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Offset is commonly used in Poisson regression to take into account different exposure (different time periods for instance): offset = log of exposure
Question: what's the typical use case of offset in logistic regression?
I assume we can't do exposure (proportional effect) in classification problems since E(y|x) can't go beyond 1 so I'm curious about why someone would need to use offset in a logisic regression.
You include an offset when you know what the coefficient of that variable should be. Typically software fixes it at unity. As you point out in Poisson regression this is often used to include the effect of the denominator when we assume that if we multiplied the denominator by a factor we would also multiply the outcome by the same factor.
One case where an offset might be used outside the Poisson special case is when you have a hypothesised value for the coefficient from theory of previous studies. If you then include your predictor variable in the regression multiplied by the theoretical value and as an offset this will have the effect of including it with the theoretical value of the coefficient. If you also include the predictor as a standard regressor you will see from testing its coefficient against zero whether the offset is sufficient (so the theoretical value is supported) or whether you can reject that.
I sometimes use an offset in a logistic regression model. The use case is where I already have a complex model, which needs to be re-estimated to cover some new data outside the realm of the original data sample (in time, or in cross section), but where, for various reasons, it is practically infeasible to re-estimate the model on the entire, expanded data set. The goal is a new model that gives good predictions on some out-of-sample data, but which gives unchanged predictions on the in-sample data.
So I take the linear predictors from the original model, specify those as an offset, and then introduce additional variables aimed at fitting the new data, in such a way that it wouldn't change the predictions on the original data.
It's admittedly ad hoc, but an awfully useful trick in practice. I have no idea what the "legitimate" use of an offset in logistic regression is, but I'm glad statistical software packages allow for it.
When the model is based on oversampled data, offset is used to correct the bias. An alternative is to use weights argument. Note however that offset produces correct probabilities by changing the intercept, usually a judgment based override. On the other hand, weights produces correct parameter estimates by countering the effect of oversampling, as if the model was based on correctly sampled data.
There are a variety of uses for offsets in logistic regression, whether for specific factors or the outputs of other models.
For specific factors, they may be included or excluded from the final implementation. Included, if the goal is fix them in the final model, whether to force assumptions (based upon past experience) or restrict influence. Excluded, if the goal is to control for those factors {maturity effect, disparate impact}.
For model outputs they can be i) part of a full development process to stage in groups of variables; ii) to incorporate new data sources in new samples without affecting the original model; iii) to validate a model's ranking abilitys pre- or post-implementation.
