Poisson regression for ordered variables

Question

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

Answer 1

Your dependent variable is not directly a count variable, since you do not know the exact count (and maybe the persons asked do not know exactly the count themselves, so there is some uncertainty in the measurement of the process). But, underlying the response variable there is a latent count variable, and you have observed an interval censored version of that. So, you could try to use Poisson regression, with a non-standard likelihood function based on interval censoring. I will write out the model below:

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.