advancing psychological measurement. Multivariate behavioral research, 54(3), 429-443.

• Thanks @Ed Rigdon. Very much appreciated! In respect to my observed averages: each of the q items are calculated as the arithmetic mean, but the number of observations each mean is based on can vary substantially. So the location of the 'true' mean is somewhat uncertain and that's why I had the idea to somehow incorporate the corresponding variances. Commented Dec 30, 2019 at 16:58
• I see, @griischdoffer. You want to capture the "mass" of observed variables represented by each composite--but without modeling the original observed variables themselves. If your factor analysis were based on covariances of the composites, rather than correlations, you would accomplish this, to an extent. Be wary of large differences (factor of 10) in the variances of the composites. Differences in variance can be based on # of items as well as covariances across items. You want eigenvalues, maybe? Commented Dec 31, 2019 at 12:03