Let Y_i be a response and X_i be a covariate vector. Poisson regression with an offset log(S_i) has the structure

Y_i ~ Po(log(mu_i)), mu_i = log(S_i) + X_i’b

The same structure is obtained when using the log of a rate (Y_i/S_i) in linear regression, i.e. that model is equivalent to

log(Y_i) ~ N(mu_i,s^2), mu_i = log(S_i) + X_i’b

Note that that coefficient for log(S_i) is fixed at 1.

Estimated coefficient is not 1

If, instead of treating log(S_i) as an offset, we use log(S_i) as an explanatory variable and find that the coefficient is significantly different from 1, what should we do?

The advice from the group was

Try both (offset and as an explanatory variable) and see if the results change the answers to the scientific questions of interest. If the results do not change, then report one and state the opposite. If the results do change, then there is probably more to write about.