genlin poisson regression with offset

[Maguin, Eugene]

I need some help understanding what the offset subcommand does when running a poisson distribution analysis in Genlin. My understanding of the offset, from Andy’s post of a few days ago and also from Wikipedia, is that it functions to convert rates to counts when the ‘base’ differs from case to case such as if a followup period differs in length across cases or the count of cars crossing an intersection differs from day to day. However, that doesn’t seem to be the case at all.

 

Thanks, Gene Maguin

 

[Andy W]

In a Poisson regression model you model the logarithm of the expected value conditional on a set of covariates:

log(E[Y]) = Intercept + B*(Other Stuff)

An "offset" is just a constant shift in the stuff on the righthand side. It is basically just a coefficient set to a particular value:

log(E[Y]) = Intercept + B*(Other Stuff) + 1*(Offset)

You can have an offset in any type of regression equation, but it is convenient for Poisson regression because it allows us to interpret the outcome in terms of rates. So consider exponentiating both sides of the equation:

E[Y] = e^(Intercept + B*(Other Stuff) + 1*(Offset))
     = e^(Intercept)*e^(B*(Other Stuff))*e^(Offset)
         
Now here is a potential confusion I've seen others make, so I will try to be clear here. When one includes an offset that is intended to be interpreted as a rate for the Poisson regression equation, it should be the LOG of the denominator value. This is a potential confusion as some software allows you to include an "exposure" variable that is not logged. SPSS forces you estimate the LOG yourself and include it as an offset in GENLIN.

So going back to the equations, lets divide each side by e^(Offset):

E[Y]/(e^(Offset)) =  e^(Intercept)*e^(B*(Other Stuff))

And lets realize if Offset above is the log of some variables, lets say "care-days", then we have the relation:

e^(Offset) = e^(log(care-days)) = care-days

Then we have our expected rate on the left hand side (conditional on the covariates, for notation simplicity I have left out the conditionals):

E[Y]/care-days = e^(Intercept)*e^(B*(Other Stuff)) = e^(Intercept + B*(Other Stuff))

So hopefully that makes it clear what an offset is, and how it allows Poisson regression equations to be interpreted in terms of rates. For your application if you had a measure of the number of individuals per month that may be reasonable to actually estimate parameter instead of forcing it to 1. E.g. you may expect in units that are more crowded for people to act out more - or obverse the staff ratio in places with fewer adolescents may be able to expend more resources (it may also be non-linear). For a start though the log of care-days are theoretically reasonable to use as an offset in the Poisson regression equation given their rate per care days interpretation.