= Scaled deviance
{title2=$D^*=D/\phi$}
For an <exponential dispersion family> with a common <dispersion parameter> $\phi$, scaled deviance compares a fitted model with the <saturated statistical model> at the same dispersion:
$$
D^*=2\{\ell(\widetilde\theta;\phi)-\ell(\widehat\theta;\phi)\}=\frac D\phi,
\qquad
D=2\sum_i\{y_i(\widetilde\theta_i-\widehat\theta_i)-b(\widetilde\theta_i)+b(\widehat\theta_i)\}.
$$
For a <normal distribution> with variance $\sigma^2$, unscaled <deviance> is the <residual sum of squares> and scaled deviance is that sum divided by $\sigma^2$. For the <binomial distribution>, $\phi=1$ and the two coincide. Unknown Gaussian dispersion requires variance estimation; raw deviance differences should not be treated as chi-squared statistics without the scale.
Back to article page
