# Offset using Poisson distribution

I get everything but the two sentences are confusing me.

Offset is a variable that serves to consider the different exposures of different observation. When properly included, the target variable is directly proportional to the exposure. So, you have mu= w*lambda where mu is target variable and lambda is exposure.

Then, you have the offset exposure that is proportional to the mean of the target variable. Offset is ln(w). mean of the target variable is just ln(mu).

Do you know what the difference between the mean of the target variable and the target variable in general?

• I'm not sure I have understood what you're after. Are you asking "what is the difference between a random variable and its mean?" Mar 30, 2023 at 2:57
• no. For offsets using Poisson distribution (GLM) the first sentence says "target variable is directly proportional to the exposure" and the second sentence says "the offset exposure is in direct proportion to the mean of the target variable". I think the target variable in the first sentence and the mean of target variable in the second sentence mean the same except the first sentence means mu=wlambda whereas the second sentence means ln(mu)=lnw+x^Tbeta. Do you think target variable in the first sentence and the mean of target variable in the second sentence are the same? Mar 30, 2023 at 3:03
• Clearly the target variable is a random quantity, so the random variable itself cannot literally be proportional to nonrandom exposure (which is treated as fixed/known) Mar 30, 2023 at 3:08
• Then, is it possible if you can create two new sentences for me assuming the two are wrong? thanks Mar 30, 2023 at 3:09
• @shawn using the law of logarithms and exponentiating, ln(mu) = ln(w) + ln(lambda) implies ln(mu) = ln(w * lambda), herefore mu = w * lambda Mar 30, 2023 at 4:01

An offset is a model term with a coefficient of 1. So, in a Poisson regression with an intercept, an offset $$\log t_i$$ (where $$t_i$$ is e.g. the observation time for record $$i=1,\ldots,I$$) and the log-link function, we have $$\log E Y_i = \beta + \log t_i$$ and $$Y_i \sim \text{Poisson}(e^{\beta + \log t_i})$$. You can re-write that as $$\log \frac{E Y_i}{t_i} = \beta.$$ I.e. the intercept (or if you add additional mode terms the regression equation) describes the logarithm of the expected number of events per unit of time.