Let's say that you were wanting to model how many times someone had to take a certain test before passing (depending on a range of predictors like practice, mock tests taken, classes attended, etc.). Let's also say that most people pass after the first attempt, but that others have to take the test several times, and that the distribution looks Poisson-ish.
If you were to model the dependent variable as the number of tests taken, your minimum count would be 1. On the other hand, if you wanted to model the dependent variable as the number of resits needed, the minimum count would be 0. Both of these seem to me to a reasonable thing to do, and the latter is just the former minus 1, i.e. is shifted.
It also seems, conceptually like this difference (tests~x1+x2+x3... vs. resits~x1+x2+x3... or tests-1~x1+x2+x3...) shouldn't really affect your ultimate conclusions: if practice decreases the number of tests, it should also decrease the number of resits, and it seems like it should do so to a similar extent.
My questions are:
What is the practical effect on the model parameters of using (a) shifted dependent variable (resits) rather than (b) the unshifted one (tests)? For instance, would you generally expect the parameter to be overestimated if using resits, or underestimated? If either, would you generally expect the difference to be substantial or minor? Or would this all depend so much on the particular data set that there's no way to tell? That is, is the conceptual similarity between tests and resits misleading, in as far as it makes me think I should get similar results for both.
What is the practical effect on the model parameters of using:
(a) a zero-truncated model - e.g., in R, I'd specify:
vglm(tests~x, data, family=pospoisson())and
(b) a left-shifted model - e.g. in R,
There is a discussion of shifting vs. truncation here but this discussion doesn't specifically address things like the model parameters and significance. It also focuses on right-shifting rather than left-shifting.
I have tried the various options above on my data and it turned out that the basic Poisson
(y~x, fam=poisson) had a lower estimate for the predictor than the zero-truncated
(y~x, fam=pospoisson), which in turn had a lower estimate than the left-shifted model
(y-1~x, fam=poisson). Bootstrapped confidence intervals suggest that these differences are not significant, though. However, doing this hasn't told me whether I can expect this to hold generally, i.e., whether the conceptual similarity between tests and resits should typically translate into similar models. In my case, left-shifting resulted in a higher parameter than the zero-truncation model, but is that generally the case? In my case, the parameters weren't significantly different, but is that generally the case? I realize that someone might be able to derive an answer to all this from first principals, mathematically, but I don't have the mathematical background to do so.
I'm asking this as a prelude to another question, here. For reasons that I'll explain in that post, I have to left-shift my response variable and I'm wanting to know whether this is in principle problematic (in which case I've just been lucky that the model parameters are quite similar).
*Edit: My data is not in the form of test-counts vs resit-counts. I'm just using these as illustrations because the conceptual similarity between tests and resits is fairly obvious. So my question is not about what regression someone should use for such variables, but is rather about what the effect of shifting vs truncating is on model parameters - would you expect a trivial difference, a significant difference, or there's no way to tell without data? Since people suggested negative binomials below, though, I'm happy to accept answers to this question concerning either Poisson or negative binomial models.