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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:

In the above definition,

That is, is the total sum of squares, is the explained sum of squares, and is the residual sum of squares.

Explanation and interpretation of R²Edit

For expository purposes, consider a linear model of the form

where Y_i is the response variable, are unknown coefficients; are p regressors, and is a mean zero error term. The coefficient of determination R² is a measure of the global fit of the model. Specifically, is an element of [0,1] and represents the proportion of variability in Y_i 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, indicates that the fitted model explains all variability in , while indicates no 'linear' relationship between the response variable and regressors. An interior value such as 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 is the square of the correlation between the regressor and response variables. More generally, is the square of the correlation between y and .

Inflation of R²Edit

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 R²_p, 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)

The optimal value of the objective is weakly smaller as additional columns of 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 depends only on y, the non-decreasing property of R₂ follows directly from the definition above.

Adjusted R²Edit

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

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 R²Edit

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

External linksEdit

• Adjusted R-Square Calculator

See alsoEdit

• Correlation

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