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Instatistics, thecoefficient of determination, denotedR² orr² and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).It is astatistic used in the context ofstatistical models whose main purpose is either theprediction of future outcomes or the testing ofhypotheses, 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.^[1][2][3]

There are several definitions ofR² that are only sometimes equivalent. Insimple linear regression (which includes anintercept),r² is simply the square of the samplecorrelation coefficient (r), between the observed outcomes and the observed predictor values.^[4] If additionalregressors are included,R² is the square of thecoefficient of multiple correlation. In both such cases, the coefficient of determination normally ranges from 0 to 1.

There are cases whereR² can yield negative values. 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,^[5] or when a non-linear function is used to fit the data.^[6] 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.

The coefficient of determination can be more intuitively informative thanMAE,MAPE,MSE, andRMSE inregression analysis evaluation, as the former can be expressed as a percentage, whereas the latter measures have arbitrary ranges. It also proved more robust for poor fits compared toSMAPE on certain test datasets.^[7]

When evaluating the goodness-of-fit of simulated (Y_pred) versus measured (Y_obs) values, it is not appropriate to base this on theR² of the linear regression (i.e.,Y_obs=m·Y_pred + b).^{[citation needed]} TheR² quantifies the degree of any linear correlation betweenY_obs andY_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).^[8][9]

Adata set hasn values markedy₁, ...,y_n (collectively known asy_i or as a vectory = [y₁, ...,y_n]^T), each associated with a fitted (or modeled, or predicted) valuef₁, ...,f_n (known asf_i, or sometimesŷ_i, as a vectorf).

Define theresiduals ase_i =y_i −f_i (forming a vectore).

If${\displaystyle \overline{y}}$ is the mean of the observed data:$${\displaystyle \overline{y}=\frac{1}{n}\sum _{i=1}^n{y}_i}$$then the variability of the data set can be measured with twosums of squares formulas:

• The sum of squares of residuals, also called theresidual sum of squares:$${\displaystyle S{S}_{\text{res}}=\sum _i(y_i-f_i{)}^2=\sum _i{e}_i^2}$$
• Thetotal sum of squares (proportional to thevariance of the data):$${\displaystyle S{S}_{\text{tot}}=\sum _i(y_i-\overline{y}{)}^2}$$

The most general definition of the coefficient of determination is$${\displaystyle R^2=1-\frac{S{S}_{\mathrm{r}\mathrm{e}\mathrm{s}}}{S{S}_{\mathrm{t}\mathrm{o}\mathrm{t}}}}$$

In the best case, the modeled values exactly match the observed values, which results in${\displaystyle S{S}_{\text{res}}=0}$ andR² = 1. A baseline model, which always predictsy, will haveR² = 0.

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):$${\displaystyle R^2=1-\text{FVU}}$$

A larger value ofR² implies a more successful regression model.^[4]: 463SupposeR² = 0.49. This implies that 49% of the variability of the dependent variable in the data set has been accounted for, and the remaining 51% of the variability is still unaccounted for. For regression models, the regression sum of squares, also called theexplained sum of squares, is defined as

${\displaystyle S{S}_{\text{reg}}=\sum _i(f_i-\overline{y}{)}^2}$

In some cases, as insimple linear regression, thetotal sum of squares equals the sum of the two other sums of squares defined above:

${\displaystyle S{S}_{\text{res}}+S{S}_{\text{reg}}=S{S}_{\text{tot}}}$

SeePartitioning 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 ofR² is equivalent to

${\displaystyle R^2=\frac{S{S}_{\text{reg}}}{S{S}_{\text{tot}}}=\frac{S{S}_{\text{reg}}/n}{S{S}_{\text{tot}}/n}}$

wheren is the number of observations (cases) on the variables.

In this formR² is expressed as the ratio of theexplained variance (variance of the model's predictions, which isSS_reg /n) to the total variance (sample variance of the dependent variable, which isSS_tot /n).

This partition of the sum of squares holds for instance when the model valuesƒ_i have been obtained bylinear regression. A mildersufficient condition reads as follows: The model has the form

${\displaystyle f_i=\hat{\alpha }+\hat{\beta }{q}_i}$

where theq_i are arbitrary values that may or may not depend oni or on other free parameters (the common choiceq_i = x_i is just one special case), and the coefficient estimates${\displaystyle \hat{\alpha }}$ and${\displaystyle \hat{\beta }}$ 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 fittedresiduals and the modelled values. In particular, under these conditions:

${\displaystyle \overline{f}=\overline{y}.}$

In linear least squaresmultiple regression (with fitted intercept and slope),R² equals${\displaystyle {\rho }^2(y,f)}$ the square of thePearson correlation coefficient between the observed${\displaystyle y}$ and modeled (predicted)${\displaystyle f}$ data values of the dependent variable.

In alinear least squares regression with a single explanator (with fitted intercept and slope), this is also equal to${\displaystyle {\rho }^2(y,x)}$ the squared Pearson correlation coefficient between the dependent variable${\displaystyle y}$ and explanatory variable${\displaystyle x}$.

It should not be confused with the correlation coefficient between twoexplanatory variables, defined as

${\displaystyle {\rho }_{\hat{\alpha },\hat{\beta }}=\frac{\mathrm{cov}\left(\hat{\alpha },\hat{\beta }\right)}{{\sigma }_{\hat{\alpha }}{\sigma }_{\hat{\beta }}},}$

where the covariance between two coefficient estimates, as well as theirstandard deviations, are obtained from thecovariance matrix of the coefficient estimates,${\displaystyle (X^{T}X{)}^{-1}}$.

Under more general modeling conditions, where the predicted values might be generated from a model different from linear least squares regression, anR² value can be calculated as the square of thecorrelation coefficient between the original${\displaystyle y}$ and modeled${\displaystyle f}$ 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 formα +βƒ_i).^{[citation needed]} According to Everitt,^[10] this usage is specifically the definition of the term "coefficient of determination": the square of the correlation between two (general) variables.

R² is a measure of thegoodness of fit of a model.^[11] In regression, theR² coefficient of determination is a statistical measure of how well the regression predictions approximate the real data points. AnR² of 1 indicates that the regression predictions perfectly fit the data.

Values ofR² outside the range 0 to 1 occur when the model fits the data worse than the worst possibleleast-squares predictor (equivalent to a horizontal hyperplane at a height equal to the mean of the observed data). This occurs when a wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvålseth^[12] 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 whereR² is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizingSS_res. In this case,R² increases as the number of variables in the model is increased (R² ismonotone increasing with the number of variables included—it will never decrease). This illustrates a drawback to one possible use ofR², where one might keep adding variables (kitchen sink regression) to increase theR² 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 probably irrelevant factors such as the first letter of the model's name or the height of the lead engineer designing the car because theR² will never decrease as variables are added and will likely experience an increase due to chance alone.

This leads to the alternative approach of looking at theadjustedR². The explanation of this statistic is almost the same asR² but it penalizes the statistic as extra variables are included in the model. For cases other than fitting by ordinary least squares, theR² statistic can be calculated as above and may still be a useful measure. If fitting is byweighted least squares orgeneralized least squares, alternative versions ofR² can be calculated appropriate to those statistical frameworks, while the "raw"R² may still be useful if it is more easily interpreted. Values forR² can be calculated for any type of predictive model, which need not have a statistical basis.

Consider a linear model withmore than a single explanatory variable, of the form

${\displaystyle Y_i={\beta }_0+\sum _{j=1}^p{\beta }_j{X}_{i,j}+{\epsilon }_i,}$

where, for theith case,${\displaystyle Y_i}$ is the response variable,${\displaystyle X_{i,1},\dots ,X_{i,p}}$ arep regressors, and${\displaystyle {\epsilon }_i}$ is a mean zeroerror term. The quantities${\displaystyle {\beta }_0,\dots ,{\beta }_p}$ are unknown coefficients, whose values are estimated byleast squares. The coefficient of determinationR² is a measure of the global fit of the model. Specifically,R² is an element of [0, 1] and represents the proportion of variability inY_i that may be attributed to some linear combination of the regressors (explanatory variables) inX.^[13]

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${\displaystyle y}$, whileR² = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope = 0, intercept = ${\displaystyle \overline{y}}$) between the response variable and regressors). An interior value such asR² = 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 toR², as to other statistical descriptions ofcorrelation 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 thePearson 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, theR² can be referred to as thecoefficient of multiple determination.

Inleast squares regression using typical data,R² is at least weakly increasing with an increase in number of regressors in the model. Because increases in the number of regressors increase the value ofR²,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, anF-test can be performed on theresidual sum of squares^{[citation needed]}, similar to the F-tests inGranger causality, though this is not always appropriate^{[further explanation needed]}. As a reminder of this, some authors denoteR² byR_q², whereq is the number of columns inX (the number of explanators including the constant).

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

${\displaystyle \underset{b}{min}S{S}_{\text{res}}(b)\Rightarrow \underset{b}{min}\sum _i(y_i-X_{i}b{)}^2}$

whereX_i is a row vector of values of explanatory variables for casei andb is a column vector of coefficients of the respective elements ofX_i.

The optimal value of the objective is weakly smaller as more explanatory variables are added and hence additional columns of${\displaystyle X}$ (the explanatory data matrix whoseith row isX_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${\displaystyle S{S}_{tot}}$ depends only ony, the non-decreasing property ofR² follows directly from the definition above.

The intuitive reason that using an additional explanatory variable cannot lower theR² is this: Minimizing${\displaystyle S{S}_{\text{res}}}$ is equivalent to maximizingR². 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 theR² unchanged. The only way that the optimization problem will give a non-zero coefficient is if doing so improves the R².

The above gives an analytical explanation of the inflation ofR². Next, an example based on ordinary least square from a geometric perspective is shown below.^[14]

A simple case to be considered first:

${\displaystyle Y={\beta }_0+{\beta }_1\cdot X_1+\epsilon }$

This equation describes theordinary least squares regression model with one regressor. The prediction is shown as the red vector in the figure on the right. Geometrically, it is the projection of true value onto a model space in${\displaystyle \mathbb{R}}$ (without intercept). The residual is shown as the red line.

${\displaystyle Y={\beta }_0+{\beta }_1\cdot X_1+{\beta }_2\cdot X_2+\epsilon }$

This equation corresponds to the ordinary least squares regression model with two regressors. The prediction is shown as the blue vector in the figure on the right. Geometrically, it is the projection of true value onto a larger model space in${\displaystyle {\mathbb{R}}^2}$ (without intercept). Noticeably, the values of${\displaystyle {\beta }_0}$ and${\displaystyle {\beta }_0}$ are not the same as in the equation for smaller model space as long as${\displaystyle X_1}$ and${\displaystyle X_2}$ are not zero vectors. Therefore, the equations are expected to yield different predictions (i.e., the blue vector is expected to be different from the red vector). The least squares regression criterion ensures that the residual is minimized. In the figure, the blue line representing the residual is orthogonal to the model space in${\displaystyle {\mathbb{R}}^2}$, giving the minimal distance from the space.

The smaller model space is a subspace of the larger one, and thereby the residual of the smaller model is guaranteed to be larger. Comparing the red and blue lines in the figure, the blue line is orthogonal to the space, and any other line would be larger than the blue one. Considering the calculation forR², a smaller value of${\displaystyle S{S}_{tot}}$ will lead to a larger value ofR², meaning that adding regressors will result in inflation ofR².

R² does not indicate whether:

• the independent variables are a cause of the changes in thedependent variable;
• omitted-variable bias exists;
• the correctregression was used;
• the most appropriate set of independent variables has been chosen;
• there iscollinearity 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;
• there are a fewoutliers in an otherwise good sample.

The use of an adjustedR² (one common notation is${\displaystyle {\overline{R}}^2}$, pronounced "R bar squared"; another is${\displaystyle R_{\text{a}}^2}$ or${\displaystyle R_{\text{adj}}^2}$) is an attempt to account for the phenomenon of theR² automatically increasing when extra explanatory variables are added to the model. There are many different ways of adjusting.^[15] By far the most used one, to the point that it is typically just referred to as adjustedR, is the correction proposed byMordecai Ezekiel.^[15][16][17] The adjustedR² is defined as

${\displaystyle {\overline{R}}^2=1-\frac{S{S}_{\text{res}}/{\text{df}}_{\text{res}}}{S{S}_{\text{tot}}/{\text{df}}_{\text{tot}}}}$

where df_res is thedegrees of freedom of the estimate of the population variance around the model, and df_tot is the degrees of freedom of the estimate of the population variance around the mean. df_res is given in terms of the sample sizen and the number of variablesp in the model,df_res =n −p − 1. df_tot is given in the same way, but withp being zero for the mean (i.e.,df_tot =n − 1).

Inserting the degrees of freedom and using the definition ofR², it can be rewritten as:

${\displaystyle {\overline{R}}^2=1-(1-R^2)\frac{n-1}{n-p-1}}$

wherep is the total number of explanatory variables in the model (excluding the intercept), andn is the sample size.

The adjustedR² can be negative, and its value will always be less than or equal to that ofR². UnlikeR², the adjustedR² increases only when the increase inR² (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 adjustedR² computed each time, the level at which adjustedR² reaches a maximum, and decreases afterward, would be the regression with the ideal combination of having the best fit without excess/unnecessary terms.

The adjustedR² can be interpreted as an instance of thebias-variance tradeoff. When we consider the performance of a model, a lower error represents a better performance. When the model becomes more complex, the variance will increase whereas the square of bias will decrease, and these two metrics add up to be the total error. Combining these two trends, the bias-variance tradeoff describes a relationship between the performance of the model and its complexity, which is shown as a u-shape curve on the right. For the adjustedR² specifically, the model complexity (i.e. number of parameters) affects theR² and the term / frac and thereby captures their attributes in the overall performance of the model.

R² can be interpreted as the variance of the model, which is influenced by the model complexity. A highR² indicates a lower bias error because the model can better explain the change of Y with predictors. For this reason, we make fewer (erroneous) assumptions, and this results in a lower bias error. Meanwhile, to accommodate fewer assumptions, the model tends to be more complex. Based on bias-variance tradeoff, a higher complexity will lead to a decrease in bias and a better performance (below the optimal line). InR², the term (1 −R²) will be lower with high complexity and resulting in a higherR², consistently indicating a better performance.

On the other hand, the term/frac term is reversely affected by the model complexity. The term/frac will increase when adding regressors (i.e., increased model complexity) and lead to worse performance. Based on bias-variance tradeoff, a higher model complexity (beyond the optimal line) leads to increasing errors and a worse performance.

Considering the calculation ofR², more parameters will increase theR² and lead to an increase inR². Nevertheless, adding more parameters will increase the term/frac and thus decreaseR². These two trends construct a reverse u-shape relationship between model complexity andR², which is in consistent with the u-shape trend of model complexity versus overall performance. UnlikeR², which will always increase when model complexity increases,R² will increase only when the bias eliminated by the added regressor is greater than the variance introduced simultaneously. UsingR² instead ofR² could thereby prevent overfitting.

Following the same logic, adjustedR² can be interpreted as a less biased estimator of the populationR², whereas the observed sampleR² is a positively biased estimate of the population value.^[18] AdjustedR² 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 thefeature selection stage of model building.^[18]

The principle behind the adjustedR² statistic can be seen by rewriting the ordinaryR² as

${\displaystyle R^2=1-\frac{{\text{VAR}}_{\text{res}}}{{\text{VAR}}_{\text{tot}}}}$

where${\displaystyle {\text{VAR}}_{\text{res}}=S{S}_{\text{res}}/n}$ and${\displaystyle {\text{VAR}}_{\text{tot}}=S{S}_{\text{tot}}/n}$ 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 statisticallyunbiased versions:${\displaystyle {\text{VAR}}_{\text{res}}=S{S}_{\text{res}}/(n-p)}$ and${\displaystyle {\text{VAR}}_{\text{tot}}=S{S}_{\text{tot}}/(n-1)}$.

Despite using unbiased estimators for the population variances of the error and the dependent variable, adjustedR² is not an unbiased estimator of the populationR²,^[18] which results by using the population variances of the errors and the dependent variable instead of estimating them.Ingram Olkin andJohn W. Pratt derived theminimum-variance unbiased estimator for the populationR²,^[19] which is known as Olkin–Pratt estimator. Comparisons of different approaches for adjustingR² concluded that in most situations either an approximate version of the Olkin–Pratt estimator^[18] or the exact Olkin–Pratt estimator^[20] should be preferred over (Ezekiel) adjustedR².

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 model.^[21][22][23] 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 partialR² is relatively straightforward after estimating two models and generating theANOVA tables for them. The calculation for the partialR² is

${\displaystyle \frac{S{S}_{\text{ res, reduced}}-S{S}_{\text{ res, full}}}{S{S}_{\text{ res, reduced}}},}$

which is analogous to the usual coefficient of determination:

${\displaystyle \frac{S{S}_{\text{tot}}-S{S}_{\text{res}}}{S{S}_{\text{tot}}}.}$

As explained above, model selection heuristics such as the adjustedR² criterion and theF-test examine whether the totalR² sufficiently increases to determine if 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 totalR² 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.^[24]

Alternatively, one can decompose a generalized version ofR² to quantify the relevance of deviating from a hypothesis.^[24] As Hoornweg (2018) shows, severalshrinkage estimators – such asBayesian linear regression,ridge regression, and the (adaptive)lasso – make use of this decomposition ofR² when they gradually shrink parameters from the unrestricted OLS solutions towards the hypothesized values. Let us first define the linear regression model as

${\displaystyle y=X\beta +\epsilon .}$

It is assumed that the matrixX is standardized with Z-scores and that the column vector${\displaystyle y}$ is centered to have a mean of zero. Let the column vector${\displaystyle {\beta }_0}$ refer to the hypothesized regression parameters and let the column vector${\displaystyle b}$ denote the estimated parameters. We can then define

${\displaystyle R^2=1-\frac{(y-Xb{)}^{\prime }(y-Xb)}{(y-X{\beta }_0{)}^{\prime }(y-X{\beta }_0)}.}$

AnR² of 75% means that the in-sample accuracy improves by 75% if the data-optimizedb solutions are used instead of the hypothesized${\displaystyle {\beta }_0}$ values. In the special case that${\displaystyle {\beta }_0}$ is a vector of zeros, we obtain the traditionalR² again.

The individual effect onR² of deviating from a hypothesis can be computed with${\displaystyle R^{\otimes }}$ ('R-outer'). This${\displaystyle p}$ times${\displaystyle p}$ matrix is given by

${\displaystyle R^{\otimes }=(X^{\prime }{\tilde{y}}_0)(X^{\prime }{\tilde{y}}_0{)}^{\prime }(X^{\prime }X{)}^{-1}({\tilde{y}}_0^{\prime }{\tilde{y}}_0{)}^{-1},}$

where${\displaystyle {\tilde{y}}_0=y-X{\beta }_0}$. The diagonal elements of${\displaystyle R^{\otimes }}$ exactly add up toR². If regressors are uncorrelated and${\displaystyle {\beta }_0}$ is a vector of zeros, then the${\displaystyle j^{\text{th}}}$ diagonal element of${\displaystyle R^{\otimes }}$ simply corresponds to ther² value between${\displaystyle x_j}$ and${\displaystyle y}$. When regressors${\displaystyle x_i}$ and${\displaystyle x_j}$ are correlated,${\displaystyle R_{ii}^{\otimes }}$ might increase at the cost of a decrease in${\displaystyle R_{jj}^{\otimes }}$. As a result, the diagonal elements of${\displaystyle R^{\otimes }}$ 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${\displaystyle R^{\otimes }}$ to quantify the relevance of deviating from a hypothesized value.^[24] Click on thelasso for an example.

In the case oflogistic regression, usually fit bymaximum likelihood, there are several choices ofpseudo-R².

One is the generalizedR² originally proposed by Cox & Snell,^[25] and independently by Magee:^[26]

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

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

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

whereD is the test statistic of thelikelihood ratio test.

Nico Nagelkerke noted that it had the following properties:^[27][22]

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${\displaystyle \mathcal{L}(\hat{\theta })}$ cannot be greater than 1,R² is between 0 and${\displaystyle R_{max}^2=1-(\mathcal{L}(0){)}^{2/n}}$: thus, Nagelkerke suggested the possibility to define a scaledR² asR²/R²_max.^[22]

Occasionally, residual statistics are used for indicating goodness of fit. Thenorm of residuals is calculated as the square-root of thesum of squares of residuals (SSR):

${\displaystyle \text{norm of residuals}=\sqrt{S{S}_{\text{res}}}=\Vert e\Vert .}$

Similarly, thereduced chi-square is calculated as the SSR divided by the degrees of freedom.

BothR² and the norm of residuals have their relative merits. Forleast squares analysisR² 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 ofR² is the${\displaystyle S{S}_{\text{tot}}}$ term acts tonormalize the value. If they_i values are all multiplied by a constant, the norm of residuals will also change by that constant butR² 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 ofy are multiplied by 1000 (for example, in anSI prefix change), thenR² remains the same, but norm of residuals = 302.

Another single-parameter indicator of fit is theRMSE of the residuals, or standard deviation of the residuals. This would have a value of 0.135 for the above example given that the fit was linear with an unforced intercept.^[28]

The creation of the coefficient of determination has been attributed to the geneticistSewall Wright and was first published in 1921.^[29]

• Anscombe's quartet
• 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
• Stepwise regression

• 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.
• Hughes, Ann; Grawoig, Dennis (1971).Statistics: A Foundation for Analysis. Reading: Addison-Wesley. pp. 344–348.ISBN 0-201-03021-7.
• Kmenta, Jan (1986).Elements of Econometrics (Second ed.). New York: Macmillan. pp. 240–243.ISBN 978-0-02-365070-3.
• Lewis-Beck, Michael S.; Skalaban, Andrew (1990). "TheR-Squared: Some Straight Talk".Political Analysis.2:153–171.doi:10.1093/pan/2.1.153.JSTOR 23317769.
• Chicco, Davide; Warrens, Matthijs J.; Jurman, Giuseppe (2021)."The coefficient of determination R-squared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation".PeerJ Computer Science.7 (e623): e623.doi:10.7717/peerj-cs.623.PMC 8279135.PMID 34307865.

• Regression diagnostics
• Statistical ratios
• Least squares

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