GLMs, link functions, offsets, and interpretation
A generalized linear model connects a response distribution, a linear predictor, and a link function. The link relates the conditional mean to the linear predictor; it does not automatically mean transforming the observed response and fitting ordinary least squares. State the response, the mean structure, and the distributional assumptions separately. Different choices can use similar predictors while implying different ranges, variance relationships, likelihoods, and interpretations. A good numerical answer depends on the specified model, not on a remembered coefficient rule used in every setting.
For a Poisson count model with a log link, the conditional mean is the exponential of the linear predictor. With a log-exposure offset whose coefficient is fixed at one, expected count equals exposure times the modeled rate. Doubling exposure with the other predictors fixed doubles expected count and leaves the modeled rate unchanged. A coefficient for a predictor represents a multiplicative effect after exponentiation, subject to coding and interactions. An offset differs from an ordinary estimated predictor because its coefficient is specified rather than freely fitted.
A binomial model with a logit link models log odds. Exponentiating a coefficient gives an odds ratio under the relevant conditioning; it is not generally a probability difference or a relative risk. To recover a probability, apply the inverse logistic transformation to the full linear predictor. A gamma model with a log link is one common positive-severity model, but a response containing zeros does not satisfy a strictly positive gamma response model without an appropriate separate treatment. A compound Poisson-gamma Tweedie model can represent a mass at zero and positive continuous values within its relevant parameter range.
Interpret categorical effects relative to their actual reference levels. If interactions are present, a main effect alone may not describe the change at every level of the interacting predictor. Correlated predictors can produce unstable individual coefficients even when overall predictions are useful. Regularization can stabilize estimation and control complexity, but a selected penalty must be tuned without using the final test set. A zero penalized coefficient is a model result, not proof that the variable has no causal or business relevance.
Check residuals, calibration, influential observations, dispersion, and performance across meaningful groups. Overdispersion means that the conditional variance exceeds the simple Poisson assumption; it does not by itself identify the single best replacement model. Compare reasonable alternatives in a validation design that reflects the intended use. Coefficients summarize modeled associations conditional on the specification. They are not automatically causal effects, approval for rating use, or a reason to ignore poor calibration. Report uncertainty and limitations alongside a clear explanation of what the fitted mean actually predicts.