# Poisson regression for ordered variables

I have three waves of data, and I am trying to estimate group-based trajectories of binge drinking across the three waves. The question asked (at all three waves was): “Over/During the past 12 months, on how many days did you drink five or more drinks in a row?” Response categories were: 0=none; 1=one or two days; 2=once a month or less (three to 12 times); 3=two or three days a month; 4=one or two days a week; 5=three to five days a week; and 6=every day or almost every day.

I am using the traj plugin in Stata, and it is limited to normally-distributed continuous variables, dichotomous variables, and zero-inflated variables. Technically, this is an ordinal variable with seven categories. So, it seems like I have two choices: (1) create a dichotomous variable at each wave; or (2) treat this as a count variable and use poisson regression. The latter approach yields much more detailed and seemingly accurate findings, but here is my question:

Does poisson regression assume that the distance between the categories (counts) is equal? Also, does anyone see any problems with treating this as a count variable? The distributin of the data takes the form of many count variables I have seen (the distribution at one of the waves is shown below).

bingedrink  |      Freq.     Percent        Cum.
------------+-----------------------------------
0 |      8,057       53.24       53.24
1 |      2,401       15.87       69.11
2 |      1,514       10.01       79.12
3 |      1,255        8.29       87.41
4 |      1,334        8.82       96.23
5 |        460        3.04       99.27
6 |        111        0.73      100.00
------------+-----------------------------------
Total |     15,132      100.00


The latent data is $Y_1, \dotsc, Y_n$ which is assumed poisson distributed with mean $\DeclareMathOperator{\E}{\mathbb{E}} \E \left\{ Y_i | x_i \right\} = e^{\eta(x_i)}$, where $x_1, \dotsc, x_n$ are the covariables and $\eta(x)$ the linear predictor (containing parameters to be estimated). Write the coresponding Poisson cumulative distribution function (cdf) as $F(y | x)$. But we do not observe $Y$, we only observe that $Y$ falls in one of a set of speciefied intervals, denote the upper interval limits as $c_j$. The intervals are then $[0,c_1], (c_1,c_2], (c_2, c_3], \dotsc$. Observation $i$ falls in interval $(c_{k_i -1},c_{k_i}]$ (where it is understood that in case $k_i=1$ the lover parenthesis changes from a $($ to a $[$). So $$P(Y_i \in (c_{k_i -1},c_{k_i}] ) = F(c_{k_i}) - F(c_{k_i -1})$$ so the likelihood function becomes $$L = \prod_{i=1}^n \left\{ F(c_{k_i}) - F(c_{k_i -1}) \right\}$$ which you can optimize directly. You mentioned stata: I do not know how to do this in stata.