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NOTES & UNIQUE PHENOMENA

Concordance Correlation for Model Performance Assessment: An Example with Reference Evapotranspiration Observations

^a USDA-ARS, National Soil Tilth Lab., 2110 University Blvd., Ames, IA 50011
^b USDA-ARS Conservation and Production Research Lab., P.O. Drawer 10, Bushland, TX 79012
^c USDA-ARS, SDI+, 13089 Wiregrass Lane, Clovis, CA 93619

^* Corresponding author (Dave.Meek{at}ars.usda.gov).

The assessment procedures for agronomic model performance are often arbitrary and unhelpful. An omnibus analysis, the concordance correlation coefficient (r_c), is widely used in many other sciences. This work illustrates model assessment with two r_c measures accompanied with a mean-difference (MD) plot and a distribution comparison. Each r_c is an adjusted value of the usual Pearson correlation coefficient, r, assuming the exact relationship observations = predictions. The adjustments use a scale shift, u, and a location shift, v. Both of these measures also can indicate the similarity of the two variables' distributions; however, a formal test, the Kolmorgov-Smirnov D statistic, is used to statistically compare the distributions. Daily evapotranspiration data (ET₀) from a published study are compared with estimates from two possible weather observation based models. Although the first model has slightly lower r than the second (0.980 vs. 0.982), its predictions reasonably agree with observations by having comparatively small location and scale shifts [(u = 0.025, v = 1.10, D = 0.12 (p 0.86)] and, consequently, a higher r_c (0.975 vs. 0.946). Results for the second model are comparatively unacceptable having larger scale and location shifts (u = 0.215, v = 1.19, D = 0.28 [p 0.04]) with the bias 0 (p 0.05) as clearly shown in the associated MD plot. Researchers should consider using r_c with an MD plot and distribution comparison in their model assessment toolkit because, together, they can provide a simple and sound probability based omnibus test as well as add useful insight.

Abbreviations: , Laio's location shift relative to scale • µ_x, prediction variable mean • µ_y, observation variable mean • A_p, Laio's bias correction factor • BV, bivariate • CIMIS, California Irrigation Management Information System • C_b, Lin's bias correction factor • d, Wilmott's index of agreement • D, Kolmorgov-Smirnov two-sample test statistic • e, Nash-Suitcliffe efficiency statistic • EDF, empirical distribution function • ET₀, daily evapotranspiration data • i, observation index • MBE, mean bias error (i.e., bias) • MD, mean-difference • MSD, mean squared deviation • n, number of paired observations • r_c, concordance correlation coefficient • u, Lin's location shift relative to scale • x, prediction variable • x_i the ith prediction • y, observation variable • y_i the ith observation

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Received for publication November 12, 2008.