Coefficient of determination

In statistics, the coefficient of determination R2 is the proportion of variability in a data set that is accounted for by a statistical model. There are several common and equivalent expressions for R2. The version most common in statistics texts is based on an analysis of variance decomposition as follows:
 * $${R^{2} = {SS_R \over SS_T} = {1-{SS_E \over SS_T}}}.$$

In the above definition,
 * $$SS_T=\sum_i (y_i-\bar{y})^2, SS_R=\sum_i (\hat{y_i}-\bar{y})^2, SS_E=\sum_i (y_i - \hat{y_i})^2.$$

That is, $$SS_{T}$$ is the total sum of squares, $$SS_{R}$$ is the explained sum of squares, and $$SS_{E}$$ is the residual sum of squares.

Explanation and interpretation of R2
For expository purposes, consider a linear model of the form
 * $${Y_i = \beta_0 + \sum_j^p {\beta_j X_{i,j}} + \epsilon_i},$$

where Yi is the response variable, $$\beta_0,...,\beta_p$$ are unknown coefficients; $$X_1,...,X_p$$ are p regressors, and $$\epsilon_i$$ is a mean zero error term. The coefficient of determination R2 is a measure of the global fit of the model. Specifically, $$R^{2}$$ is an element of [0,1] and represents the proportion of variability in Yi that may be attributed to some linear combination of the regressors (explanatory variables) in X.

More simply, R2 is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, $$R^{2}=1$$ indicates that the fitted model explains all variability in $$y$$, while $$R^{2}=0$$ indicates no 'linear' relationship between the response variable and regressors. An interior value such as $$R^{2}=0.7$$ may be interpreted as follows: "Approximately seventy percent of the variation in the response variable can be explained by the explanatory variable. The remaining thirty percent can be explained by unknown, lurking variables or inherent variability."

If there is just one scalar-valued regressor, then $$R^{2}$$ is the square of the correlation between the regressor and response variables. More generally, $$R^{2}$$ is the square of the correlation between y and $$\hat{y}$$.

Inflation of R2
In least squares regression, R2 is weakly increasing in the number of regressors in the model. As such, R2 cannot be used as a meaningful comparison of models with different numbers of covariants. As a reminder of this, some authors denote R2 by R2p, where p is the number of columns in X

Demonstration of this property is trivial. To begin, recall that the objective of least squares regression is (in matrix notation)


 * $$\min_b SS_E(b) \Rightarrow \min_b \sum_i (y_i - X_ib)^2$$

The optimal value of the objective is weakly smaller as additional columns of $$X$$ are added, by the fact that relatively unconstrained minimization leads to a solution which is weakly smaller than relatively constrained minimization. Given the previous conclusion and noting that $$SS_T$$ depends only on y, the non-decreasing property of R2 follows directly from the definition above.

Adjusted R2
Adjusted R2 is a modification of R2 that adjusts for the number of explanatory terms in a model. Unlike R2, the adjusted R2 increases only if the new term improves the model more than would be expected by chance. The adjusted R2 can be negative, and will always be less than R2. The adjusted R2 is defined as


 * $${1-(1-R^{2}){n-1 \over n-p-1}}$$

where p is the total number of regressors in the linear model, and n is sample size.

Adjusted R2 does not have the same interpretation as R2. As such, care must be taken in interpreting and reporting this statistic. Adjusted R2 is particularly useful in the Feature selection stage of model building.

Notes on interpreting R2
$$R^{2}$$ does NOT tell whether:
 * the independent variables are a true cause of the changes in the dependent variable
 * omitted-variable bias exists; or
 * the most appropriate set of independent variables has been chosen