Reconceptualizing the p-value from a likelihood ratio test: a probabilistic pairwise comparison of models based on Kullback-Leibler discrepancy measures

Abstract

Discrepancy measures are often employed in problems involving the selection and assessment of statistical models. A discrepancy gauges the separation between a fitted candidate model and the underlying generating model. In this work, we consider pairwise comparisons of fitted models based on a probabilistic evaluation of the ordering of the constituent discrepancies. An estimator of the probability is derived using the bootstrap. In the framework of hypothesis testing, nested models are often compared on the basis of the p-value. Specifically, the simpler null model is favored unless the p-value is sufficiently small, in which case the null model is rejected and the more general alternative model is retained. Using suitably defined discrepancy measures, we mathematically show that, in general settings, the likelihood ratio test p-value is approximated by the bootstrapped discrepancy comparison probability (BDCP). We argue that the connection between the p-value and the BDCP leads to potentially new insights regarding the utility and limitations of the p-value. The BDCP framework also facilitates discrepancy-based inferences in settings beyond the limited confines of nested model hypothesis testing.

Keywords:

• Bootstrap
• discrepancy functions
• hypothesis testing
• model evaluation
• model selection

Acknowledgments

We wish to thank the referees for their valuable feedback, which served to substantially improve the original version of this manuscript. The content of this manuscript, including incorporated figures and tables, has been derived from the PhD thesis of the first author [26]. This content has not been published elsewhere.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The R software that was used to produce the results for the simulation study and the application can be obtained by request from the first author, and is available on github at https://github.com/briedle-lilly/bdcp.