In
statisticsStatistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the coefficient of determination R
^{2} 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. It provides a measure of how well future outcomes are likely to be predicted by the model.
There are several different definitions of R
^{2} which are only sometimes equivalent. One class of such cases includes that of
linear regressionIn statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
. In this case, if an intercept is included then R
^{2} is simply the square of the sample
correlation coefficientIn statistics, the Pearson productmoment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
between the outcomes and their predicted values, or in the case of
simple linear regressionIn statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable. In other words, simple linear regression fits a straight line through the set of n points in such a way that makes the sum of squared residuals of the model as...
, 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 R
^{2} 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 modelfitting procedure using those data, and where linear regression is conducted without including an intercept. Additionally, negative values of R
^{2} may occur when fitting nonlinear trends to data. 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 fitThe goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g...
analysis.
Definitions
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 squaresThe partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion...
:
the
total sum of squaresIn statistical data analysis the total sum of squares is a quantity that appears as part of a standard way of presenting results of such analyses...
(proportional to the sample variance);
the regression sum of squares, also called the
explained sum of squaresIn statistics, the explained sum of squares is a quantity used in describing how well a model, often a regression model, represents the data being modelled...
.
, the sum of squares of residuals, also called the
residual sum of squaresIn statistics, the residual sum of squares is the sum of squares of residuals. It is also known as the sum of squared residuals or the sum of squared errors of prediction . It is a measure of the discrepancy between the data and an estimation model...
.
In the above
is the mean of the observed data:
where n is the number of observations.
The notations
and
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
Relation to unexplained variance
In a general form, R
^{2} 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 unexplainedIn statistics, the fraction of variance unexplained in the context of a regression task is the fraction of variance of the regressand Y which cannot be explained, i.e., which is not correctly predicted, by the explanatory variables X....
.
As explained variance
In some cases the
total sum of squaresIn statistical data analysis the total sum of squares is a quantity that appears as part of a standard way of presenting results of such analyses...
equals the sum of the two other sums of squares defined above,
See
sum of squaresThe partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion...
for a derivation of this result for one case where the relation holds. When this relation does hold, the above definition of R
^{2} is equivalent to
In this form R
^{2} 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 regressionIn statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
. A milder sufficient condition reads as follows: The model has the form
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
residualsIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
and the modelled values. In particular, under these conditions:
As squared correlation coefficient
Similarly, after least squares regression with a constant+linear model (i.e.,
simple linear regressionIn statistics, simple linear regression is the least squares estimator of a linear regression model with a single explanatory variable. In other words, simple linear regression fits a straight line through the set of n points in such a way that makes the sum of squared residuals of the model as...
), R
^{2} equals the square of the
correlation coefficientIn statistics, the Pearson productmoment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
between the observed and modeled (predicted) data values.
Under general conditions, an R
^{2} value is sometimes calculated as the square of the
correlation coefficientIn statistics, the Pearson productmoment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
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
R
^{2} is a statistic that will give some information about the
goodness of fitThe goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g...
of a model. In regression, the R
^{2} coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R
^{2} of 1.0 indicates that the regression line perfectly fits the data.
Values of R
^{2} 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 R
^{2} 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 R
^{2} is used, the predictors are calculated by ordinary leastsquares regression: that is, by minimizing SS
_{err}. In this case Rsquared increases as we increase the number of variables in the model (R
^{2} will not decrease). This illustrates a drawback to one possible use of R
^{2}, 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 R
^{2}. The explanation of this statistic is almost the same as R
^{2} but it penalizes the statistic as extra variables are included in the model. For cases other than fitting by ordinary least squares, the R
^{2} statistic can be calculated as above and may still be a useful measure. If fitting is by weighted least squares or
generalized least squaresIn statistics, generalized least squares is a technique for estimating the unknown parameters in a linear regression model. The GLS is applied when the variances of the observations are unequal , or when there is a certain degree of correlation between the observations...
, alternative versions of R
^{2} can be calculated appropriate to those statistical frameworks, while the "raw" R
^{2} may still be useful if it is more easily interpreted. Values for R
^{2} 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
where, for the ith case,
is the response variable,
are p regressors, and
is a mean zero
errorIn statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...
term. The quantities
are unknown coefficients, whose values are determined by
least squaresThe method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
. The coefficient of determination R
^{2} 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 Y
_{i} that may be attributed to some linear combination of the regressors (explanatory variables) in X.
R
^{2} 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
, while R
^{2} = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope=0, intercept=
) 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 variableIn statistics, a confounding variable is an extraneous variable in a statistical model that correlates with both the dependent variable and the independent variable...
s or inherent variability."
A caution that applies to R
^{2}, as to other statistical descriptions of
correlationIn statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....
and association is that "
correlation does not imply causation"Correlation does not imply causation" is a phrase used in science and statistics to emphasize that correlation between two variables does not automatically imply that one causes the other "Correlation does not imply causation" (related to "ignoring a common cause" and questionable cause) is a...
." 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, R
^{2} is the square of the
Pearson productmoment correlation coefficientIn statistics, the Pearson productmoment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
relating the regressor and the response variable. More generally, R
^{2} is the square of the correlation between the constructed predictor and the response variable.
Inflation of R^{2}
In
least squaresThe method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
regression, R
^{2} is weakly increasing in the number of regressors in the model. As such, R
^{2} alone cannot be used as a meaningful comparison of models with different numbers of independent variables. For a meaningful comparison between two models, an
FtestAn Ftest is any statistical test in which the test statistic has an Fdistribution under the null hypothesis.It is most often used when comparing statistical models that have been fit to a data set, in order to identify the model that best fits the population from which the data were sampled. ...
can be performed on the
residual sum of squaresIn statistics, the residual sum of squares is the sum of squares of residuals. It is also known as the sum of squared residuals or the sum of squared errors of prediction . It is a measure of the discrepancy between the data and an estimation model...
, similar to the Ftests in
Granger causalityThe Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another. Ordinarily, regressions reflect "mere" correlations, but Clive Granger, who won a Nobel Prize in Economics, argued that there is an interpretation of a set of tests...
. As a reminder of this, some authors denote R
^{2} by R
^{2}_{p}, where p is the number of columns in X
To demonstrate this property, first recall that the objective of least squares regression is:
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 nondecreasing property of R
^{2} follows directly from the definition above.
The intuitive reason that using an additional explanatory variable cannot lower the R
^{2} is this: Minimizing
is equivalent to maximizing R
^{2}. 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 R
^{2} unchanged. The only way that the optimization problem will give a nonzero coefficient is if doing so improves the R
^{2}.
Adjusted R^{2}
Adjusted R
^{2} (often written as
and pronounced "R bar squared") is a modification due to Theil of R
^{2} that adjusts for the number of explanatory terms in a model. Unlike R
^{2}, the adjusted R
^{2} increases only if the new term improves the model more than would be expected by chance. The adjusted R
^{2} can be negative, and will always be less than or equal to R
^{2}. The adjusted R
^{2} is defined as
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 freedomIn statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...
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 R
^{2} statistic can be seen by rewriting the ordinary R
^{2} as
where
and
are estimates of the variances of the errors and of the observations, respectively. These estimates are replaced by statistically unbiased versions:
and
.
Adjusted R
^{2} does not have the same interpretation as R
^{2}. As such, care must be taken in interpreting and reporting this statistic. Adjusted R
^{2} is particularly useful in the
Feature selectionIn machine learning and statistics, feature selection, also known as variable selection, feature reduction, attribute selection or variable subset selection, is the technique of selecting a subset of relevant features for building robust learning models...
stage of model building..
The use of an adjusted R
^{2} is an attempt to take account of the phenomenon of statistical
shrinkageIn statistics, shrinkage has two meanings:*In relation to the general observation that, in regression analysis, a fitted relationship appears to perform less well on a new data set than on the data set used for fitting. In particular the value of the coefficient of determination 'shrinks'...
.
Generalized R^{2}
Nagelkerke (1991) generalizes the definition of the coefficient of determination:
 A generalized coefficient of determination should be consistent with the classical coefficient of determination when both can be computed;
 Its value should also be maximised by the maximum likelihood estimation of a model;
 It should be, at least asymptotically, independent of the sample size;
 Its interpretation should be the proportion of the variation explained by the model;
 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;
 It should not have any unit.
The generalized R² has all of these properties.

where L(0) is the likelihood of the model with only the intercept,
is the likelihood of the estimated model and n is the sample size.
However, in the case of a logistic model, where
cannot be greater than 1, R² is between 0 and
: thus, it is possible to define a scaled R² as R²/R²
_{max}.
See also
 Goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g...
 Fraction of variance unexplained
In statistics, the fraction of variance unexplained in the context of a regression task is the fraction of variance of the regressand Y which cannot be explained, i.e., which is not correctly predicted, by the explanatory variables X....
 Pearson productmoment correlation coefficient
In statistics, the Pearson productmoment correlation coefficient is a measure of the correlation between two variables X and Y, giving a value between +1 and −1 inclusive...
 Nash–Sutcliffe model efficiency coefficient (hydrological applications
Hydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...
)
 Regression model validation
 Proportional reduction in loss
Proportional reduction in loss refers to a general framework for developing and evaluating measures of the reliability of particular ways of making observations which are possibly subject to errors of all types...
 Root mean square deviation
The rootmeansquare deviation is the measure of the average distance between the atoms of superimposed proteins...
 Multiple correlation
In statistics, multiple correlation is a linear relationship among more than two variables. It is measured by the coefficient of multiple determination, denoted as R2, which is a measure of the fit of a linear regression...