It appears that in Poisson regression, using an identity link function means your betas will be rate differences, and using a log link function your (exponentiated) betas will be rate ratios.
You can use an offset when using a log link function because log(events/time) = log(events) – log(time) and you can then move log(time) to the predictors side of the equation to make it the offset. But when using an identity link function, the outcome is just "events/time" with no log, so you can't use logarithm math to turn the division into a subtraction and make it easily moveable to the other side.
At first I thought you could just make sure your rates have the same denominator before you fit the model, the same as you would do if calculating a rate difference by hand (eg. if you are comparing 2 events in 50 person-years vs. 7 events in 100 person-years, convert them to 4/100 and 7/100, and then for Poisson regression drop the "per hundred person-years" because it now cancels out) but then I realized that because the Poisson distribution only has a single parameter for center and variance, 4 events in 100 person-years would not have the same variance as 2 events in 50 person-years (I think — if I'm wrong on this, please explain how/why!).
So is it just not possible to use an identity link function if you need an offset, or is there a way to do it? And if there is a way to do it, what does the equation look like?
Best Answer
Try this: Suppose your has $Y_i$, $X_{i1}$, $X_{i2}$, and $N_i$. where $Y_i$ is # of events, $X_{i1}$ and $X_{i2}$ are covariates and $N_{i}$ is person-time, where $i$ indicates the subject. Then $Y_i$ ~ Poisson($\pi_iN_i$), where $\pi_i= \beta_0 + \beta_1X_{i1} + \beta_2X_{i2}$. (based on your requirement of identity link). So $Y_i$ ~ Poisson($\beta_0N_i + \beta_1X_{i1}N_i + \beta_2X_{i2}N_i$). So you need to generate two new variables $Z_{i1} = X_{i1}N_i$ and $Z_{i2} = X_{i2}N_i$. Then fit a Poisson regression model with identity link function on $N_i$, $Z_{i1}$ and $Z_{i2}$ WITHOUT intercept. In this new model, the regression coefficient for covariate $N_i$ is the original intercept, so no additional intercept is needed. Then you get what you wanted.