Dos and don'ts of reduced chi-squared

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.

An Expert Evaluation of "Dos and Don'ts of Reduced Chi-Squared"

The paper titled "Dos and Don'ts of Reduced Chi-Squared" by Andrae et al. critically examines the usage of reduced chi-squared ($\chi^2_{\text{red}}$) in the assessment, comparison, and convergence diagnostics of models, particularly within the field of astronomy. It identifies crucial misconceptions and limitations related to $\chi^2_{\text{red}}$, focusing on two primary issues: the estimation of degrees of freedom and the inherent uncertainty in the value of $\chi^2_{\text{red}}$ itself.

Key Issues in the Use of Reduced Chi-Squared

1. Degrees of Freedom Estimation:

   • The paper emphasizes that calculating the number of degrees of freedom is often inaccurately assumed to be simply the number of data points minus the number of model parameters ($N-P$). Andrae et al. discuss how this is true only for linear models with linearly independent basis functions. For more complex nonlinear models, the number of degrees of freedom varies and is not constant, contravening the common assumption of $N-P$. This has significant implications for models in fields that routinely use nonlinear models.

2. Uncertainty in $\chi^2$ Values:

   • The variability of $\chi^2$ due to the stochastic nature of the data is another critical issue discussed. The paper quantitatively explains the variance in $\chi^2_{\text{red}}$, showing that even for large datasets, this uncertainty can obscure meaningful comparisons or conclusions. For a dataset with $N=1,000$, a standard deviation of approximately $0.045$ illustrates that the $\chi^2_{\text{red}}$ value cannot be solely relied upon for definitive model comparison or convergence assessment, as uncertainties accumulate and affect interpretation.

Alternative Methods and Recommendations

Acknowledging these challenges with $\chi^2_{\text{red}}$, the authors advocate for alternative methods that can offer more reliable assessments:

• Residual Analysis: A straightforward yet effective approach involves evaluating the distribution of normalised residuals against a Gaussian distribution. This approach facilitates identification of statistically meaningful deviations that imply a model misfit.

• Cross-validation: While computationally intensive, especially in iterations like leave-one-out cross-validation, this method leverages predictive capability rather than just goodness of fit, providing unbiased model comparison. It is particularly valuable when data errors are well-characterized.

• Bootstrapping: This is another robust method, providing model validation without requiring complete knowledge of the data's error distribution, albeit at the cost of computational efficiency.

Implications and Concluding Remarks

The findings highlight significant implications for researchers, emphasizing a more cautious and informed use of $\chi^2_{\text{red}}$. They suggest that researchers need to adopt complementary statistical techniques to resolve the pitfalls associated with $\chi^2_{\text{red}}$.

Beyond providing clarity on specific statistical misuses, the paper calls for a broader re-evaluation of conventional metrics in data analysis, particularly in complex datasets where nonlinear models predominate. This manuscript does not undermine the utility of minimizing $\chi^2$ for fitting models to data, a practice validated when Gaussian errors are assumed. However, it stresses the importance of integrating additional statistical foundations when engaging in tasks like model selection or convergence diagnostics.

The paper's directives underscore a crucial paradigm shift in adhering to rigorous statistical protocols when utilizing $\chi^2_{\text{red}}$, recommending an expanded toolkit for researchers to improve model evaluation and analytical rigour in quantitative research fields.