In statistics, the coefficient of determination R² is the proportion of variability in a data set that is accounted for by a statistical model. This article is about the field of statistics. ...

There is no consensus about the exact definition of R². Only in the case of linear regression, all definitions are equivalent. In this case, R² is simply the square of a correlation coefficient. In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xi, i = 1, ..., p, and a random term ε. The model can be written as Example of linear regression with one dependent and one independent variable. ... In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...

Definitions

A data set has values y_i each of which has an associated modelled value f_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: Sum of squares is a concept that permeates much of inferential statistics and descriptive statistics. ...

, the total sum of squares (proportional to the sample variance);

, the regression sum of squares, also called the explained sum of squares.

, the sum of squared errors, also called the residual sum of squares.

In the above, and are the means of the observed data and modelled (predicted) values respectively.

Note: the notations SS_R and SS_E should be avoided because their meaning is exchanged in some texts.

The most general definition of the coefficient of determination is

.

Relation to unexplained variance

In the 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. In statistics, the fraction of variance unexplained (or FVU) in the context of a regression task is the amount of variance of the regressand Y which cannot be explained, i. ...

As explained variance

In some cases the total sum of squares equals the sum of the two other sums of squares defined above, The value of the total sum of squares [TSS] depends on the data being analyzed and the test that is being done. ...

.

Then, the above definition of R² is equivalent to

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). Explained variance is part of is part of the variance of any residual that can be be attributed to a specific condition (cause). ...

This equivalence holds for instance when the model values f_i have been obtained by linear regression. A milder sufficient condition reads as follows: The model has the form In statistics, linear regression is a regression method that models the relationship between a dependent variable Y, independent variables Xi, i = 1, ..., p, and a random term ε. The model can be written as Example of linear regression with one dependent and one independent variable. ... In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ...

where the g_i are arbitrary values that may or may not depend on i or on other free parameters (the common choice g_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: In statistics and optimization, the concepts of error and residual are easily confused with each other. ...

.

As squared correlation coefficient

Similarly, after least squares regression with a constant+linear model, R² equals the square of the correlation coefficient between the observed and modelled (predicted) data values. In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random variables. ...

Under general conditions, an R² value is sometimes calculated as the square of the correlation coefficient between the original and modelled data values. In this case, the value is not directly a measure of how good the modelled values are, but rather a measure of how good a predictor might be constructed from the modelled values (by creating a revised predictor of the form α+βf_i). According to Everitt (2002, p78), this usage is specifically the definition of the term "coefficient of determination": the square of the correlation between two (general) variables. In probability theory and statistics, correlation, also called correlation coefficient, is a numeric measure of the strength of linear relationship between two random 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. Goodness of fit means how well a statistical model fits a set of observations. ...

In some (but not all) instances where R² is used, the predictors are calculated by ordinary least-squares regression: that is, by minimising SS_err. In this case R-squared increases as we increase the number of variables in the model (R-squared 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 R-squared 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. However, the conclusion that that R-squared increases with extra variables no longer holds, but downward variations are usually small. 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 regression analysis, least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the residuals (the difference between the predicted and observed values) squared. ... Weighted least squares is a method of regression, similar to least squares in that it uses the same minimization of the sum of the residuals: However, instead of weighting all points equally, they are weighted such that points with a greater weight contribute more to the fit: Often, wi is... In statistics the linear model can be expressed by saying where Y is an n×1 column vector of random variables, X is an n×p matrix of known (i. ...

In a linear model

For expository purposes, consider a linear model of the form

where, for the i'th case, Y_i is the response variable, are p regressors, and is a mean zero error term. The quantities 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. In statistics and optimization, the concepts of error and residual are easily confused with each other. ... In regression analysis, least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the residuals (the difference between the predicted and observed values) squared. ... In statistics, an explanatory variable (also regressor or independent variable) is a variable in a regression model which appears on the right hand side of the equation. ...

More simply, 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 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." This article contains information that has not been verified and thus might not be reliable. ...

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. Several sets of (x, y) points, with the correlation coefficient of x and y for each set. ... Correlation does not imply causation is a phrase used in the sciences and statistics to emphasize that correlation between two variables does not imply there is a cause-and-effect relationship between the two. ...

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. In statistics, the Pearson product-moment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ...

Inflation of R²

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 independent variables. As a reminder of this, some authors denote R² by R²_p, where p is the number of columns in X In regression analysis, least squares, also known as ordinary least squares analysis, is a method for linear regression that determines the values of unknown quantities in a statistical model by minimizing the sum of the residuals (the difference between the predicted and observed values) squared. ...

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 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.

Notes on interpreting R²

R² does NOT tell 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
• the model might be improved by using transformed versions of the existing set of independent variables

In experimental design, a dependent variable (also known as response variable, responding variable or regressand) is a factor whose values in different treatment conditions are compared. ... Omitted-variable bias is the bias that appears in an estimate of a parameter if a regression run does not have the appropriate form and data for other parameters. ... In statistics, regression analysis examines the relation of a dependent variable (response variable) to specified independent variables (explanatory variables). ...

Adjusted R²

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 or equal to R². The adjusted R² is defined as In statistics, an explanatory variable (also regressor or independent variable) is a variable in a regression model which appears on the right hand side of the equation. ...

where p is the total number of regressors in the linear model (but not counting the constant term), and n is sample size.

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

where VAR_E = SS_E / n and VAR_T = SS_T / n are estimates of the variances of the errors and of the observations, respectively. These estimates are replaced by notionally "unbiased" versions: VAR_E = SS_E / (n − p − 1) and VAR_T = SS_T / (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. Feature selection, also known as variable selection, feature reduction, attribute selection or variable subset selection, is the technique, commonly used in machine learning, of selecting a subset of relevant features for building robust learning models. ...

Adjusted R² is not always better than R²: adjusted R² will be more useful only if the R² is calculated based on a sample, not the entire population. For example, if our unit of analysis is a state, and we have data for all counties, then adjusted R² will not yield any more useful information than R². The unit of analysis is the major entity that is being analyzing in the study. ... For other uses, see State (disambiguation). ...

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 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 R square have all the preceding 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, R² is between 0 and .

Thus, we define the maxed-rescaled R square . ^[1]

See also

• Goodness of fit
• Correlation
• Fraction of variance unexplained
• Pearson product-moment correlation coefficient
• Nash-Sutcliffe efficiency coefficient (Hydrological applications)

Goodness of fit means how well a statistical model fits a set of observations. ... Several sets of (x, y) points, with the correlation coefficient of x and y for each set. ... In statistics, the fraction of variance unexplained (or FVU) in the context of a regression task is the amount of variance of the regressand Y which cannot be explained, i. ... In statistics, the Pearson product-moment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ... Water covers 70% of the Earths surface. ...

Notes

1. ^ N. Nagelkerke, “A Note on a General Definition of the Coefficient of Determination,” Biometrika, vol. 78, no. 3, pp. 691-692, 1991.

References

External links

• Adjusted R-Square Calculator
• Rules for Cheaters: How to Get a High R squared