Dos and don'ts of reduced chi-squared

TL;DR

This paper critically examines the limitations of using reduced chi-squared in astronomy, highlighting issues with degrees of freedom estimation and noise impact, and recommends more reliable alternatives especially for nonlinear models.

Contribution

It clarifies the pitfalls of reduced chi-squared, especially for nonlinear models, and advocates for adopting more robust model assessment methods.

Findings

Reduced chi-squared is only reliable for linear models.

Noise significantly affects reduced chi-squared, especially in small data sets.

Alternative methods are recommended for nonlinear models.

Abstract

Reduced chi-squared is a very popular method for model assessment, model comparison, convergence diagnostic, and error estimation in astronomy. In this manuscript, we discuss the pitfalls involved in using reduced chi-squared. There are two independent problems: (a) The number of degrees of freedom can only be estimated for linear models. Concerning nonlinear models, the number of degrees of freedom is unknown, i.e., it is not possible to compute the value of reduced chi-squared. (b) Due to random noise in the data, also the value of reduced chi-squared itself is subject to noise, i.e., the value is uncertain. This uncertainty impairs the usefulness of reduced chi-squared for differentiating between models or assessing convergence of a minimisation procedure. The impact of noise on the value of reduced chi-squared is surprisingly large, in particular for small data sets, which are very common in astrophysical problems. We conclude that reduced chi-squared can only be used with due caution for linear models, whereas it must not be used for nonlinear models at all. Finally, we recommend more sophisticated and reliable methods, which are also applicable to nonlinear models.