Dos and don'ts of reduced chi-squared

By R. Andrae, T. Schulze-Hartung, Peter Melchior. 2010

249 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/353308ea0f2e5be88f397cbd5c0528bc0069208b

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In Applied Mathematics

Th. Henning

In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole (2018, methodology): “For linear models the formula presented in Andrae et al. (2010) can be used to estimate the significance.”
Collider limits on new physics within micrOMEGAs4.3 (2016, unknown): “[36] for a detailed discussion of this problem.”
Neutron star mass and radius measurements from atmospheric model fits to X-ray burst cooling tail spectra (2017, methodology): “Although we reiterate our caveats about the use of χ2, particularly for model comparisons (see also Andrae et al. 2010), we note that the Model D fit to 4U 1702−429 has χ2/d.o.f. = 2541.0/2122 (see also Table 2), whereas for a simple blackbody fit χ2/d.o.f. = 2716.3/2010.”