Generalized linear model offset ($g({\mu }_i)=x_i^T\beta +o_i$)

An offset is a known term in a generalized linear model's linear predictor, with coefficient fixed to one. With a log link and exposure $n_i$, choosing $o_i=\log n_i$ gives a mean count $n_i\exp (x_i^T\beta )$. Its role differs from that of a fitted explanatory-variable coefficient.