GLM - Count Target

Question 1

Poisson - Exposures

Answer 1

A Poisson random variable counts within a given interval, where scaling the interval results in scaling the mean by the same amount. The predictors only explain the Poisson mean per exposure and an offset is how weights are accounted for. The offset can be viewed as a predictor that does not require a coefficient estimate which means its impact on the target is known. mu^ = w*e^(xTB)

A Poisson random variable counts within a given interval, where scaling the interval results in scaling the mean by the same amount.
->An exposure is a chosen interval measure. Mean = u_i = w_i*lambda_i
->w_i is the exposure for the ith observation
->lambda_i is the rate (i.e. mean per exposure unit) for the ith observation

Using a log link: ln(mu_i) = ln(w_i) + xTB
->The predictors only explain the Poisson mean per exposure. An offset is the term that adds to the regular xTB in order to form g(mu) (ln(w) here)

Offset can be viewed as functionally the same as a predictor, except it does not have a corresponding B that needs estimating, hence its impact on the target is considered known.

mu^ = w*e^(xTB)

A poisson regression that does not incorporate exposures is identical to one that assumes all w_i’s are equal