Applied work involving statistical modeling frequently leads to situations where models must be compared which are not related to one another by parameter restrictions. In such a situation, log-likelihood ratios of pairs of estimated models do not have a chi-square limiting distribution, and statisticians making model selection decisions frequently resort to rather complicated and subjective comparisons of residuals or other model artifacts to accomplish the selection. In this paper, we give some theoretical background for the use of the usual log-likelihood ratios for non-nested comparisons. The practical importance of this capability is magnified by the fact that maximized likelihood values are usually available from the software used for estimation. Thus comparisons can often be made quickly. This encourages inventiveness and experimentation by the modeler.

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