# Negative Binomial Regression?

I have a dependent count variable that measures the number of days spent in a hospital (LOS) for a group of patients who received two different medical interventions upon hospitalization. I'm trying to examine the effect of the treatment on the dependent variable, LOS while controlling for other variables. Would it be appropriate to use a negative binomial model here? Normally, I'd think yes, but I'm a bit confused since everything I've read on Poisson and Negative Binomial regression says that I need to be using counts that have the same time period (or use an offset). But in my case, the time, or days spent in the hospital, is my dependent variable. Given this, would it still be appropriate to use negative binomial regression? If it helps in providing an answer, I'm using the following SAS code (but am not sure if it's appropriate):

proc genmod data=work.hosp;
class trt gender;
model los = trt admissiondept age severity_index gender injscore / dist=negbin link=log type3;
run;

• "examine the effect of the treatment on the dependent variable while controlling for LOS"... If LOS is your dependent variable, you can't be controlling for it, for then it would appear on both sides of the equation. Is it your DV (in which case you're not controlling for it) or is it a covariate (in which case, it's not your dependent variable)? This confusion in your question cannot lead to good answers. – Glen_b Jan 2 '15 at 3:13
• Thanks, Glen_b for pointing that out. You are exactly right. I've edited my question accordingly. The LOS is my DV -- I'm not controlling for it. I guess that's what happens when you make a most at 3:30 in the morning. ;-) – StatsStudent Jan 12 '15 at 8:34
• Just a note on the "offset" - this would be appropriate for a rate type variable. You offset to account for higher/lower "exposure" to an event happening. For example, modelling number of times the nurse was called - you would include an offset. The LOS doesn't need one. One way to see this is try to answer - model LOS per what? Difficult to fill in the "what". – probabilityislogic Jan 12 '15 at 8:44

Assuming LOS is intended as a DV and not a covariate, "Length of stay" is not really a count (in the required sense), but a (possibly discretized) duration. You wouldn't normally use a count model for that.

I'd be inclined toward using a survival model; that will also enable you to cope with the likely censoring (for people who are still in hospital when you stop taking data, for example -- you can't just leave them out because their duration wasn't finished, otherwise you'll be biasing against people with long LOS).

• The data contains only patients who were discharged from the facility, so there shouldn't be any censoring. – StatsStudent Jan 12 '15 at 8:37
• That's the problem that concerns me. If some people were not discharged, by only including those who were discharged you bias your results. – Glen_b Jan 12 '15 at 9:17
• Let's say one of the two groups tends to have larger values than the other, but (as a result) has some undischarged patients. Then you'll lose the information in the censored observations that were left out. – Glen_b Jan 12 '15 at 9:23
• The analysis is only among those who were discharged. "Discharged patients" is the population of interest. I am performing a separate Survival Analysis of all patients. – StatsStudent Jan 12 '15 at 21:17

The advice you refers (that the count need to refer to the same length time intervals) seems to be irrelevant here. That is meant for a situation where you are counting number of point events within some time interval. But your response variable is a duration, so a completely different situation. So I think you could try with a Poisson (or negative binomial) regression, and then validate it with residual plots and so on.

Negative binomial would still be appropriate. Poisson would be as well, if your data meets the equidispersion assumption. (If it does not, stick with nbreg.)