How is the intercept computed in the GLM fit ?

Question

Hi!

I have been reading the code used by R to fit a generalized linear model (GLM), since the source-code of R is freely available. The algorithm used is called iteratively reweighted least squares (IRLS), which is a fairly documented algorithm. For each iteration, there is a call to a Fortran function to solve the weighted least squares problem.

From the end-user's viewpoint, for a logistic regression for instance, a call in R looks just like this:

y <- rbinom(100, 1, 0.5)
x <- rnorm(100)
glm(y~x, family=binomial)$coefficients

And if you do not want to use an intercept, either of these calls is okay:

glm(y~x-1, family=binomial)$coefficients
glm(y~x+0, family=binomial)$coefficients

However, I cannot manage to understand how the formula, i.e. y~x or y~x-1, is making sense in the code and being understood as for whether to use an intercept or not. I was looking for a part of the code where a column of ones would be bound to x, but it seems there is none.

Thanks.

PS: As far as I have read, the boolean intercept which appears in the function called glm.fit is not the same as the intercept which I am referring to. And it is the same for the offset.

The documentation about glm and glm.fit is here.

Answer 1

You are probably looking in the wrong place. Usually, model.matrix() is called first in the fitting functions:

> D <- data.frame(x1=1:4, x2=4:1)
> model.matrix(~ x1 + x2, D)
  (Intercept) x1 x2
1           1  1  4
2           1  2  3
3           1  3  2
4           1  4  1
attr(,"assign")
[1] 0 1 2
> model.matrix(~ x1 + x2 -1 , D)
  x1 x2
1  1  4
2  2  3
3  3  2
4  4  1
attr(,"assign")
[1] 1 2
> 

and it is the output of model.matrix() which is passed down to Fortran. That is the case for lm() and other model fitters.

For glm(), it is different and only model.frame() is called which does not add an intercept column. Why that is so has to do with the difference between generalized linear models and standard linear models and beyond the scope of this posting.