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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?

gung - Reinstate Monica
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Anna Efron
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    The square root is the variance stabilizing transformation for the poisson distribution – kjetil b halvorsen Oct 19 '17 at 19:36
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    Seet [Why is the square root transformation recommended for count data?](https://stats.stackexchange.com/questions/46418/why-is-the-square-root-transformation-recommended-for-count-data) – Glen_b Oct 20 '17 at 06:04

1 Answers1

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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.

Benjamin Christoffersen
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