Archive for reduced chi-squared

Dos and Don’ts of reduced chi-squared

Posted in Bad Statistics, The Universe and Stuff with tags , , on April 26, 2019 by telescoper

Yesterday I saw a tweet about an arXiv paper and thought I’d share it here. The paper, I mean. It’s not new but I’ve never seen it before and I think it’s well worth reading. The abstract of the paper is:

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.

I added the link at the beginning; you can download a PDF of the paper here.

I’ve never really understood why this statistic (together with related frequentist-inspired ideas) is treated with such reverence by astronomers, so this paper offers a valuable critique to those tempted to rely on it blindly.