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I have 6 models with -2 log likelihood as an indication of model fit. The fit increases from model 1 to 6, with model 1 having the worst fit (lowest -2 log likelihood) and 6 the best fit. How do I test whether the fit of one model is significantly better than the other(s)?

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I think log-likelihood does not have distribution so there is no such test for log-likelihood value. You can compare with other criteria such as AIC and BIC altogether and make a discrete choice which model would be the best.

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If the models are nested then you can you use Wilks' theorem to perform a likelihood-ratio test.

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