You can conduct a LR test based on log-likelihoods if the two distributions are nested (i.e., if one is a special case of the other). For instance, the Poisson is a special case of the negative binomial (as 1/k =0, negative binomial = Poisson). In this example, the negative binomial has one more parameter than the Poisson (many sources use k as the overdispersion parameter of the negative binomial, but sas uses scale = 1/k in several procedures). The df for the LR is 1 because of the difference of parameters. LR is -2 times the difference in log-likelihoods. Under the null hypothesis (H0: distribution is the simpler one), the test statistic nominally has a chi-squared distribution. Caution: when the scale parameter is on boundary in order to get the simpler distribution, then the the test statistic may have a more complex distribution than a simple chi-squared (with 1 df). For instance, scale parameter ranges from 0 to infinity, and scale=0 gives you the simpler distribution. Thus, the more complex test statistic distribution. Many ignore this issue. Be careful with different procedures. Some programs may not give the the actual log-likelihood. For instance, many log-likelihoods can be written as sum of terms, where some terms invovle parameters and data, and some terms involve only the data (not the parameters). To be computationally efficient, the term not involving parameters may not be calculated or displayed. This is fine when one is comparing log-likelihoods all for the same distribution (with the same procedure), but could cause trouble if you are comparing distributions. Be careful with different procdures. If you use GLIMMIX (say, with different choices of distributions), make sure you are not using one of the conditional log-likelihood methods (rspl, mspl, ...). You need to be using the actual log-likelihood (method=quad).
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