Evaluation of the Nash–Sutcliffe Efficiency Index

TLDR

The Nash–Sutcliffe efficiency index is widely used to assess hydrologic model fit, yet its statistical significance and the factors that produce poor sample values remain poorly understood. The study aims to approximate Ef’s sampling distribution, develop hypothesis‑testing and confidence‑interval methods, and identify how sample size, outliers, magnitude bias, time‑offset bias, and sampling interval affect its values. The authors used real hydrologic datasets and simulated analyses to approximate Ef’s sampling distribution and assess the impact of these factors. Analyses revealed that outliers and both magnitude and time‑offset biases substantially degrade Ef, while the recording time step is largely irrelevant unless the sample size is small, indicating that Ef can be a reliable goodness‑of‑fit metric when properly interpreted.

Abstract

The Nash–Sutcliffe efficiency index (Ef) is a widely used and potentially reliable statistic for assessing the goodness of fit of hydrologic models; however, a method for estimating the statistical significance of sample values has not been documented. Also, factors that contribute to poor sample values are not well understood. This research focuses on the interpretation of sample values of Ef. Specifically, the objectives were to present an approximation of the sampling distribution of the index; provide a method for conducting hypothesis tests and computing confidence intervals for sample values; and identify the effects of factors that influence sample values of Ef including the sample size, outliers, bias in magnitude, time-offset bias of hydrograph models, and the sampling interval of hydrologic data. Actual hydrologic data and hypothetical analyses were used to show these effects. The analyses show that outliers can significantly influence sample values of Ef. Time-offset bias and bias in magnitude can have an adverse effect on Ef. The time step at which the data are recorded appears to be an insignificant factor unless the sample size is small. The Nash–Sutcliffe index can be a reliable goodness-of-fit statistic if it is properly interpreted.