You can build CFA models in BUGS (or any of the other Bayesian MCMC engines). There exist several articles and a book by Sik-yum Lee on that topic. Once you've created a Gaussian CFA that way, it should be straight-forward to change the residual edistribution and introduce a link function (since you are familiar with GLM).

Note the blavaan package, which produces BUGS or Stan code, which you can tweak to your liking.

A valuable question, Oliver.

Many researchers raise the issue of non-normal data in their analyses.

CFA, in itself, does not require data that are normally distributed. However, interpretation of significance levels, and related output, often does assume normally-distributed coefficients and error terms. For example, James Arbuckle explained, somewhere deep in the documentation for his SEM program AMOS, that he decided to use the term CR (Critical Ratio), instead of t-test, because t-tests do assume normally-distributed parameter estimates.

Many years ago, my evil PhD supervisor required me and other candidates to run loads of experiments with synthetic data to test the output of various analysis techniques. We found that, as with other regression-like methods, EFA and CFA are extraordinarily robust against non-normal data. We only found strange results when the data were extremely skewed. My guess is that you will get quite acceptable results if your gamma-distributed data have shape parameter k>=2.

Dear colleagues

Many thanks for your suggestions, this is much appreciated!

Martin Schmettow : Thanks, Martin, for your idea, I will check out the book by Sik-Yum Lee. I was not planning on using a Bayesian approach, but I am curious to see whether it can solve the problem.

Hume F. Winzar : Thanks for the information that the CFA itself is quite robust in case of non-normal data. The literature I have consulted is somewhat confusing on the question whether normality is a requirement for SEM techniques or not: Gorsuch (2015, p. 136) and McCoach et al. (2013, p. 229) list normality as an assumption of both CFA and SEM techniques, but Bollen (1989, p. 79) argues that normality is not required. It is quite possible that the former two refer to inference on coefficients, whereas the latter to the estimation of the coefficients themselves. If you know a reference that puts this discussion to bed, your input is very welcome.

I will look up how Arbuckle calculates the CR. This seems to imply that he found a way of testing his coefficients without assuming normality.

Thanks also for relaying the k>2 threshold. Unfortunately, my distributions are very skewed: attached you find histograms of the six indicators with estimated gamma densities. Indicators 1-3 (top, red) are supposed to load on Factor 1, Indicators 4-6 (bottom, blue) on Factor 2. This data is the response of 163 participants to a stimulus with expected low scores on Factor 1 and high scores on Factor 2 - the data shows this quite nicely. Yet the k's are between 0.18 and 1.48, so the data tends to stick close to zero and is very unsymmetric.

I wonder whether a normal transformation could work in order to carry out the CFA:

Gamma(k,theta) -> U(0,1) -> N(0,1),

using the cdfs of the estimated empirical gamma distributions and the quantile function of the standard normal.

Bollen, K. A. (1989). Structural equations with latent variables. New York: J. Wiley and Sons.

Gorsuch, R. L. (1983). Factor Analysis (2. Aufl.). Hillsdale, N.J: Lawrence Erlbaum Associates.

McCoach, D. B., Gable, R. K., & Madura, J. P. (2013). Instrument Development in the Affective Domain : School and Corporate Applications (3rd ed. 2013). New York, NY: Springer New York.

The Critical Ratio (CR) does not have a special formula. It's the same as a regular t-test: CR=t=(Estimate)/(stdError_of_Estimate). Arbuckle suggested using the term CR, instead of t-test, because users cannot be confident that parameter estimates are normally distributed, even when the original data are normally distributed. So, the usual cutoff for "statistical significance", 1.96, should be regarded with caution.

I don't have a direct reference of the Arbuckle comment. I just recall it from long ago when AMOS was a standalone program, before IBM/SPSS purchased it, and Jim Arbuckle was communicating with early adopters and documenting the program as he went.

It looks like a data transformation is warranted. A qube-root of your data might do the job. (Wilson & Hilferty, 1931)

https://water.usgs.gov/osw/bulletin17b/Wilson_Hilferty_1931.pdf

Thank you, Hume F. Winzar , for expanding on the Critical Ratio subject - I understand that the CR is more about giving the t statistic a different name in order to avoid the implication of normality. Thank you also for the cube-root idea and the truly classic Wilson & Hilferty quote! Unfortunately, ladder of powers transformations do not work with distributions that have their mode at one of the extremes. However, the same can be said about my Gamma(k,theta) -> U(0,1) -> N(0,1) transformation idea yesterday: no monotonic function will move the mode to the center of the distribution.