Correlation, often denoted by "R", actually shares two definitions that are logically separable. This snippet demonstrates that.

Let \(\mathbf{y}\) be a true outcome vector and \(\widehat{\mathbf{y}}\) be a vector of predictions of \(\mathbf{y}\). Define the (centered) sum of products \(\text{SP}(\mathbf{z}, \mathbf{w}) := (\mathbf{z}-\overline{\mathbf{z}})^\top(\mathbf{w}-\overline{\mathbf{w}})\), and define the curried function \(\text{SP}(\mathbf{z}) := \text{SP}(\mathbf{z}, \mathbf{z})\).

Consider two estimands. First, the coefficient of determination¹:

\begin{align} R^2 := 1 - \frac{\text{SP}(\mathbf{y}-\widehat{\mathbf{y}})}{\text{SP}(\mathbf{y})} = \frac{\text{SP}(\widehat{\mathbf{y}})}{\text{SP}(\mathbf{y})} \tag{1} \end{align}

Second, the Pearson correlation between outcomes and prediction:

\begin{align} \widehat{\rho} := \frac{\text{SP}(\mathbf{y},\ \widehat{\mathbf{y}})}{\sqrt{\text{SP}(\mathbf{y})\text{SP}(\widehat{\mathbf{y}})}} \end{align}

To summarize, \(\widehat{\rho}\) uses both quantities from \(R^2\), in addition to a third term that does not strongly depend on the other two². So the two estimands are logically separable.

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