The Winsteps program can analyze Poisson count data, with a little work.

 

Poisson counts are a rating (or partial credit) scale with pre-set structure. The structure measures are loge(n), n=1 upwards.

You can define a structure anchor file in this way:

 

XWIDE=2

STKEEP=YES

CODES=0001020304050607080901011121314.......979899

SAFILE=*

0  0

1 0                ; the value corresponding to log(1) - the pivot point for the item measure

2  .693           ; the value corresponding to loge(2)

3  1.099          ; the value corresponding to loge(3)

.....

99  4.595         ; the value corresponding to loge(99)

*

 

Arrange that the observations have an upper limit much less than 99, or extend the range of CODES= and SAFILE= to be considerably wider than the observations.

 

You will need to multiply all Poisson structure measures by a constant to adjust the "natural" form of the Poisson counts to the actual discrimination of  your empirical Poisson process.You need to adjust the constant so that the average overall mean-square of the analysis is about 1.0. See RMT 14:2 about using mean-squares to adjust logit user-scaling.  (The Facets program does this automatically, if so instructed.)

 

But my experience with running Poisson counts in the Facets program (which supports them directly) is that most "Poisson count" data do not match the Poisson process well, and are more usefully parameterized as a rating (or partial credit) scale. There is nearly always some other aspect of the situation that perturbs the pure Poisson process.