Coefficient of determination

In statistics, the coefficient of determination R² is used in the context of statistical models whose main purpose is the prediction of future outcomes on the basis of other related information. It is the proportion of variability in a data set that is accounted for by the statistical model.^[1] It provides a measure of how well future outcomes are likely to be predicted by the model.

There are several different definitions of R² which are only sometimes equivalent. One class of such cases includes that of linear regression. In this case, if an intercept is included then R² is simply the square of the sample correlation coefficient between the outcomes and their predicted values, or in the case of simple linear regression, between the outcomes and the values of the single regressor being used for prediction. In such cases, the coefficient of determination ranges from 0 to 1. Important cases where the computational definition of R² can yield negative values, depending on the definition used, arise where the predictions which are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data, and where linear regression is conducted without including an intercept. Additionally, negative values of R² may occur when fitting non-linear trends to data.^[2] In these instances, the mean of the data provides a fit to the data that is superior to that of the trend under this goodness of fit analysis.

Definitions

$R^2 = 1 - \frac{\color{blue}{SS_\text{err}}}{\color{red}{SS_\text{tot}}}$
The better the linear regression (on the right) fits the data in comparison to the simple average (on the left graph), the closer the value of R² is to one. The areas of the blue squares represent the squared residuals with respect to the linear regression. The areas of the red squares represent the squared residuals with respect to the average value.

A data set has values y_i, each of which has an associated modelled value f_i (also sometimes referred to as ŷ_i). Here, the values y_i are called the observed values and the modelled values f_i are sometimes called the predicted values.

The "variability" of the data set is measured through different sums of squares:

$SS_\text{tot}=\sum_i (y_i-\bar{y})^2,$ the total sum of squares (proportional to the sample variance);

$SS_\text{reg}=\sum_i (f_i -\bar{y})^2,$ the regression sum of squares, also called the explained sum of squares.

$SS_\text{err}=\sum_i (y_i - f_i)^2\,$, the sum of squares of residuals, also called the residual sum of squares.

In the above $\bar{y}$ is the mean of the observed data:

$\bar{y}=\frac{1}{n}\sum_i^n y_i$

where n is the number of observations.

The notations SS_R and SS_E should be avoided, since in some texts their meaning is reversed to Residual sum of squares and Explained sum of squares, respectively.

The most general definition of the coefficient of determination is

$R^2 \equiv 1 - {SS_{\rm err}\over SS_{\rm tot}}.\,$

Relation to unexplained variance

In a general form, R² can be seen to be related to the unexplained variance, since the second term compares the unexplained variance (variance of the model's errors) with the total variance (of the data). See fraction of variance unexplained.

As explained variance

In some cases the total sum of squares equals the sum of the two other sums of squares defined above,

$SS_{\rm err}+SS_{\rm reg}=SS_{\rm tot}. \,$

See sum of squares for a derivation of this result for one case where the relation holds. When this relation does hold, the above definition of R² is equivalent to

$R^{2} = {SS_{\rm reg} \over SS_{\rm tot} }.$

In this form R² is given directly in terms of the explained variance: it compares the explained variance (variance of the model's predictions) with the total variance (of the data).

This partition of the sum of squares holds for instance when the model values ƒ_i have been obtained by linear regression. A milder sufficient condition reads as follows: The model has the form

$f_i=\alpha+\beta q_i \,$

where the q_i are arbitrary values that may or may not depend on i or on other free parameters (the common choice q_i = x_i is just one special case), and the coefficients α and β are obtained by minimizing the residual sum of squares.

This set of conditions is an important one and it has a number of implications for the properties of the fitted residuals and the modelled values. In particular, under these conditions:

$\bar{f}=\bar{y}.\,$

As squared correlation coefficient

Similarly, after least squares regression with a constant+linear model (i.e., simple linear regression), R² equals the square of the correlation coefficient between the observed and modeled (predicted) data values.

Under general conditions, an R² value is sometimes calculated as the square of the correlation coefficient between the original and modeled data values. In this case, the value is not directly a measure of how good the modeled values are, but rather a measure of how good a predictor might be constructed from the modeled values (by creating a revised predictor of the form α + βƒ_i). According to Everitt (2002, p. 78), this usage is specifically the definition of the term "coefficient of determination": the square of the correlation between two (general) variables.

Interpretation

R² is a statistic that will give some information about the goodness of fit of a model. In regression, the R² coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R² of 1.0 indicates that the regression line perfectly fits the data.

Values of R² outside the range 0 to 1 can occur where it is used to measure the agreement between observed and modelled values and where the "modelled" values are not obtained by linear regression and depending on which formulation of R² is used. If the first formula above is used, values can never be greater than one. If the second expression is used, there are no constraints on the values obtainable.

In many (but not all) instances where R² is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing SS_err. In this case R-squared increases as we increase the number of variables in the model (R² will not decrease). This illustrates a drawback to one possible use of R², where one might try to include more variables in the model until "there is no more improvement". This leads to the alternative approach of looking at the adjusted R². The explanation of this statistic is almost the same as R² but it penalizes the statistic as extra variables are included in the model. For cases other than fitting by ordinary least squares, the R² statistic can be calculated as above and may still be a useful measure. If fitting is by weighted least squares or generalized least squares, alternative versions of R² can be calculated appropriate to those statistical frameworks, while the "raw" R² may still be useful if it is more easily interpreted. Values for R² can be calculated for any type of predictive model, which need not have a statistical basis.

In a linear model

Consider a linear model of the form

${Y_i = \beta_0 + \sum_{j=1}^p {\beta_j X_{i,j}} + \varepsilon_i},$

where, for the ith case, Y_i is the response variable, $X_{i,1},\dots,X_{i,p}$ are p regressors, and ε_i is a mean zero error term. The quantities $\beta_0,\dots,\beta_p$ are unknown coefficients, whose values are determined by least squares. The coefficient of determination R² is a measure of the global fit of the model. Specifically, R² 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.

R² is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, R² = 1 indicates that the fitted model explains all variability in y, while R² = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope=0, intercept=$\bar{y}$) between the response variable and regressors). An interior value such as R² = 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."

A caution that applies to R², as to other statistical descriptions of correlation and association is that "correlation does not imply causation." In other words, while correlations may provide valuable clues regarding causal relationships among variables, a high correlation between two variables does not represent adequate evidence that changing one variable has resulted, or may result, from changes of other variables.

In case of a single regressor, fitted by least squares, R² is the square of the Pearson product-moment correlation coefficient relating the regressor and the response variable. More generally, R² is the square of the correlation between the constructed predictor and the response variable.

Inflation of R²

In least squares regression, R² is weakly increasing in the number of regressors in the model. As such, R² alone cannot be used as a meaningful comparison of models with different numbers of independent variables. For a meaningful comparison between two models, an F-test can be performed on the residual sum of squares, similar to the F-tests in Granger causality. As a reminder of this, some authors denote R² by R²_p, where p is the number of columns in X

To demonstrate this property, first recall that the objective of least squares regression is:

$\min_b SS_\text{err}(b) \Rightarrow \min_b \sum_i (y_i - X_ib)^2\,$

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_tot depends only on y, the non-decreasing property of R² follows directly from the definition above.

The intuitive reason that using an additional explanatory variable cannot lower the R² is this: Minimizing SS_err is equivalent to maximizing R². When the extra variable is included, the data always have the option of giving it an estimated coefficient of zero, leaving the predicted values and the R² unchanged. The only way that the optimization problem will give a non-zero coefficient is if doing so improves the R².

Notes on interpreting R²

R² does not indicate whether:

• the independent variables are a true cause of the changes in the dependent variable;
• omitted-variable bias exists;
• the correct regression was used;
• the most appropriate set of independent variables has been chosen;
• there is collinearity present in the data on the explanatory variables;
• the model might be improved by using transformed versions of the existing set of independent variables.

Adjusted R²

Adjusted R² (often written as $\bar R^2$ and pronounced "R bar squared") is a modification due to Theil^[3] 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 or equal to R². The adjusted R² is defined as

$\bar R^2 = {1-(1-R^{2}){n-1 \over n-p-1}} = {1-{SS_\text{err} \over SS_\text{tot}}}{df_t \over df_e}$

where p is the total number of regressors in the linear model (but not counting the constant term), n is the sample size, df_t is the degrees of freedom n– 1 of the estimate of the population variance of the dependent variable, and df_e is the degrees of freedom n – p – 1 of the estimate of the underlying population error variance.

The principle behind the Adjusted R² statistic can be seen by rewriting the ordinary R² as

${R^{2} = {1-{\textit{VAR}_\text{err} \over \textit{VAR}_\text{tot}}}}$

where VAR_err = SS_err / n and VAR_tot = SS_tot / n are estimates of the variances of the errors and of the observations, respectively. These estimates are replaced by statistically unbiased versions: VAR_err = SS_err / (n − p − 1) and VAR_tot = SS_tot / (n − 1).

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..

The use of an adjusted R² is an attempt to take account of the phenomenon of statistical shrinkage.^[4]

Generalized R²

Nagelkerke (1991) generalizes the definition of the coefficient of determination:

1. A generalized coefficient of determination should be consistent with the classical coefficient of determination when both can be computed;
2. Its value should also be maximised by the maximum likelihood estimation of a model;
3. It should be, at least asymptotically, independent of the sample size;
4. Its interpretation should be the proportion of the variation explained by the model;
5. It should be between 0 and 1, with 0 denoting that model does not explain any variation and 1 denoting that it perfectly explains the observed variation;
6. It should not have any unit.

The generalized R² has all of these properties.

$R^{2} = 1 - \left({ L(0) \over L(\hat{\theta})}\right)^{2/n}$

where L(0) is the likelihood of the model with only the intercept, ${L(\hat{\theta})}$ is the likelihood of the estimated model and n is the sample size.

However, in the case of a logistic model, where $L(\hat{\theta})$ cannot be greater than 1, R² is between 0 and $R^2_\max = 1- (L(0))^{2/n}$: thus, it is possible to define a scaled R² as R²/R²_max.^[5]

See also

• Goodness of fit
• Fraction of variance unexplained
• Pearson product-moment correlation coefficient
• Nash–Sutcliffe model efficiency coefficient (hydrological applications)
• Regression model validation
• Proportional reduction in loss
• Root mean square deviation
• Multiple correlation

Notes

1. ^ Steel, R. G. D. and Torrie, J. H., Principles and Procedures of Statistics, New York: McGraw-Hill, 1960, pp. 187, 287.
2. ^ Cameron, A.C., Windmeijer, F.A.G., (1997)."An R-squared measure of goodness of fit for some common nonlinear regression models." Journal of Econometrics, Volume 77, Issue 2, April 1997, Pages 329-342.
3. ^ Theil, Henri (1961). Economic Forecasts and Policy. Holland, Amsterdam: North. 
4. ^ Everitt, B.S. (2002) The Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-x (See entries for "Shrinkage", "Shrinkage formulae")
5. ^ N. Nagelkerke, "A Note on a General Definition of the Coefficient of Determination," Biometrika, vol. 78, no. 3, pp. 691–692, 1991.

References

• Draper, N.R. and Smith, H. (1998). Applied Regression Analysis. Wiley-Interscience. ISBN 0-471-17082-8
• Everitt, B.S. (2002). Cambridge Dictionary of Statistics (2nd Edition). CUP. ISBN 0-521-81099-x
• Nagelkerke, Nico J.D. (1992) Maximum Likelihood Estimation of Functional Relationships, Pays-Bas, Lecture Notes in Statistics, Volume 69, 110p ISBN 0-387-97721-X.
• Glantz, S.A. and Slinker, B.K., (1990). Primer of Applied Regression and Analysis of Variance. McGraw-Hill. ISBN 0-07-023407-8.