Generalized linear models (GLM)

Gist

When an ordinary Linear regression (LM) regression doesn't cut it - maybe your response variable are whole integers or proportions bounded by $[0,1]$. Who you gonna call? Generalized linear model (GLM)

Mathematics

For an outcome $Y$, it is assumed to be generated from a distribution in the Exponential Family. The conditional mean $\mu$ depends on the independent variable $X$. Here:

where the expected value of $Y$ conditioned on $X$ is equal to $\mu$ which is also equal to a link function of the linear predictor: $XB$.

Why isn't just using a transformation better?

An interesting thing I found online was the question of: Okay, why can't we just log-transform the response variable like:

The answer is that it it has to do with Jensen's inequality. The transformed model is actually estimating $E(log(Y)|X)$

Fitting

Done through Maximum likelihood

Binomial

Poisson

The link function for Poisson is a natural log

glm(Y~X, family = "poisson")

Note

• You can add an offset to account for different "efforts". The offset function tells the function to NOT estimate a coefficient for it.

Interpretation

References