Hypothesis Testing for the Difference between Two Nash–Sutcliffe Efficiencies for Comparing Hydrological Model Performance

The Nash–Sutcliffe efficiency (NSE) is now the most widely used criterion for measuring the goodness of fit between the hydrological model simulation and corresponding observation. Because there is substantial sampling uncertainty regarding hydrological simulation and observation, the NSE is a random variable. A probability density function (PDF) of NSE variable was derived based on the assumption of the simple linear regression model between the observation and simulation from the hydrological model. To avoid a subjective interpretation of the hydrological model performance, the confidence interval of the NSE variable was determined by its PDF. Because the difference in NSE variables ($\mathrm{\Phi }$) between two time periods or two hydrological models is often used for comparing their performances and can also be taken as a random variable, hypothesis testing should be implemented to determine whether the difference is adequate or whether the difference is a chance variation. Because the PDF of the difference can be derived based on the joint PDF of the every NSE random variables, a procedure of the hypothesis testing for the difference between two NSE variables is then proposed for comparing the performances between the hydrological models or between different time periods for a hydrological model. The proposed hypothesis testing has been applied in the abcd and dynamic water balance model (DWBM) hydrological models as case studies to illustrate the procedure of assessing the performances of a hydrological model and justifying its superiority to another model according to both the estimated NSE value and its confidence level due to the sampling uncertainty. Therefore, the proposed hypothesis testing can provide hydrological model end-users with rational model assessment.