Coefficient of determination

Not to be confused with Coefficient of variation or Coefficient of correlation.

Ordinary least squares regression of Okun's law. Since the regression line does not miss any of the points by very much, the R² of the regression is relatively high.

Comparison of the Theil–Sen estimator (black) and simple linear regression (blue) for a set of points with outliers. Because of the many outliers, neither of the regression lines fits the data well, as measured by the fact that neither gives a very high R².

In statistics, the coefficient of determination, denoted R² or r² and pronounced "R squared", is the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model.

There are several definitions of R² that are only sometimes equivalent. One class of such cases includes that of simple linear regression where r² is used instead of R². When an intercept is included, then r² is simply the square of the sample correlation coefficient (i.e., r) between the observed outcomes and the observed predictor values. If additional regressors are included, R² is the square of the coefficient of multiple correlation. In both such cases, the coefficient of determination ranges from 0 to 1.

There are cases where the computational definition of R² can yield negative values, depending on the definition used. This can arise when the predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data. Even if a model-fitting procedure has been used, R² may still be negative, for example when linear regression is conducted without including an intercept, or when a non-linear function is used to fit the data. In cases where negative values arise, the mean of the data provides a better fit to the outcomes than do the fitted function values, according to this particular criterion. Since the most general definition of the coefficient of determination is also known as the Nash–Sutcliffe model efficiency coefficient, this last notation is preferred in many fields, because denoting a goodness-of-fit indicator that can vary from -∞ to 1 (i.e., it can yield negative values) with a squared letter is confusing.

When evaluating the goodness-of-fit of simulated (Y_pred) vs. measured (Y_obs) values, it is not appropriate to base this on the R² of the linear regression (i.e., Y_obs= m·Y_pred + b). The R² quantifies the degree of any linear correlation between Y_obs and Y_pred, while for the goodness-of-fit evaluation only one specific linear correlation should be taken into consideration: Y_obs = 1·Y_pred + 0 (i.e., the 1:1 line).

Definitions

$R^{2}=1-{\frac {\color {blue}{SS_{\text{res}}}}{\color {red}{SS_{\text{tot}}}}}$
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 $R^{2}$ is to 1. 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 n values marked y₁,...,y_n (collectively known as y_i or as a vector y = [y₁,...,y_n]^T), each associated with a predicted (or modeled) value f₁,...,f_n (known as f_i, or sometimes ŷ_i, as a vector f).

Define the residuals as e_i = y_i − f_i (forming a vector e).

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

${\bar {y}}={\frac {1}{n}}\sum _{i=1}^{n}y_{i}$

then the variability of the data set can be measured using three sums of squares formulas:

• The total sum of squares (proportional to the variance of the data):

$SS_{\text{tot}}=\sum _{i}(y_{i}-{\bar {y}})^{2},$

• The regression sum of squares, also called the explained sum of squares:

$SS_{\text{reg}}=\sum _{i}(f_{i}-{\bar {y}})^{2},$

• The sum of squares of residuals, also called the residual sum of squares:

$SS_{\text{res}}=\sum _{i}(y_{i}-f_{i})^{2}=\sum _{i}e_{i}^{2}\,$

The most general definition of the coefficient of determination is

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

Relation to unexplained variance

Main article: Fraction of variance unexplained

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

${\begin{aligned}R^{2}=1-{\text{FVU}}\end{aligned}}$

As explained variance

Suppose R² = 0.49. This implies that 49% of the variability of the dependent variable has been accounted for, and the remaining 51% of the variability is still unaccounted for. In some cases the total sum of squares equals the sum of the two other sums of squares defined above,

$SS_{\text{res}}+SS_{\text{reg}}=SS_{\text{tot}}.\,$

See Partitioning in the general OLS model 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

$R^{2}={\frac {SS_{\text{reg}}}{SS_{\text{tot}}}}={\frac {SS_{\text{reg}}/n}{SS_{\text{tot}}/n}}$

where n is the number of observations (cases) on the variables.

In this form R² is expressed as the ratio of the explained variance (variance of the model's predictions, which is SS_reg / n) to the total variance (sample variance of the dependent variable, which is SS_tot / n).

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

$f_{i}={\hat {\alpha }}+{\hat {\beta }}q_{i}\,$

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 coefficient estimates ${\hat {\alpha }}$ and ${\hat {\beta }}$ 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

In linear least squares multiple regression with an estimated intercept term, R² equals the square of the Pearson correlation coefficient between the observed $y$ and modeled (predicted) $f$ data values of the dependent variable.

In a linear least squares regression with an intercept term and a single explanator, this is also equal to the squared Pearson correlation coefficient of the dependent variable $y$ and explanatory variable $x.$

It should not be confused with the correlation between two coefficient estimates, defined as

$\rho _{{\hat {\alpha }},{\hat {\beta }}}={\mathrm {cov} ({\hat {\alpha }},{\hat {\beta }}) \over \sigma _{\hat {\alpha }}\sigma _{\hat {\beta }}},$

where the covariance between two coefficient estimates, as well as their standard deviations, are obtained from the covariance matrix of the coefficient estimates.

Under more general modeling conditions, where the predicted values might be generated from a model different from linear least squares regression, an R² value can be calculated as the square of the correlation coefficient between the original $y$ and modeled $f$ 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 (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 predictions approximate the real data points. An R² of 1 indicates that the regression predictions perfectly fit the data.

Values of R² outside the range 0 to 1 can occur when the model fits the data worse than a horizontal hyperplane. This would occur when the wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvålseth is used (this is the equation used most often), R² can be less than zero. If equation 2 of Kvålseth is used, R² can be greater than one.

In all instances where R² is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing SS_res. In this case R² increases as we increase the number of variables in the model (R² is monotone increasing with the number of variables included—i.e., it will never decrease). This illustrates a drawback to one possible use of R², where one might keep adding variables (Kitchen sink regression) to increase the R² value. For example, if one is trying to predict the sales of a model of car from the car's gas mileage, price, and engine power, one can include such irrelevant factors as the first letter of the model's name or the height of the lead engineer designing the car because the R² will never decrease as variables are added and will probably experience an increase due to chance alone.

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 non-simple linear model

Consider a linear model with more than a single explanatory variable, 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 $\varepsilon _{i}$ is a mean zero error term. The quantities $\beta _{0},\dots ,\beta _{p}$ are unknown coefficients, whose values are estimated 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: "Seventy percent of the variance in the response variable can be explained by the explanatory variables. The remaining thirty percent can be attributed to 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 sometimes provide valuable clues in uncovering causal relationships among variables, a non-zero estimated correlation between two variables is not, on its own, evidence that changing the value of one variable would result in changes in the values of other variables. For example, the practice of carrying matches (or a lighter) is correlated with incidence of lung cancer, but carrying matches does not cause cancer (in the standard sense of "cause").

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. With more than one regressor, the R² can be referred to as the coefficient of multiple determination.

Inflation of R²

In least squares regression, R² is weakly increasing with increases in the number of regressors in the model. Because increases in the number of regressors increase the value of R², R² alone cannot be used as a meaningful comparison of models with very 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, though this is not always appropriate. As a reminder of this, some authors denote R² by R_q², where q is the number of columns in X (the number of explanators including the constant).

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

$\min _{b}SS_{\text{res}}(b)\Rightarrow \min _{b}\sum _{i}(y_{i}-X_{i}b)^{2}\,$

where X_i is a row vector of values of explanatory variables for case i and b is a column vector of coefficients of the respective elements of X_i.

The optimal value of the objective is weakly smaller as more explanatory variables are added and hence additional columns of $X$ (the explanatory data matrix whose ith row is X_i) are added, by the fact that less constrained minimization leads to an optimal cost which is weakly smaller than more constrained minimization does. 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 $SS_{\text{res}}$ 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.

Caveats

R² does not indicate whether:

• the independent variables are a 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;
• there are enough data points to make a solid conclusion.

Extensions

Adjusted R²

See also: Effect size § Omega-squared (ω2)

The use of an adjusted R² (one common notation is ${\bar {R}}^{2}$, pronounced "R bar squared"; another is $R_{\text{adj}}^{2}$) is an attempt to take account of the phenomenon of the R² automatically and spuriously increasing when extra explanatory variables are added to the model. It is a modification due to Henri Theil of R² that adjusts for the number of explanatory terms in a model relative to the number of data points. The adjusted R² can be negative, and its value will always be less than or equal to that of R². Unlike R², the adjusted R² increases only when the increase in R² (due to the inclusion of a new explanatory variable) is more than one would expect to see by chance. If a set of explanatory variables with a predetermined hierarchy of importance are introduced into a regression one at a time, with the adjusted R² computed each time, the level at which adjusted R² reaches a maximum, and decreases afterward, would be the regression with the ideal combination of having the best fit without excess/unnecessary terms. The adjusted R² is defined as

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

where p is the total number of explanatory variables in the model (not including the constant term), and n is the sample size.

Adjusted R² can also be written as

${\bar {R}}^{2}={1-{SS_{\text{res}}/{\text{df}}_{e} \over SS_{\text{tot}}/{\text{df}}_{t}}}$

where 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

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

where ${\text{VAR}}_{\text{res}}=SS_{\text{res}}/n$ and ${\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/n$ are the sample variances of the estimated residuals and the dependent variable respectively, which can be seen as biased estimates of the population variances of the errors and of the dependent variable. These estimates are replaced by statistically unbiased versions: ${\text{VAR}}_{\text{res}}=SS_{\text{res}}/(n-p-1)$ and ${\text{VAR}}_{\text{tot}}=SS_{\text{tot}}/(n-1)$.

Adjusted R² can be interpreted as an unbiased (or less biased) estimator of the population R², whereas the observed sample R² is a positively biased estimate of the population value. Adjusted R² is more appropriate when evaluating model fit (the variance in the dependent variable accounted for by the independent variables) and in comparing alternative models in the feature selection stage of model building.

Coefficient of partial determination

See also: Partial correlation

The coefficient of partial determination can be defined as the proportion of variation that cannot be explained in a reduced model, but can be explained by the predictors specified in a full(er) model. This coefficient is used to provide insight into whether or not one or more additional predictors may be useful in a more fully specified regression model.

The calculation for the partial R² is relatively straightforward after estimating two models and generating the ANOVA tables for them. The calculation for the partial R² is

${\frac {SS_{\text{ res, reduced}}-SS_{\text{ res, full}}}{SS_{\text{ res, reduced}}}},$

which is analogous to the usual coefficient of determination:

${\frac {SS_{\text{tot}}-SS_{\text{res}}}{SS_{\text{tot}}}}.$

Generalizing and decomposing $R^{2}$

As explained above, model selection heuristics such as the Adjusted $R^{2}$ criterion and the F-test examine whether the total $R^{2}$ sufficiently increases to determine whether a new regressor should be added to the model. If a regressor is added to the model that is highly correlated with other regressors which have already been included, then the total $R^{2}$ will hardly increase even if the new regressor is of relevance. As a result, the above-mentioned heuristics will ignore relevant regressors when cross-correlations are high.

Geometric representation of $r^{2}$.

Alternatively, one can decompose a generalized version of $R^{2}$ to quantify the relevance of deviating from a hypothesis. As Hoornweg (2018) shows, several shrinkage estimators - such as Bayesian linear regression, ridge regression, and the (adaptive) lasso - make use of this decomposition of $R^{2}$ when they gradually shrink parameters from the unrestricted OLS solutions towards the hypothesized values. Let us first define the linear regression model as

$y=X\beta +\epsilon .$

It is assumed that the matrix $X$ is standardized with Z-scores and that the column vector $y$ is centered to have a mean of zero. Let the column vector $\beta _{0}$ refer to the hypothesized regression parameters and let the column vector $b$ denote the estimated parameters. We can then define

$R^{2}=1-{\frac {(y-Xb)'(y-Xb)}{(y-X\beta _{0})'(y-X\beta _{0})}}.$

An $R^{2}$ of 75% means that the in-sample accuracy improves by 75% if the data-optimized $b$ solutions are used instead of the hypothesized $\beta _{0}$ values. In the special case that $\beta _{0}$ is a vector of zeros, we obtain the traditional $R^{2}$ again.

The individual effect on $R^{2}$ of deviating from a hypothesis can be computed with $R^{\otimes }$ ('R-outer'). This $p$ times $p$ matrix is given by

$R^{\otimes }=(X'{\tilde {y}}_{0})(X'{\tilde {y}}_{0})'(X'X)^{-1}({\tilde {y}}_{0}'{\tilde {y}}_{0})^{-1},$

where ${\tilde {y}}_{0}=y-X\beta _{0}$. The diagonal elements of $R^{\otimes }$ exactly add up to $R^{2}$. If regressors are uncorrelated and $\beta _{0}$ is a vector of zeros, then the $j^{th}$ diagonal element of $R^{\otimes }$ simply corresponds to the $r^{2}$ value between $x_{j}$ and $y$. When regressors $x_{i}$ and $x_{j}$ are correlated, $R_{ii}^{\otimes }$ might increase at the cost of a decrease in $R_{jj}^{\otimes }$. As a result, the diagonal elements of $R^{\otimes }$ may be smaller than 0 and, in more exceptional cases, larger than 1. To deal with such uncertainties, several shrinkage estimators implicitly take a weighted average of the diagonal elements of $R^{\otimes }$ to quantify the relevance of deviating from a hypothesized value. Click on the lasso for an example.

$R^{2}$ in logistic regression

In the case of logistic regression, usually fit by maximum likelihood, there are several choices of pseudo-R².

One is the generalized R² originally proposed by Cox & Snell, and independently by Magee:

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

where ${\mathcal {L}}(0)$ is the likelihood of the model with only the intercept, ${{\mathcal {L}}({\hat {\theta }})}$ is the likelihood of the estimated model (i.e., the model with a given set of parameter estimates) and n is the sample size. It is easily rewritten to:

$R^{2}=1-e^{{\frac {2}{n}}(ln({\mathcal {L}}(0))-ln({\mathcal {L}}({\hat {\theta }}))}=1-e^{-{\frac {D}{n}}}$

where D is the test statistic of the likelihood ratio test.

Nagelkerke noted that it had the following properties:

1. It is consistent with the classical coefficient of determination when both can be computed;
2. Its value is maximised by the maximum likelihood estimation of a model;
3. It is asymptotically independent of the sample size;
4. The interpretation is the proportion of the variation explained by the model;
5. The values are 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 does not have any unit.

However, in the case of a logistic model, where ${\mathcal {L}}({\hat {\theta }})$ cannot be greater than 1, R² is between 0 and $R_{\max }^{2}=1-({\mathcal {L}}(0))^{2/n}$: thus, Nagelkerke suggested the possibility to define a scaled R² as R²/R²_max.

Comparison with norm of residuals

Occasionally, the norm of residuals is used for indicating goodness of fit. This term is calculated as the square-root of the sum of squared residuals:

${\text{norm of residuals}}={\sqrt {SS_{\text{res}}}}=\|e\|.$

Both R² and the norm of residuals have their relative merits. For least squares analysis R² varies between 0 and 1, with larger numbers indicating better fits and 1 representing a perfect fit. The norm of residuals varies from 0 to infinity with smaller numbers indicating better fits and zero indicating a perfect fit. One advantage and disadvantage of R² is the $SS_{\text{tot}}$ term acts to normalize the value. If the y_i values are all multiplied by a constant, the norm of residuals will also change by that constant but R² will stay the same. As a basic example, for the linear least squares fit to the set of data:

$x=1,\ 2,\ 3,\ 4,\ 5$

$y=1.9,\ 3.7,\ 5.8,\ 8.0,\ 9.6$

R² = 0.998, and norm of residuals = 0.302. If all values of y are multiplied by 1000 (for example, in an SI prefix change), then R² remains the same, but norm of residuals = 302.

Other single parameter indicators of fit include the standard deviation of the residuals, or the RMSE of the residuals. These would have values of 0.151 and 0.174 respectively for the above example given that the fit was linear with an unforced intercept.

History

The creation of the coefficient of determination has been attributed to the geneticist Sewall Wright and was first published in 1921.

See also

• Fraction of variance unexplained
• Goodness of fit
• Nash–Sutcliffe model efficiency coefficient (hydrological applications)
• Pearson product-moment correlation coefficient
• Proportional reduction in loss
• Regression model validation
• Root mean square deviation
• t-test of $H_{0}\colon R^{2}=0.$

References

• Gujarati, Damodar N.; Porter, Dawn C. (2009). Basic Econometrics (Fifth ed.). New York: McGraw-Hill/Irwin. pp. 73–78. ISBN 978-0-07-337577-9.
• Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 240–243. ISBN 978-0-02-365070-3.