I am working with count data from a psychological experiment. Data are from several subtests, i.e. scores on these tests. The way these scores are generated is clearly not Gaussian (not only because they are discrete, but also since the scores are basically sums of correctly solved tasks).
As a baseline I worked with models using the Poisson distribution. For example the Rasch (1960) Poisson counts model (which is a type of GLM), for which I also estimated the dispersion from the data (1.01, bootstrapped CI for the dispersion parameter includes 1). To go one step further I would like to fit factor analytic models. I am aware of work of Wedel et al (2003, Journal of Multivariate Analysis), who deal with the situation with and without overdispersion. Also MPlus is able to handle the overdispersed situation.
However my data shows only little dispersion (in fact the raw scores on some subtests are clearly underdispersed). I went ahead anyway and tried to use MPlus for (ML) factor analysis with Poisson errors and log-link. I used MPlus mainly for convenience, since it is the only package that handles count data and factor analysis (feel free to correct me here).
Using MPlus did not work well with my real data, but did OK with simulated data, at least when the latent factors had unit variance. Factor loadings are harder to recover when the variance is reduced (say to 0.25^2) and is not working (at least with MPlus) when the variance is very low (say 0.1^2).
I suspsect that factor analytic models for count data need at least some variance in the underlying (latent) factors so that they can be estimated, since otherwise the observed variability is attributed to the Poisson (neg-Bin, zero-inflated Poisson, ...) distribution rather than to the latent factor. I could not find a reference for this claim and my own simulations are rather limited.
1) Is there any reference for this phenomenon? Is it maybe trivial from another point of view?
2) Clearly I could treat the data as continuous, use standard exploratory and confirmatory factor analysis, but I feel this is neither correct nor satisfying. What are alternative strategies?