I am writing a report and am unsure about whether is correct to say one 'measures' r-squared or whether one 'estimates' it. I know the two words have two different semantic meanings, probably related to whether you are identifying the 'true' value or not, but as a non-statistician I am finding it hard to decide which is more appropriate.
Or perhaps neither is suitable and a different word is better.
I have searched around a fair amount and couldn't find an obvious answer.
The coefficient of determination $R^2$ is the square of the multiple correlation coefficient (see this related answer), which is a function of the sample coefficients between the variables. Consequently, it is something that you "measure" or "compute" from the data, rather than something you estimate. However, it is possible to define an analogous value for the square of the true multiple correlation coefficient and treat the coefficient of determination as an estimate for this, so you could reasonably say that you "measure" the coefficient of determination, but you "estimate" the underlying value of the square of the true multiple correlation.
Here is a more formal version of this breakdown. Suppose we first define the true correlation values and sample correlation values (using the Pearson coefficient) for all the variables in the problem. We will label the true correlation values $\rho_i = \mathbb{Corr}(Y,X_i)$ and $\rho_{i,j} = \mathbb{Corr}(X_i,X_j)$ and the sample correlation values $r_i = \mathbb{Corr}(\mathbf{y},\mathbf{x}_i)$ and $r_{i,j} = \mathbb{Corr}(\mathbf{x}_i,\mathbf{x}_j)$, where the latter denote sample correlation between observed vectors of values. Now define the true version and sample version of the goodness of fit vector and design correlation matrix respectively by:
$$\mathbf{GOF} \ \mathbf{vector} \quad \quad \quad \quad \quad \quad \quad \quad \quad \mathbf{DC} \ \mathbf{matrix} \quad \quad \quad \quad \\[12pt] \boldsymbol{\rho}_{\mathbf{y},\mathbf{x}} = \begin{bmatrix} \rho_1 \\ \rho_2 \\ \vdots \\ \rho_m \end{bmatrix} \quad \quad \quad \boldsymbol{\rho}_{\mathbf{x},\mathbf{x}} = \begin{bmatrix} \rho_{1,1} & \rho_{1,2} & \cdots & \rho_{1,m} \\ \rho_{2,1} & \rho_{2,2} & \cdots & \rho_{2,m} \\ \vdots & \vdots & \ddots & \vdots \\ \rho_{m,1} & \rho_{m,2} & \cdots & \rho_{m,m} \\ \end{bmatrix}, \\[40pt] \boldsymbol{r}_{\mathbf{y},\mathbf{x}} = \begin{bmatrix} r_1 \\ r_2 \\ \vdots \\ r_m \end{bmatrix} \quad \quad \quad \boldsymbol{r}_{\mathbf{x},\mathbf{x}} = \begin{bmatrix} r_{1,1} & r_{1,2} & \cdots & r_{1,m} \\ r_{2,1} & r_{2,2} & \cdots & r_{2,m} \\ \vdots & \vdots & \ddots & \vdots \\ r_{m,1} & r_{m,2} & \cdots & r_{m,m} \\ \end{bmatrix}.$$
The parameter version and sample version of the coefficient of determination are then given by:
$$\begin{matrix} \text{Regression model parameter (unnamed)} \quad \quad \quad \quad \quad \phi^2 = \boldsymbol{\rho}_{\mathbf{y},\mathbf{x}}^\text{T} \boldsymbol{\rho}_{\mathbf{x},\mathbf{x}}^{-1} \boldsymbol{\rho}_{\mathbf{y},\mathbf{x}}, \\[6pt] \text{Coefficient of Determination} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad R^2 = \boldsymbol{r}_{\mathbf{y},\mathbf{x}}^\text{T} \boldsymbol{r}_{\mathbf{x},\mathbf{x}}^{-1} \boldsymbol{r}_{\mathbf{y},\mathbf{x}}. \\[6pt] \end{matrix}$$
Now, the value $R^2$ is a statistic that can be computed from the sample, whereas the parameter $\phi^2$ is an unobservable aspect of the regression model that can only be estimated. We can of course use the coefficient of determination $R^2$ to estimate the unknown parameter $\phi^2$.
Although some theory relates $R^2$ to a certain "true" value, it is mostly not used and interpreted as an estimator of that value, but rather as a characteristic of an empirical fit, and as such I'd say it is "computed" and not "estimated". I wouldn't say "measured" either (which goes against my intuition but I have difficulties explaining why), although one could probably defend both "measured" and "estimated" if necessary.
To me, "estimates" sounds weird because the r-squares is not a representation of some hidden object. "Measures" could be okay but I don't hear that so often. I would prefer "calculates" or "computes".
