Strange variance weights for Poisson GLM for square root link

Question

Is there a reason why all the variance weights from a Poisson GLM are equal when a square root link is used? That is (on R):

glm(Y~X1+X2+X3+X4, family=poisson(link=sqrt), data=df)$weights

Always returns equal weights for all observations when a square root link is used; but different weights when the canonical (log) link is used. Does anyone know why?

Answer 1

The weights in the glm function are

$$ w_i = \left.\frac{(\partial \mu_i/\partial\eta_i)^2}{\text{var}(\mu_i)}\right|_{\mu_i=h(\eta_i) = \eta_i^2} $$

So if $\mu_i = \eta_i^2$ and you recall that $\text{var}(\mu_i)=\mu_i$ then $\partial \mu_i/\partial\eta_i = 2\eta_i$ so $w_i = 4$. This is what you get from glm

> counts <- c(18,17,15,20,10,20,25,13,12)
> outcome <- gl(3,1,9)
> treatment <- gl(3,3)
> glm.D93 <- glm(counts ~ outcome + treatment, family = poisson(link = "sqrt"))
> 
> glm.D93$weights
1 2 3 4 5 6 7 8 9 
4 4 4 4 4 4 4 4 4   

On the other hand, the log-link function has $\partial \mu_i/\partial\eta_i = \exp(\eta_i)$ and thus you get different weights.