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$begingroup$The deviance of a model is defined as:$D = -2(loglikelihood(model) - loglikelihood(saturated.model)) $
So I tried to compare the result of this formula with the output of the deviance() function with different models:
Linear regression model:
299 $neq $ Inf
Something is wrong
GLM: Poisson regression
14 = 14
Ok, it works!
GLM: Gamma regression
0.16 $neq $ 758
There must be a mistake somewhere
Shouldn't the results be the same?
kjetil b halvorsen1 Answer
$begingroup$I'm also trying to understand how the dispersion parameter should be included in the formula to compute the deviance of a linear model from its log-likelihood, and I cannot find any reference in books about GLMs. This is really a mystery to me. Even if this is quite a basic point, the fact that nobody answered seems to indicate it's obscure for many people...
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My small contribution: since the log-likelihood of the saturated model is Inf, we may also start with a simpler question, and compute the difference in deviance between two non-saturated models. But this still doesn't work:
It seems obvious to me that the dispersion parameter should be included in the formula, but how? For a start, there are two estimations of the dispersion, one for m1 and one for m2.