# Strange variance weights for Poisson GLM for square root link

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? • The square root is the variance stabilizing transformation for the poisson distribution – kjetil b halvorsen Oct 19 '17 at 19:36 • – Glen_b Oct 20 '17 at 6:04 ## 1 Answer 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.

• +1, excellent answer, clear, to the point, well formatted, etc. – gung - Reinstate Monica Oct 19 '17 at 23:55