Goodness of Fit in GLM

A fitted GLM predicts means for grouped data (e.g., success rates across 20 treatment groups). Two statistics—Pearson's X² and deviance—both test whether observed counts match fitted means. They can disagree when group sizes or variances differ.

The idea

Pearson's X² = Σ (observed - fitted)² / fitted sums squared standardised residuals. Deviance = 2 Σ observed × log(observed / fitted) sums likelihood ratios. Both follow χ² with df = groups - parameters under the null hypothesis that the model fits.

X² weights each group by fitted variance. Deviance weights by observed counts. When group sizes vary widely, X² down-weights small groups; deviance treats them equally if counts are similar. Neither is universally better—compare both and check residual plots.

The maths

Try it

Fitted probability: 0.50

Pearson X²

19.70

Deviance

49.29

df = 4

At probability 0.50: Pearson X² = 19.70, Deviance = 49.29

Interactive demonstration of statistical models concepts

Where I used it

Easy to get wrong

Sources

MAST90139 Statistical Modelling (2024). Deviance and Pearson X² for model diagnostics.