Wilks's Theorem, Global Fits, and Neutrino Oscillations

Tests of models for new physics appearing in neutrino experiments often involve global fits to a quantum mechanical effect called neutrino oscillations. This paper introduces students to methods commonly used in these global fits starting from an understanding of more conventional fitting methods using log-likelihood and $\chi^2$ minimization. Specifically, we discuss how the $\Delta\chi^2$, which compares the $\chi^2$ of the fit with the new physics to the $\chi^2$ of the Standard Model prediction, is often interpreted using Wilks's theorem. This paper uses toy models to explore the properties of $\Delta\chi^2$ as a test statistic for oscillating functions. The statistics of such models are shown to deviate from Wilks's theorem. Tests for new physics also often examine data subsets for "tension" called the "parameter goodness of fit". In this paper, we explain this approach and use toy models to examine the validity of the probabilities from this test also. Although we have chosen a specific scenario -- neutrino oscillations -- to illustrate important points, students should keep in mind that these points are widely applicable when fitting multiple data sets to complex functions.