# Factor Analysis of Count Data

I am new to factor analysis. I inherited a project at work from another team. They took 9 variables that are all Poisson-distributed count random variables and ran a "regular" factor analysis in Stata/SPSS (they did not specifically account for this being count data). The four factors they obtained are each normally distributed continuous variables.

My question is:

Is what they did correct, or are there special factor analysis methods for count data?

Should I be concerned that the factors are a totally different distribution than the original data?

The next step in the project is to set up regression models for each of the factors as the DV. Obviously, if the factors had been in the same family as the original count data, I would be running Poisson or NBD regressions, but after this transformation, I am running linear regressions. Not sure which is the right way to go.

• I think that one may use classic (linear) factor analysis with count data. But... think where the origin of the data cloud should be placed. Is it correct to center the variables (to place origin at the centroid) when the data are counts (and so count 0 looks as the natural origin)? What do you think about it? Wouldn't SSCP or cosine matrix be the better input in your factor analysis than their corresponding centered versions - covariance or correlation matrix? Commented Jun 26, 2015 at 9:51
• If you need 4 factors to represent 9 variables, you aren't gaining much and you may be giving up a lot. So ... I'm not sure that I would use the factors rather than the original variables - unless the factors make a lot of sense. Commented Dec 23, 2018 at 0:48

Poisson data is normally distributed when the rate parameter $$> 10$$ish. If even one of the count variables included in the factor analysis has an average rate close to 10, it seems straightforward that the resulting mixture of all those count variables could take on this distribution given the right circumstances.

Furthermore, even binary random variables, when several are averaged will begin to approximate a normal distribution

Still, if the variables input into the factor analysis don't relate linearly to each other, they're not going to load heavily, and you could miss out on predictive power. I.e., at times, $$X$$ might not load, but $$X^2$$ or $$\text{log}(X)$$ could