TABLE 1.

Common data types, canonical link, and typical variance-stabilizing link functions (up to an affine transformation) for each marginal variance function

Family	Mean-variance	Canonical link	Variance-stabilizing link
Gaussian	$h(x)\propto 1$	$g(x)=x$	$g(x)=x$
Binomial	$h(x)\propto x(1-x)$	$g(x)=\log \{x/(1-x)\}$	$g(x)=\arcsin (\sqrt{x})$
Negative binomial	$h(x)\propto x(1+x)$	$g(x)=x/(1+x)$	$g(x)=\log \left\{x+\sqrt{1+x^2}\right\}$
Poisson	$h(x)\propto \hspace{0.17em}x$	$g(x)=\log (x)$	$g(x)=\sqrt{x}$
Gamma	$h(x)\propto \hspace{0.17em}{x}^2$	$g(x)=1/x$	$g(x)=\log (x)$
Inverse Gaussian	$h(x)\propto \hspace{0.17em}{x}^3$	$g(x)=1/x^2$	$g(x)=1/\sqrt{x}$