Is there a Regression Model for "Negative Count Data?"? This is a question I have always been curious about.
Suppose there are medical patients and they recommended to get a medical exam done on the first of January. However, the patients can also get this medical exam done before the first of January - and they can also get this medical exam after the first of January. We have medical information (i.e. covariates) on these patients (e.g. age, gender, weight, etc.). The goal is to create a regression model which models how early different patient cohorts (e.g. men vs. women) will get this medical exam or how late will they get this medical exam.
In this example - I imagined that each day before the first of January could be considered as a "negative count" and each day after the first of January could be a "positive count".
Had there just been "positive counts", I would have stuck with the "Poisson GLM" for this problem. However, since there are also "negative counts", I have been looking for a type of regression model for this problem and could not find one. Initially I had thought that perhaps the a Negative Binomial GLM could have been useful - but now I realize that is meant to account for "overdispersion" and not for negative counts. Another idea that I had was just to assume the count data is continuous data and to just round the results - but I feel that this is disingenuous and would likely cause problems down the road.
Could someone please comment on this and suggest a type of model for this problem?

A simple approach could be to change all the values of the dependent variable into positive, by adding a suitable number to all (eg by adding 10: -5 would become +5, and +3 would become +13).
You could then analyze them with standard models (even Poisson). Still, I believe that a linear regression model could be sufficiently robust.
Another approach could be running a multivariate model such that you specify 2 dependent variables: count (in absolute terms) and sign (eg 0 indicating a negative count and 1 indicating a positive count). As sensitivity analysis you could run a traditional multivariable model putting the sign as a covariate (another independent variable).
