In statistics, the coefficient of determination, denoted R2 or r2 and pronounced R squared,
indicates how well data fit a statistical model – sometimes simply a line or curve.
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, as 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. 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. If additional explanators 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. Important cases where the computational definition
of R2 can yield negative values, depending on the definition used, arise where the predictions
that 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 functions
to 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.
Definitions
A data set has values yi, each of which has an associated modelled value fi. Here, the
values yi are called the observed values and the modelled values fi are sometimes called
the predicted values. In what follows is the mean of the observed
data:
where n is the number of observations. The "variability" of the data set is measured
through different sums of squares: the total sum of squares;
the regression sum of squares, also called the explained sum of squares.
, the sum of squares of residuals, also called the residual sum of squares.
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, R2 can be seen to be related
to the unexplained variance, since the second term compares the unexplained variance with
the total variance. 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,
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
In this form R2 is expressed as the ratio of the explained variance to the total variance.
This partition of the sum of squares holds for instance when the model values ƒi have
been obtained by linear regression. A milder sufficient condition reads as follows: The
model has the form
where the qi are arbitrary values that may or may not depend on i or on other free parameters,
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:
As squared correlation coefficient Similarly, in linear least squares regression
with an estimated intercept term, R2 equals the square of the Pearson correlation coefficient
between the observed and modeled data values of the dependent variable.
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 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.
According to Everitt, this usage is specifically the definition of the term "coefficient of
determination": the square of the correlation between two 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 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 modeled values and where the "modeled" 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 be greater than one. If the second expression is used, there are no
constraints on the values obtainable. In many instances where R2 is used, the predictors
are calculated by ordinary least-squares regression: that is, by minimizing SSerr. In this case
R2 increases as we increase the number of variables in the model. This illustrates a
drawback to one possible use of R2, where one might keep adding variables 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 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 error
term. The quantities 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 Yi that may
be attributed to some linear combination of the regressors 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 , while R2 = 0 indicates no 'linear' relationship between the response variable
and regressors). An interior value such as R2 = 0.7 may be interpreted as follows:
"Seventy percent of the variation 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 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. 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 Rp2, where p is the number of columns in X.
To demonstrate this property, first recall that the objective of least squares linear
regression is:
The optimal value of the objective is weakly smaller as additional columns of 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 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 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 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. Adjusted R2
The use of an adjusted R2 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 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
when a new explanator is included only if the new explanator improves the R2 more than
would be expected 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
where p is the total number of regressors in the linear model, and n is the sample size.
Adjusted R2 can also be written as
where dft is the degrees of freedom n– 1 of the estimate of the population variance
of the dependent variable, and dfe 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
where and 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: and . Adjusted R2 does not have the same interpretation
as R2—while R2 is a measure of fit, adjusted R2 is instead a comparative measure of suitability
of alternative nested sets of explanators. 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. Generalized R2
The generalized R² was originally proposed by Cox & Snell:
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.
Nagelkerke noted that it had the following properties:
It's consistent with the classical coefficient of determination when both can be computed;
Its value is maximised by the maximum likelihood estimation of a model;
It is asymptotically independent of the sample size;
The interpretation is the proportion of the variation explained by the model;
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;
It does not have any unit. However, in the case of a logistic model,
where cannot be greater than 1, R² is between 0 and : thus, Nagelkerke suggests the possibility
to define a scaled R² as R²/R²max. Comparison with norm of residuals
Occasionally the norm of residuals is used for indicating goodness of fit. This term
is encountered in MATLAB and is calculated by
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 represents a
perfect fit. 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 term acts to normalize the value. If the yi 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:
R2 = 0.997, and norm of residuals = 0.302. If all values of y are multiplied by 1000,
then R2 remains the same, but norm of residuals = 302.
See also Fraction of variance unexplained
Goodness of fit Nash–Sutcliffe model efficiency coefficient
Pearson product-moment correlation coefficient Proportional reduction in loss
Regression model validation Root mean square deviation
Notes
References