The R Book

Link Function

One of the difficult things to grasp about GLMs is the relationship between the values of the response variable (as measured in the data and predicted by the model in fitted values) and the linear predictor. The thing to remember is that the link function relates the mean value of y to its linear predictor. In symbols, this means that

which is simple, but needs thinking about. The linear predictor, η, emerges from the linear model as a sum of the terms for each of the p parameters. This is not a value of y (except in the special case of the identity link that we have been using (implicitly) up to now). The value of η is obtained by transforming the value of y by the link function, and the predicted value of y is obtained by applying the inverse link function to η.

The most frequently used link functions are shown below. An important criterion in the choice of link function is to ensure that the fitted values stay within reasonable bounds. We would want to ensure, for example, that counts were all greater than or equal to 0 (negative count data would be nonsense). Similarly, if the response variable was the proportion of individuals that died, then the fitted values would have to lie between 0 and 1 (fitted values greater than 1 or less than 0 would be meaningless). In the first case, a log link is appropriate because the fitted values are antilogs of the linear predictor, ...