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I am developing a negative binomial regression model that investigates factors impacting the death rate of animals that are fed. The number of deaths was counted over the course of a feeding season, however, the feeding season length (and therefore the number of days over which dead animals were counted) varied. Therefore, my response variable is calculated as a daily death rate: number of dead animals/population size/season length.

I suspect that death rate will be higher in longer feeding seasons due to animals being congregated for longer periods of time, increasing disease transmission. Therefore, I would like to include season length as both an independent variable and as an offset term. Is this an acceptable thing to do?

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In the situation you are describing, there should be no problem. It is just the case that one variable (here: length of feeding season) plays two different roles, once the denominator in a rate and once as a potential explanatory variable. Then it is good that the two roles are separate in the model, it makes interpretation easier, and, more importantly, you can treat them differently. Maybe, for instance, the act of season length as explanatory variable is nonlinear, then you could use it with a quadratic term, or maybe even with a spline.

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