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?

You say that the goal of your analysis is "...to create a regression model which models how early different patient cohorts ... will get this medical exam".  If that is the goal then the response variable of interest is the time/date of the first medical exam for each patient (with right-censoring if they don't get the exam within the observation period).  Converting this information to counts obscures the basic information that you say is the goal of your analysis, so this is not a count regression at all.
When modelling the time-to-event in such cases, it is usual to use a survival model, particularly if there is a possibility of censored observations (e.g., people who didn't have a medical exam at all in the observation period).  It is possible to build survival models that incorporate explanatory variables that impact the hazard function applying to the time-to-event.  Unless there are compelling reasons to the contrary, this is the type of model I would recommend for this type of analysis.
