 Individual Bernoulli deviance need not have a chi-squared calibration

A logistic regression fitted to ungrouped Bernoulli observations has residual deviance $D=-2\ell (\hat{\beta })$. Comparing it automatically with ${\chi }_{n-p}^2$ is generally invalid: the saturated dimension grows with $n$, with one observation per fitted probability. If the true model has only an intercept and success probability $1/2$, then $D/n\to 2\log 2$, whereas ${\chi }_{n-1}^2/n\to 1$. Fixed-dimensional nested-model likelihood-ratio tests can still use Wilks theorem under regularity. For absolute fit, use meaningful grouping, residual diagnostics, or a fitted-model parametric bootstrap.