Understanding where the link function is computed in Poisson regression

Question

From Poisson Regression by hand this 'manual' Poisson coefficient function is provided:

LogLike <- function(y,x, par) {
  beta <- par
  # the deterministic part of the model:
  lambda <- exp(beta%*%t(x))
  # and here comes the negative log-likelihood of the whole dataset, given     the
  # model:
  LL <- -sum(dpois(y, lambda, log = TRUE))
  return(LL)
}


PoisMod<-function(formula, data){

  # # definiere Regressionsformel
  form <- formula(formula)
  # 
  # # dataFrame wird erzeugt 
   model <- model.frame(formula, data = data)
  # 
  # # Designmatrix erzeugt
  x <- model.matrix(formula,data = data)
  # 
  # # Response Variable erzeugt
   y <- model.response(model)

  par <- rep(0,ncol(x))
  erg <- list(optim(par=par,fn=LogLike,x=x,y=y)$par)
  return(erg)
}

PoisMod(breaks~wool+tension, as.data.frame(daten))
glm(breaks~wool+tension, family = "poisson", data = as.data.frame(daten))

Can any one tell me exactly where the link function is computed here? What would this code look like with an identity link function? I have basic understanding from YouTube videos etc, but no one explains the actual computation.

How would this code look like with offset and weights?

in the `dpois()` function there is the argument `log=TRUE`. Normally Poisson GLMs use the log link function, so I believe an identity link function could be achieved by setting `log=FALSE` in the `dpois()` call — sjp, Jan 30 '22 at 16:36
The `log=TRUE` in the `dpois()` function simply produces individual log-likelihoods instead of likelihoods. Where the link function comes in is `exp(beta %*% t(x))` the inverse of the log link is `exp()`. If you were using an identity link, you would have `lambda = beta %*% t(x)` — DaveArmstrong, Jan 30 '22 at 18:09
thank you so much! This really makes is easier to understand. I have a follow up question on weights and offsets. Maybe you are able to simply explain this aswell? — MariusJ, Jan 30 '22 at 18:50

DaveArmstrong · Answer 1 · 2022-01-30T21:51:59.340

The GLM is specified as:

or equivalently as

In the case of the Poisson model, the canonical link is , so

You can see that in your code where lambda = exp(beta %*% t(x)) If you wanted to estimate a model with an identity link, you would use lambda = beta %*% t(x) because in that case

Addressing the follow-up questions in the comments, with weights, often these are used to weight the likelihood contribution for each observation. So, imagine you had a variable wt that you passed into your function LogLike(), you could modify it by multiplying the likelihood contribution by the weight:

LL <- -sum(dpois(y, lambda, log = TRUE)*wt)

An offset is just a variable whose coefficient is forced to be 1 (see this post for a discussion). Lets say you had a variable called offset that you wanted to include in your model. Again, you could pass that argument to LogLike() and you would need to modify the function by:

  lambda <- exp(beta%*%t(x) + offset)

Following the example in the CrossValidated post I linked, offset = log(time). Which offset is correct is more of a theoretical or substantive matter.

Thank you so much! The details in GLM is so much easier to understand with code example. I'm sure many other students like me will appreciate this post. — MariusJ, Jan 31 '22 at 10:15

Understanding where the link function is computed in Poisson regression

1 Answers1