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 How the Likelihod ratio test works to compare models

The likelihood ratio test compares the fits of two nested models fit by Poisson regression. Nested means one model (the simpler one, model 1 below) is a special case of the other model (the more complicated one; model 2 below).

The Χ2 ratio quantifies the relative goodness of fit of the two models:

Χ2=2ln(LR)

If the simpler model is correct you expect to get an LR ratio near 0, so Q will be near 2. If the ratio is much greater than 2.0, there are two possibilities:

The more complicated model is correct.

The simpler model is correct, but random scatter led the more complicated model to fit better. The P value tells you how rare this coincidence would be.

The P value is computed from Χ2 using the chi-square distribution. The degrees of freedom equals the difference between the df of the two models.

The P value answers this question:

If model 1 is really correct, what is the chance that you would randomly obtain data that fits model 2 so much better?