Generalized linear models (GLMs)

Let's say, I want to establish a relation between the distribution of $$Y$$ and $$p$$ covariates. I denote the covariates with $$x_1,x_2,...,x_p$$. Then, I denote the linear predictor with $$\eta$$ , where $$\eta = \beta_0 + \beta_1 x_1 + ~... ~+\beta_p x_p$$ .

Thus, $$E \left[Y|X \right]=\mu$$, which is connected with the linear predictor by the response function $$h$$, where $$\mu=h(\eta)$$. Finally, I call $$g$$ the inverse of $$h$$, where $$g= h^{-1}$$. It must be noted that $$\eta=g(\mu)$$, and $$g$$ is called the link function.

Why does the link function $$g$$ make the model linear? The question seems a bit trivial, but is the answer that $$\eta=g(\mu)$$ is just a definition, in the sense that the linear predictor is linear because $$g$$ is a linear function?

• It's called linear because the cheeky statisticians apply the link function on the left side of the equation, so viewed from that angle, the right side (linear predictor) remains a polynomial. However, when considering the effect of parameter changes on the expectation value of the response, a GLM is not linear. Commented Nov 27, 2022 at 19:14

Some clarifications:

$$\bullet$$ The linear predictor

$$\eta_i =\sum_{j=1}^p\beta_jx_{ij}\tag 1\label 1$$

is linear in the parameters.

$$\bullet$$ $$g(\cdot),$$ aka the link function that links $$\eta_i$$ to $$\mu_i$$ by $$\eta_i=g(\mu_i),$$ i.e., $$\mu_i\mapsto \mathbf x_i^\top\boldsymbol\beta$$, is monotonic and differentiable.

So, $$g\left(\mathbb E[Y]\right)$$ is linear in the parameters. This is the basis of GLM.