# GLM with Gamma distribution: Choosing between two link functions

I need to perform a GLM based analysis on a purely positive, continuous, and highliy right skewed (inflated around low values) outcome variable. I tested several combinations of distributions and link functions with the Akaike and Bayesian Information Criterions as proposed in this textbook: glm_log_gam glm_sqrt_gam glm_log_gam glm_sqrt_gau glm_log_gau glm_log_igau glm_log_poi glm_sqrt_poi

The Gamma distribution with both square root and log link returned almost identical AIC and BIC values (both far superior to the other combinations):

Here are the P-P plots for the two (using deviance residuals):

Now I am wondering how to determine the more appropriate/ "best fit" for my model. In case it might be of any help, here are descriptive statistics about my regressand:

• The log-link would always be the first choice for a gamma glm because it transforms the expected values to an unconstrained linear predictor on the whole real line. A sqrt link is almost never defensible. However I disagree with the advice of the textbook you are using. I believe that the glm family and link-function should be chosen on more general considerations, then the linear model should be built to fit the data. The shotgun approach you are using assumes that the terms in the linear predictor are fixed regardless of the family or link, which I believe does not make scientific sense. Apr 23, 2022 at 7:28
• If your linear model is equivalent to a one-way layout, then all link functions will lead to exactly the same fitted values, AIC and BIC. If your results are identical for the two link functions, that might be the reason. This will arise if your model has no continuous covariates and either has just one explanatory factor or has multiple factors with all factorial interactions. Apr 23, 2022 at 10:39

One of the advantages of the log link is that it stabilizes the variance of data with a constant coefficient of variation. By doing so, one could run ordinary least squares on the log-transformed data. However, your intercepts would be biased by the offset -0.5*$$v$$ where $$v$$ is the coefficient of variation. Restating what Gordon touched upon in the comments, links such as the sqrt link are often not justifiable. Even the canonical inverse link is not practical.