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A Bayesian deviance is minus twice the log-likelihood, with any chosen additive data-only constant held consistent across the models being compared. Its Bayesian posterior expectation measures average fit. In the deviance information criterion, evaluating it at a parameter's posterior mean also enters the effective complexity penalty. This likelihood-based convention differs by a data-only constant from a saturated-model exponential-family deviance when a common saturated model exists.

The null hypothesis isagainst the alternative that at least one coefficient is nonzero. The code uses the scaled reduction in exponential-family deviance,and compares it with ${\chi }_3^2$.

That chi-squared calibration is appropriate when the dispersion is known, and is asymptotically valid after consistent dispersion estimation. With unknown Gamma dispersion, the standard finite-sample GLM comparison instead usesagainst an $F_{3,57}$ distribution. Its p-value is approximately $0.034$, so the correctly calibrated test still rejects at the 5% level and gives evidence that component type or position contributes to mean failure time.

A one-parameter exponential family with natural statistic $Y$ has density or probability mass functionwhere $\theta$ is the natural parameter of an exponential family and $K$ is the cumulant function of an exponential family. Its mean parameter is

The exponential-family deviance from ${\theta }_1$ to ${\theta }_2$ is twice the Kullback-Leibler divergence:The carrier $h(Y)$ cancels from the likelihood ratio, so, with ${\mu }_1={\mathbb{E}}_{{\theta }_1}Y$,

For the Poisson distribution with mean $\mu$,ThereforeWriting ${\mu }_2={\mu }_1+\delta$ and applying the Taylor series of $\log (1+\delta /{\mu }_1)$ givesso the second-order approximation is