# Coefficient of determination

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

## Contents

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. [1] [2] [3]

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 only 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. [4] If additional regressors are included, R2 is the square of the coefficient of multiple correlation. In both such cases, the coefficient of determination normally 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, [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 than MAE, MAPE, MSE, and RMSE in regression 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 to SMAPE on the test datasets in the article. [7]

When evaluating the goodness-of-fit of simulated (Ypred) vs. measured (Yobs) values, it is not appropriate to base this on the R2 of the linear regression (i.e., Yobs= m·Ypred + b).[ citation needed ] The R2 quantifies the degree of any linear correlation between Yobs and Ypred, while for the goodness-of-fit evaluation only one specific linear correlation should be taken into consideration: Yobs = 1·Ypred + 0 (i.e., the 1:1 line). [8] [9]

## Definitions

A data set has n values marked y1,...,yn (collectively known as yi or as a vector y = [y1,...,yn]T), each associated with a fitted (or modeled, or predicted) value f1,...,fn (known as fi, or sometimes ŷi, as a vector f).

Define the residuals as ei = yifi (forming a vector e).

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

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

then the variability of the data set can be measured with two sums of squares formulas:

• The sum of squares of residuals, also called the residual sum of squares:
${\displaystyle SS_{\text{res}}=\sum _{i}(y_{i}-f_{i})^{2}=\sum _{i}e_{i}^{2}\,}$
• The total sum of squares (proportional to the variance of the data):
${\displaystyle SS_{\text{tot}}=\sum _{i}(y_{i}-{\bar {y}})^{2}}$

The most general definition of the coefficient of determination is

${\displaystyle R^{2}=1-{SS_{\rm {res}} \over SS_{\rm {tot}}}}$

In the best case, the modeled values exactly match the observed values, which results in ${\displaystyle SS_{\text{res}}=0}$ and ${\displaystyle R^{2}=1}$. A baseline model, which always predicts ${\displaystyle {\bar {y}}}$, will have ${\displaystyle R^{2}=0}$. Models that have worse predictions than this baseline will have a negative ${\displaystyle R^{2}}$.

### Relation to unexplained variance

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):

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

### As explained variance

Suppose R2 = 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 the explained sum of squares, is defined as

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

In some cases, as in simple linear regression, the total sum of squares equals the sum of the two other sums of squares defined above:

${\displaystyle SS_{\text{res}}+SS_{\text{reg}}=SS_{\text{tot}}}$

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

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

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 SSreg / n) to the total variance (sample variance of the dependent variable, which is SStot / n).

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

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

where the qi are arbitrary values that may or may not depend on i or on other free parameters (the common choice qi = xi is just one special case), and the coefficient estimates ${\displaystyle {\widehat {\alpha }}}$ and ${\displaystyle {\widehat {\beta }}}$ 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:

${\displaystyle {\bar {f}}={\bar {y}}.\,}$

### 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 ${\displaystyle y}$ and modeled (predicted) ${\displaystyle f}$ 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 ${\displaystyle y}$ and explanatory variable ${\displaystyle x.}$

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

${\displaystyle \rho _{{\widehat {\alpha }},{\widehat {\beta }}}={\operatorname {cov} \left({\widehat {\alpha }},{\widehat {\beta }}\right) \over \sigma _{\widehat {\alpha }}\sigma _{\widehat {\beta }}},}$

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 ${\displaystyle y}$ 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 form α + βƒi).[ 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.

## Interpretation

R2 is a measure of the goodness of fit of a model. [11] 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 occur when the model fits the data worse than the worst possible least-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 where R2 is used, the predictors are calculated by ordinary least-squares regression: that is, by minimizing SSres. In this case, R2 increases as the number of variables in the model is increased (R2 is monotone increasing with the number of variables included—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 likely 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 multiple linear model

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

${\displaystyle Y_{i}=\beta _{0}+\sum _{j=1}^{p}\beta _{j}X_{i,j}+\varepsilon _{i},}$

where, for the ith case, ${\displaystyle {Y_{i}}}$ is the response variable, ${\displaystyle X_{i,1},\dots ,X_{i,p}}$ are p regressors, and ${\displaystyle \varepsilon _{i}}$ is a mean zero error term. The quantities ${\displaystyle \beta _{0},\dots ,\beta _{p}}$ 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 Yi that may be attributed to some linear combination of the regressors (explanatory variables) in X. [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 ${\displaystyle y}$, 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 = ${\displaystyle {\bar {y}}}$) 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 using typical data, R2 is at least 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 [ citation needed ], similar to the F-tests in Granger causality, though this is not always appropriate[ further explanation needed ]. As a reminder of this, some authors denote R2 by Rq2, 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

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

where Xi 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 Xi.

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 whose ith row is Xi) 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 ${\displaystyle SS_{tot}}$ 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 ${\displaystyle SS_{\text{res}}}$ 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

The use of an adjusted R2 (one common notation is ${\displaystyle {\bar {R}}^{2}}$, 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 the R2 automatically increasing when extra explanatory variables are added to the model. There are many different ways of adjusting ( [14] ). By far the most used one, to the point that it is typically just referred to as adjusted R, is the correction proposed by Mordecai Ezekiel. [14] [15] The adjusted R2 is defined as

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

where dfres is the degrees of freedom of the estimate of the population variance around the model, and dftot is the degrees of freedom of the estimate of the population variance around the mean. dfres is given in terms of the sample size n and the number of variables p in the model, dfres =n-p. dftot is given in the same way, but with p being unity for the mean, i.e. dftot = n-1.

Inserting the degrees of freedom and using the definition of R^2, it can be rewritten as:

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

where p is the total number of explanatory variables in the model, and n is the sample size.

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.

Adjusted R2 can be interpreted as a less biased estimator of the population R2, whereas the observed sample R2 is a positively biased estimate of the population value. [16] 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. [16]

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

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

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

Despite using unbiased estimators for the population variances of the error and the dependent variable, adjusted R2 is not an unbiased estimator of the population R2, [16] which results by using the population variances of the errors and the dependent variable instead of estimating them. Ingram Olkin and John W. Pratt derived the Minimum-variance unbiased estimator for the population R2, [17] which is known as Olkin-Pratt estimator. Comparisons of different approaches for adjusting R2 concluded that in most situations either an approximate version of the Olkin-Pratt estimator [16] or the exact Olkin-Pratt estimator [18] should be preferred over (Ezekiel) adjusted R2.

### Coefficient of partial determination

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. [19] [20] [21] 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

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

which is analogous to the usual coefficient of determination:

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

### Generalizing and decomposing R2 [22]

As explained above, model selection heuristics such as the Adjusted ${\displaystyle R^{2}}$ criterion and the F-test examine whether the total ${\displaystyle R^{2}}$ 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 total ${\displaystyle R^{2}}$ 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.

Alternatively, one can decompose a generalized version of ${\displaystyle R^{2}}$ to quantify the relevance of deviating from a hypothesis. [22] As Hoornweg (2018) shows, several shrinkage estimators – such as Bayesian linear regression, ridge regression, and the (adaptive) lasso – make use of this decomposition of ${\displaystyle R^{2}}$ when they gradually shrink parameters from the unrestricted OLS solutions towards the hypothesized values. Let us first define the linear regression model as

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

It is assumed that the matrix ${\displaystyle X}$ 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

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

An ${\displaystyle R^{2}}$ of 75% means that the in-sample accuracy improves by 75% if the data-optimized ${\displaystyle b}$ 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 traditional ${\displaystyle R^{2}}$ again.

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

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

where ${\displaystyle {\tilde {y}}_{0}=y-X\beta _{0}}$. The diagonal elements of ${\displaystyle R^{\otimes }}$ exactly add up to ${\displaystyle R^{2}}$. 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 the ${\displaystyle r^{2}}$ 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. [22] Click on the lasso for an example.

### R2 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, [23] and independently by Magee: [24]

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

where ${\displaystyle {\mathcal {L}}(0)}$ is the likelihood of the model with only the intercept, ${\displaystyle {{\mathcal {L}}({\widehat {\theta }})}}$ 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:

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

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

Nagelkerke [25] 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 ${\displaystyle {\mathcal {L}}({\widehat {\theta }})}$ 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 scaled R2 as R2/R2max. [26]

## 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 squares of residuals:

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

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 ${\displaystyle SS_{\text{tot}}}$ 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:

 x y 1 2 3 4 5 1.9 3.7 5.8 8 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.

Another single-parameter indicator of fit is the RMSE 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. [27]

## History

The creation of the coefficient of determination has been attributed to the geneticist Sewall Wright and was first published in 1921. [28]

## Notes

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7. 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:. PMC  . PMID   34307865.
8. Legates, D.R.; McCabe, G.J. (1999). "Evaluating the use of "goodness-of-fit" measures in hydrologic and hydroclimatic model validation". Water Resour. Res. 35 (1): 233–241. Bibcode:1999WRR....35..233L. doi:.
9. Ritter, A.; Muñoz-Carpena, R. (2013). "Performance evaluation of hydrological models: statistical significance for reducing subjectivity in goodness-of-fit assessments". Journal of Hydrology. 480 (1): 33–45. Bibcode:2013JHyd..480...33R. doi:10.1016/j.jhydrol.2012.12.004.
10. Everitt, B. S. (2002). Cambridge Dictionary of Statistics (2nd ed.). CUP. p. 78. ISBN   978-0-521-81099-9.
11. Casella, Georges (2002). Statistical inference (Second ed.). Pacific Grove, Calif.: Duxbury/Thomson Learning. p. 556. ISBN   9788131503942.
12. Kvalseth, Tarald O. (1985). "Cautionary Note about R2". The American Statistician. 39 (4): 279–285. doi:10.2307/2683704. JSTOR   2683704.
13. Raju, Nambury S.; Bilgic, Reyhan; Edwards, Jack E.; Fleer, Paul F. (1997). "Methodology review: Estimation of population validity and cross-validity, and the use of equal weights in prediction". Applied Psychological Measurement. 21 (4): 291–305. doi:10.1177/01466216970214001. ISSN   0146-6216. S2CID   122308344.
14. Yin, Ping; Fan, Xitao (January 2001). "Estimating R2 Shrinkage in Multiple Regression: A Comparison of Different Analytical Methods". The Journal of Experimental Education. 69 (2): 203–224. doi:10.1080/00220970109600656. ISSN   0022-0973. S2CID   121614674 . Retrieved 2021-04-23.
15. Shieh, Gwowen (2008-04-01). "Improved shrinkage estimation of squared multiple correlation coefficient and squared cross-validity coefficient". Organizational Research Methods. 11 (2): 387–407. doi:10.1177/1094428106292901. ISSN   1094-4281. S2CID   55098407.
16. Olkin, Ingram; Pratt, John W. (March 1958). "Unbiased estimation of certain correlation coefficients". The Annals of Mathematical Statistics. 29 (1): 201–211. doi:. ISSN   0003-4851.
17. Karch, Julian (2020-09-29). "Improving on Adjusted R-Squared". Collabra: Psychology. 6 (45). doi:. ISSN   2474-7394.
18. Richard Anderson-Sprecher, "Model Comparisons and R2", The American Statistician , Volume 48, Issue 2, 1994, pp. 113–117.
19. (generalized to Maximum Likelihood) N. J. D. Nagelkerke, "A Note on a General Definition of the Coefficient of Determination", Biometrika , Vol. 78, No. 3. (Sep., 1991), pp. 691–692.
20. Hoornweg, Victor (2018). "Part II: On Keeping Parameters Fixed". Science: Under Submission. Hoornweg Press. ISBN   978-90-829188-0-9.
21. Cox, D. D.; Snell, E. J. (1989). The Analysis of Binary Data (2nd ed.). Chapman and Hall.
22. Magee, L. (1990). "R2 measures based on Wald and likelihood ratio joint significance tests". The American Statistician. Vol. 44. pp. 250–3. doi:10.1080/00031305.1990.10475731.
23. Nagelkerke, Nico J. D. (1992). Maximum Likelihood Estimation of Functional Relationships, Pays-Bas. Lecture Notes in Statistics. Vol. 69. ISBN   978-0-387-97721-8.
24. Nagelkerke, N. J. D. (1991). "A Note on a General Definition of the Coefficient of Determination". Biometrika. 78 (3): 691–2. doi:10.1093/biomet/78.3.691. JSTOR   2337038.
25. OriginLab webpage, http://www.originlab.com/doc/Origin-Help/LR-Algorithm. Retrieved February 9, 2016.
26. Wright, Sewall (January 1921). "Correlation and causation". Journal of Agricultural Research. 20: 557–585.

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In statistics, the Pearson correlation coefficient ― also known as Pearson's r, the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ― is a measure of linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationship or correlation. As a simple example, one would expect the age and height of a sample of teenagers from a high school to have a Pearson correlation coefficient significantly greater than 0, but less than 1.

In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.

In applied statistics, total least squares is a type of errors-in-variables regression, a least squares data modeling technique in which observational errors on both dependent and independent variables are taken into account. It is a generalization of Deming regression and also of orthogonal regression, and can be applied to both linear and non-linear models.

In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.

In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function of the independent variable.

In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals. It is a measure of the discrepancy between the data and an estimation model, such as a linear regression. A small RSS indicates a tight fit of the model to the data. It is used as an optimality criterion in parameter selection and model selection.

In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective simple refers to the fact that the outcome variable is related to a single predictor.

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1936.

In statistics, the Durbin–Watson statistic is a test statistic used to detect the presence of autocorrelation at lag 1 in the residuals from a regression analysis. It is named after James Durbin and Geoffrey Watson. The small sample distribution of this ratio was derived by John von Neumann. Durbin and Watson applied this statistic to the residuals from least squares regressions, and developed bounds tests for the null hypothesis that the errors are serially uncorrelated against the alternative that they follow a first order autoregressive process. Note that the distribution of this test statistic does not depend on the estimated regression coefficients and the variance of the errors.

In statistics, the fraction of variance unexplained (FVU) 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.

In statistics, the variance inflation factor (VIF) is the ratio (quotient) of the variance of estimating some parameter in a model that includes multiple other terms (parameters) by the variance of a model constructed using only one term. It quantifies the severity of multicollinearity in an ordinary least squares regression analysis. It provides an index that measures how much the variance of an estimated regression coefficient is increased because of collinearity. Cuthbert Daniel claims to have invented the concept behind the variance inflation factor, but did not come up with the name.

In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). More specifically, PCR is used for estimating the unknown regression coefficients in a standard linear regression model.

In applied statistics, a partial regression plot attempts to show the effect of adding another variable to a model that already has one or more independent variables. Partial regression plots are also referred to as added variable plots, adjusted variable plots, and individual coefficient plots.

In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares.

In statistics and in particular in regression analysis, leverage is a measure of how far away the independent variable values of an observation are from those of the other observations. High-leverage points, if any, are outliers with respect to the independent variables. That is, high-leverage points have no neighboring points in space, where is the number of independent variables in a regression model. This makes the fitted model likely to pass close to a high leverage observation. Hence high-leverage points have the potential to cause large changes in the parameter estimates when they are deleted i.e., to be influential points. Although an influential point will typically have high leverage, a high leverage point is not necessarily an influential point. The leverage is typically defined as the diagonal elements of the hat matrix.

In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x). Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression.

Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.