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I've got a dataset with the dependent variable being a proportion in (0, 1), and 6 categorical predictors all with between 2-5 levels. I've fit a beta regression in R using the betareg package, and now want to determine the importance of each predictor.

If this were a standard linear regression I'd use an ANOVA and look at the F-test scores. However, there isn't an ANOVA method available for betareg objects. Instead, the documentation recommends using likelihood ratio tests.

What I've done is then calculate the log likelihood for the full model minus each of the predictors in turn. The bigger the difference between the loglikelihood of the full model and the model minus a predictor, the more 'important' that predictor is to the final model.

This is useful in that I get a measure of variable importance, but due to the large sample size (240 000 rows), the p-values are miniscule when I run a likelihood ratio tests for each reduced model compared to the full model.

Is there a more formal way of testing the variable importance in this scenario, or is this difference in loglikelihood as good as I can get?

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If all terms you omitted correspond to the same number of parameters in the model, then you can also compare the values of the likelihood ratio statistics instead of the corresponding p-values. However, if the associated numbers of parameters differ (e.g., because one variable appears only in the mean equation while another appears in both mean and dispersion equation - or one term is a factor with more than two levels etc.), then using the p-values is better. To make it easier to deal with the tiny values, you could also use log-p-values.

Another route would be to use information criteria such as AIC or BIC to rank the variables.

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