= Solution
For $Y_\mu\sim\operatorname{Poisson}(\mu)$, the <delta method> with $g(y)=2\sqrt y$ gives
$$
\operatorname{Var}(g(Y_\mu))
\approx g'(\mu)^2\operatorname{Var}(Y_\mu)
=\frac1\mu\mu=1.
$$
Thus the transformation stabilizes variance at large $\mu$.
A Gaussian linear model for $\sqrt Y$ treats the transformed observations as having additive constant-variance errors and models $\mathbb E(\sqrt Y\mid X)$ as linear. A <Poisson regression> with square-root link retains the Poisson likelihood and models
$$
\sqrt{\mathbb E(Y\mid X)}=X\beta.
$$
The two procedures therefore have different likelihoods, fitted means, and constraints; transforming the response is not the same as transforming its conditional mean.
Back to article page
