The presented table in the following paper summarizes the model performance ratings based on common statistical measures for simulations of flow and sediment yield:

Streamflow and Sediment Yield Prediction for Watershed Prioritization in the Upper Blue Nile River Basin, Ethiopia

Ayele et al 2017

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&cad=rja&uact=8&ved=0ahUKEwi6ro_OuOLYAhUCDZoKHSbVBAQQFghAMAU&url=http%3A%2F%2Fwww.mdpi.com%2F2073-4441%2F9%2F10%2F782%2Fpdf&usg=AOvVaw2P8GwI07yCLTdL8zFV5TJz

Hello Dear Friend,

Hope everything is going well with you.

The statistically significance of independent variables is the main criteria on the relationships between the variables, even for low R-squared value. Here is some more good documents about model efficiency assessment.

The presented table in the following paper summarizes the model performance ratings based on common statistical measures for simulations of flow and sediment yield:

Streamflow and Sediment Yield Prediction for Watershed Prioritization in the Upper Blue Nile River Basin, Ethiopia

Ayele et al 2017

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&cad=rja&uact=8&ved=0ahUKEwi6ro_OuOLYAhUCDZoKHSbVBAQQFghAMAU&url=http%3A%2F%2Fwww.mdpi.com%2F2073-4441%2F9%2F10%2F782%2Fpdf&usg=AOvVaw2P8GwI07yCLTdL8zFV5TJz

Raoof Answer is enough.

Dear Raoof and Najah,

Thanks for your answers.

Different efficiency criteria are available for hydrologic model assessment and R2 is one such criteria. The value of R2 ranges between 0 and 1. 0 corresponds to no correlation whereas 1 provides perfect correlation between the observed and the prediction. But the model which systematically underpredicts or overpredicts also result in R2 values closer to 1. So simply a value of R2 does not guarantee perfect efficiency. Sometimes a graphical comparison is also made between observed and computed values. The closeness of both the series to the 1:1 line indicate better correlation. So Mr. Ansari is advised to refer the following two papers for better understanding.

Krause,P., Boyle,, D. P. Base, F. (2005). Comparison of different efficiency criteria for hydrological model assessment. Advances in Geoscience, 5, 89-97.

McCuen, R. H. (2016). Assessment of Hydrological and statistical significance. Journal of Hydrological Engineering. ASCE, 21(4), 02516001-2.

Dear Dr. Mo Ansari!

Let me try to answer on your question from more general (not only

river flow discharge statistics) point of view. A real significance of any models is determined not only by R2 and other numerical measures, like those listed in references provided by Dr. Raoof Mostafazadeh. Actually R2 itself yields a little without information about a sample size (n). For example, if you have only 2 points (n = 2), then you always will have R2 = 1.

Having in your hands both R2 and n, you can estimate a significance level p which is a function of both R2 and n. What value of p may be considered as good, i.e. meaning a close correlation between predicted and measured values? In different publications there are different values, from 0.001 to 0.05. Some additional information about this issue you can find in https://en.wikipedia.org/wiki/Statistical_significance.

Equations for calculation of p you can find in Spiegel, M.R., Stephens, L.J. 2008. Theory and Problems of Statistics, 4th ed. Schaum’s Outline Series, McGraw-Hill, New York or in our paper Article Assessment of the water quality components in turbid estuari...

Good luck!

Leonid

Returning to your example: what does it mean R2 = 0.7? Let's check the p-value at n = 5, 6, and 11. A Microsoft Excel provides (by help of the function TDIST and Eq. 35 in the above-mentioned paper) values of p: 0.077, 0.038, and 0.001, respectively. Which value is good enough? For my opinion, first two are not very well, but the last value is absolutely perfect.

You can see a very fruitful discussion on this issue at the brilliant paper by David Colquhoun: http://rsos.royalsocietypublishing.org/content/1/3/140216, where he come to conclusion that if p = 0.05, then "you have at the least a 26% chance of being wrong, and often a much bigger chance". However, if p = 0.001, then such chance decreased up to 1.8% only that is probably absolutely tolerate for practical aims.

And also put your attention on an important comment by Dr. Mahendra Kumar Bhuyan concerning a closeness of predicted vs. observed values to the 1:1 line.

I agree with Mahendra, Raoof and Najah.

Other indexes for fit evaluation can be found in:

González-Zeas D., Garrote L., Iglesias A. and Sordo-Ward A., 2012. Improving runoff estimates from regional climate models: A performance analysis in Spain. Hydrology and Earth System Sciences, 16, 1709-1723. DOI:10.5194/hess-16-1709-2012.

Chávez A., Lama B., Garrote L., Martín-Carrasco F., Sordo-Ward A. and Mediero L., 2013. Characterisation of the sensitivity of water resources systems to climate change. Water Resources Management, (2013) 27(12):4237-4258, DOI 10.1007/s11269-013-0404-2.

Flores-Montoya I., Sordo-Ward A., Mediero L., Garrote L., 2016. Fully Stochastic Distributed Methodology for Multivariate Flood Frequency. Water, 8, 225, 1-20, doi:10.3390/w8060225

I would recommend to look at the Nash-Sutcliffe Coefficient of Efficiency. There is a way to also assess the statistical significance:

Ritter, A. and R. Munoz-Carpena, 2013. Performance evaluation of hydrological models: Statistical significance for reducing subjectivity in goodness-of-fit assessments. Journal of Hydrology, Vol 480, pp 33-45.

doi: Article Performance evaluation of hydrological models: Statistical s...

The application to perform some of this analysis, FITEVAL, is available here: http://abe.ufl.edu/carpena/software/FITEVAL.shtml

I think Geraldine Klarenberg has provided you a good answer. If you use NSE (Nash-Sutcliffe Coefficient of Efficiency), you can easily understand the matter

There is a classification based on NSE, so you can look at the study by Ritter, A. and R. Munoz-Carpena, 2013.

Application of the concept in the article: Modeling the Changes in Water Balance Components of Highly Irrigated Western Part of Bangladesh, Hydrology and Earth System Sciences Discussions,

DOI10.5194/hess-2017-523

Coefficient of determination (R^2)

• The coefficient of determination is a measure of the amount of variance in the dependent variable explained by the independent variable(s). A value of one (1) means perfect explanation and is not encountered in reality due to ever present error. A value of .91 means that 91% of the variance in the dependent variable is explained by the independent variables.

• The amount of variation explained by the regression model should be more than the variation explained by the average. Thus, R2 should be greater than zero.

• R2 is impacted by two facets of the data:

o the number of independent variables relative to the sample size. For this reason, analysts should use the adjusted coefficient of determination, which adjusts for inflation in R2 from overfitting the data.

o the number of independent variables included in the analysis. As you increase the number of independent variables in the model, you increase the R2 automatically because the sum of squared errors by regression begins to approach the sum of squared errors about the average.

Dear Dr Haitham Hmoud Alshibly,

I cannot agree with your statement that "As you increase the number of independent variables in the model, you increase the R2 automatically". At n = 2, R2 is always equal to 1, and at n > 2, R2 is always < 1.

Another example. Let's R2 = 0.7. Then at n = 10, a significance level p = 0.00254. If we fix this value of p (or, by another words, we fix a level of correlation) then at n = 3, R2 should be equal to 0.9999841 to keep the same p = 0.00254.

Thus, at given level of p, increasing of n leads to decreasing the R2.

Does this mean that we should want to use small values of n? Of course, not! At small values of n we generally have vary unstable situation with p; however, increasing n, p will be lead to constant (stable) value, even though R2 will be decreased.

There are lots of good responses and information shared by others in this set of answers. I have just two additional thoughts that I didn't see were raised by others. First, examine the model residuals graphically. Is there evidence of bias, or do the errors appear to be randomly distributed? This will help you decide on the quality of the model. Second, one can calculate the prediction intervals (5% and 95%) of a model. Given the expected uncertainty in your predictions from this interval, is your model adequate for your needs? The answer to this question will help you determine whether the model is good enough.

In the studies concerned with the streamflow forecasting , the Coefficient of determination (alone) does not assurance ideal efficiency, so I think that you must try to use another tests such as Nash-Sutcliffe efficiency .

Other indexes for streamflow fit evaluation can be found in:

Mohamed , T.M., Etuk , E.H, “Application of linear stochastic models to monthly streamflow data of Rahad River, ” Sudan, Int. J. Hydrology Science and Technology, 7(2), 197-212,2017.

R2=0.7 could be acceptable in most of professional works.

Thanks all of you.

Sorry for replying late.

I will use your experience for my future researches.

The closer R-square to 1 is the better.

I agree with Dr. Najah M. L. Al Maimuri .