Term for exp(beta) from a Gamma-GLM

Question

I have read a lot about interpretation of coefficients from Gamma-GLMs (using a log-link function), e. g. from this thread How to interpret parameters in GLM with family=Gamma , and found this to be very helpful.

However, I wonder what is the correct scientific term for the exp(coefficient) from a Gamma-GLM? From my understanding this would describe the ratio between two means, and could therefore be interpreted as a mean-ratio. This should be somewhat similar like when you would compare two odds in a logistic regression and call it an odds-ratio. Is this correct or is there any other more suitable term?

Answer 1

There is no scientific term in common usage for the exponential of a GLM coefficient. In some cases exp(coefficient) might represent the log-ratio of two group means, but not always. It depends on the experimental design and on how the GLM model has been parametrized.

In the simplest case that Group is a factor will two levels (A and B say) and a gamma GLM is fit by

fit <- glm(y ~ Group, family=Gamma(link="log"))

then exponentiating the second coefficient, exp(coef(fit)["GroupB"]), would indeed yield the ratio $\mu_B/\mu_A$ where $\mu_A$ and $\mu_B$ are the expected values for groups A and B respectively. In the bioinformatics contexts that I work in, exp(coef(fit))[2] might be called the fold-change and coef(fit)[2] the log-ratio or log-fold-change.

My advice would be to choose terminology appropriate for your specific context.