Poisson model offset

Question

I'm using the dataset road (in the MASS package) that concerns the annual deaths in road accidents for some of the US states. It includes:

• deaths: number of deaths;
• drivers: number of drivers (in 10,000s);
• popden: population density in people per square mile;
• rural: length of rural roads, in 1000s of miles.
• temp: average daily maximum temperature in January.
• fuel: fuel consumption in 10,000,000 US gallons per year.

I have part of the R

Answer 1

With a log-link the linear model is for the log of the conditional mean of the Poisson ($\log(\mu)=\eta=X\beta$). The GLM offset applies on the scale of the linear predictors (in the model it's essentially another predictor which has its coefficient set to 1).

However, the exposure variable (number of drivers) is expected to be multiplicative in the mean (more drivers -> more people exposed to the risk of death), so exposure is additive in the log-mean; that is, you want to shift the log-mean by the log of the number of drivers.

The number of drivers is 10000 times the value of the variable drivers (not that the 10000 matters -- it really just scales the death rate to be per driver rather than per 10000 drivers by shifting the log-mean by log(10000); outside understanding that effect -- while necessary for correct interpretation of the intercept and the fitted values -- nothing substantive about the rest of the model fit changes by including or excluding the 10000).