# How do the $r^{2}$, amount of variance explained and concordance correlation coefficient relate?

Let us define the $$r^{2}$$ as: $$r^{2} = \left(\frac{\sum_{i=1}^{n}(obs_{i} - \overline{obs})(pred_{i}-\overline{pred})}{\sqrt{\sum_{i=1}^{n}(obs_{i} - \overline{obs})^{2}}\sqrt{\sum_{i=1}^{n}(pred_{i} - \overline{pred})^{2}}}\right)^{2},$$ the amount of variance explained (AVE) as: $$\mathrm{AVE} = 1-\frac{\sum_{i=1}^{n}(obs_{i} - pred_{i})^{2}}{\sum_{i=1}^{n}(obs_{i}-\overline{obs})^{2}},$$ and the Lin's concordance coefficient ($$\rho_{c}$$) as: $$\rho_{c} = \frac{2r\sigma_{pred}\sigma_{obs}}{\sigma_{obs}^{2}\sigma_{pred}^{2}+(\mu_{obs} - \mu_{pred})^{2}} = rC_{b},$$ where $$r$$ is the Pearson's correlation coefficient, $$\sigma$$ is the variance ($$r\sigma_{pred}\sigma_{obs}$$ is the covariance between observed and predicted values) and $$\mu$$ is the mean. The $$\rho_{c}$$ reduces to $$rC_{b}$$ where $$C_{b}$$ is the bias correction factor defined as: $$\begin{equation} C_{b} = \frac{2}{v+1/v+u^{2}}, \end{equation}$$ with $$v = \sigma_{x}/\sigma_{y}$$ being the scale shift and $$u=(\mu_{x}-\mu_{y})/\sqrt{\sigma_{y}\sigma_{x}}$$ being the location shift relative to the scale.

In my field (environmental science) there is a lot of confusion between the three. I often see people assessing the prediction accuracy of their model by the $$r^{2}$$. This does not make a lot of sense if the model is not a linear regression (e.g. a ML model) and I always suggest to use the AVE instead. Imagine a scatterplot between the predicted and measured values, the AVE shows the amount of variance explained by the model with respect to the 1:1 line, not to any regression line as it is for the $$r^{2}$$.

But then people might say that there is redundancy between the AVE and $$\rho_{c}$$, because both show the agreement of predicted versus observed with respect to the 1:1 line. From the equation above it appears that the $$\rho_{c}$$ is simply a Pearson's $$r$$ times a bias correction factor.

I made a few numerical tests to try to understand how the AVE and the $$\rho_{c}$$ relate. Under perfect correlation (i.e. $$r=1$$) and increasing shift with respect to the 1:1 line, the $$AVE = log(\rho_{c})+b$$ where $$b$$ is a constant. Both the AVE and the $$\rho_{c}$$ almost linearly decrease with increase of the shift from the 1:1 line.

Under three different degrees of correlation, for weak ($$r=0.22$$), moderate ($$r=0.75$$) and strong ($$r=0.98$$) correlation between the predicted and observed values, and increasing shift with respect to the 1:1 line, I obtain the following figure. The low values of AVE and $$\rho_{c}$$ are of course related to large shift from the 1:1 line. My questions are as follows:

1. Is there redundancy between the AVE and $$\rho_{c}$$ when one wishes to assess the prediction accuracy of a model with respect to some observations?

2. Is there a direct relation between the AVE and the $$\rho_{c}$$ that I have missed. How to explain the figure above?

3. Is there a case showing a high $$\rho_{c}$$ value and low AVE?