In statistics, the coefficient of determination, denoted R^{2} or r^{2} and pronounced "R squared", is the proportion of the variance in the dependent variable that is predictable from the independent variable(s).
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 R^{2} that are only sometimes equivalent. One class of such cases includes that of simple linear regression where r^{2} is used instead of R^{2}. When an intercept is included, then r^{2} 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, R^{2} 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 R^{2} 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 modelfitting procedure using those data. Even if a modelfitting procedure has been used, R^{2} may still be negative, for example when linear regression is conducted without including an intercept,^{[5]} or when a nonlinear 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.^{[7]} Since the most general definition of the coefficient of determination is also known as the Nash–Sutcliffe model efficiency coefficient, this last notation is preferred in many fields, because denoting a goodnessoffit indicator that can vary from ∞ to 1 (i.e., it can yield negative values) with a squared letter is confusing.
When evaluating the goodnessoffit of simulated (Y_{pred}) vs. measured (Y_{obs}) values, it is not appropriate to base this on the R^{2} of the linear regression (i.e., Y_{obs}= m·Y_{pred} + b). The R^{2} quantifies the degree of any linear correlation between Y_{obs} and Y_{pred}, while for the goodnessoffit 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]}
YouTube Encyclopedic

1/5Views:573 83796 92714 81974 888176 513

✪ Rsquared or coefficient of determination  Regression  Probability and Statistics  Khan Academy

✪ Finding and Interpreting the Coefficient of Determination

✪ Coefficient Of Determination (r squared)

✪ Coefficient of Determination: Find and Define

✪ What does r squared tell us? What does it all mean
Transcription
In the last few videos, we saw that if we had n points, each of them have x and ycoordinates. Let me draw n of those points. So let's call this point one. It has coordinates x1, y1. You have the second point over here. It had coordinates x2, y2. And we keep putting points up here and eventually we get to the nth point. That has coordinates xn, yn. What we saw is that there is a line that we can find that minimizes the squared distance. This line right here, I'll call it y, is equal to mx plus b. There's some line that minimizes the square distance to the points. And let me just review what those squared distances are. Sometimes, it's called the squared error. So this is the error between the line and point one. So I'll call that error one. This is the error between the line and point two. We'll call this error two. This is the error between the line and point n. So if you wanted the total error, if you want the total squared error this is actually how we started off this whole discussion the total squared error between the points and the line, you literally just take the y value each point. So for example, you would take y1. That's this value right over here, you take y1 minus the y value at this point in the line. Well, that point in the line is, essentially, the y value you get when you substitute x1 into this equation. So I'll just substitute x1 into this equation. So minus m x1 plus b. This right here, that is the this y value right over here. That is m x1 b. I don't want to my get my graph too cluttered. So I'll just delete that there. That is error one right over there. And we want the squared errors between each of the points of the line. So that's the first one. Then you do the same thing for the second point. And we started our discussion this way. y2 minus m x2 plus b squared, all the way I'll do dot dot dot to show that there are a bunch of these that we have to do until we get to the nth point all the way to yn minus m xn plus b squared. And now that we actually know how to find these m's and b's, I showed you the formula. And in fact, we've proved the formula. We can find this line. And if we want to say, well, how much error is there? We can then calculate it. Because we now know the m's and the b's. So we can calculate it for certain set of data. Now, what I want to do is kind of come up with a more meaningful estimate of how good this line is fitting the data points that we have. And to do that, we're going to ask ourselves the question, what percentage of the variation in y is described by the variation in x? So let's think about this. How much of the total variation in y there's obviously variation in y. This y value is over here. This point's y value is over here. There is clearly a bunch of variation in the y. But how much of that is essentially described by the variation in x? Or described by the line? So let's think about that. First, let's think about what the total variation is. How much of the total variation in y? So let's just figure out what the total variation in y is. It's really just a tool for measuring. When we think about variation, and this is even true when we thought about variance, which was the mean variation in y. If you think about the squared distance from some central tendency, and the best central measure we can have of y is the arithmetic mean. So we could just say, the total variation in y is just going to be the sum of the distances of each of the y's. So you get y1 minus the mean of all the y's squared. Plus y2 minus the mean of all the y's squared. Plus, and you just keep going all the way to the nth y value. To yn minus the mean of all the y's squared. This gives you the total variation in y. You can just take out all the y values. Find their mean. It'll be some value, maybe it's right over here someplace. And so you can even visualize it the same way we visualized the squared error from the line. So if you visualize it, you can imagine a line that's y is equal to the mean of y. Which would look just like that. And what we're measuring over here, this error right over here, is the square of this distance right over here. Between this point vertically and this line. The second one is going to be this distance. Just right up to the line. And the nth one is going to be the distance from there all the way to the line right over there. And there are these other points in between. This is the total variation in y. Makes sense. If you divide this by n, you're going to get what we typically associate as the variance of y, which is kind of the average squared distance. Now, we have the total squared distance. So what we want to do is how much of the total variation in y is described by the variation in x? So maybe we can think of it this way. So our denominator, we want what percentage of the total variation in y? Let me write it this way. Let me call this the squared error from the average. Maybe I'll call this the squared error from the mean of y. And this is really the total variation in y. So let's put that as the denominator. The total variation in y, which is the squared error from the mean of the y's. Now we want to what percentage of this is described by the variation in x. Now, what is not described by the variation in x? We want to how much is described by the variation in x. But what if we want how much of the total variation is not described by the regression line? Well, we already have a measure for that. We have the squared error of the line. This tells us the square of the distances from each point to our line. So it is exactly this measure. It tells us how much of the total variation is not described by the regression line. So if you want to know what percentage of the total variation is not described by the regression line, it would just be the squared error of the line, because this is the total variation not described by the regression line, divided by the total variation. So let me make it clear. This, right over here, tells us what percentage of the total variation is not described by the variation in x. Or by the regression line. So to answer our question, what percentage is described by the variation? Well, the rest of it has to be described by the variation in x. Because our question is what percent of the total variation is described by the variation in x. This is the percentage that is not described. So if this number is 30% if 30% of the variation in y is not described by the line, then the remainder will be described by the line. So we could essentially just subtract this from 1. So if we take 1 minus the squared error between our data points and the line over the squared error between the y's and the mean y, this actually tells us what percentage of total variation is described by the line. You can either view it's described by the line or by the variation in x. And this number right here, this is called the coefficient of determination. It's just what statisticians have decided to name it. And it's also called Rsquared. You might have even heard that term when people talk about regression. Now let's think about it. If the squared error of the line is really small what does that mean? It means that these errors, right over here, are really small. Which means that the line is a really good fit. So let me write it over here. If the squared error of the line is small, it tells us that the line is a good fit. Now, what would happen over here? Well, if this number is really small, this is going to be a very small fraction over here. 1 minus a very small fraction is going to be a number close to 1. So then, our Rsquared will be close to 1, which tells us that a lot of the variation in y is described by the variation in x. Which makes sense, because the line is a good fit. You take the opposite case. If the squared error of the line is huge, then that means there's a lot of error between the data points and the line. So if this number is huge, then this number over here is going to be huge. Or it's going to be a percentage close to 1. And 1 minus that is going to be close to 0. And so if the squared error of the line is large, this whole thing's going to be close to 1. And if this whole thing is close to 1, the whole coefficient of determination, the whole Rsquared, is going to be close to 0, which makes sense. That tells us that very little of the total variation in y is described by the variation in x, or described by the line. Well, anyway, everything I've been dealing with so far has been a little bit in the abstract. In the next video, we'll actually look at some data samples and calculate their regression line. And also calculate the Rsquared, and see how good of a fit it really is.
Contents
Definitions
A data set has n values marked y_{1},...,y_{n} (collectively known as y_{i} or as a vector y = [y_{1},...,y_{n}]^{T}), each associated with a fitted (or modeled, or predicted) value f_{1},...,f_{n} (known as f_{i}, or sometimes ŷ_{i}, as a vector f).
Define the residuals as e_{i} = y_{i} − f_{i} (forming a vector e).
If is the mean of the observed data:
then the variability of the data set can be measured using three sums of squares formulas:
 The total sum of squares (proportional to the variance of the data):
 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 most general definition of the coefficient of determination is
Relation to unexplained variance
In a general form, R^{2} can be seen to be related to the 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):
As explained variance
Suppose R^{2} = 0.49. This implies that 49% of the variability of the dependent variable has been accounted for, and the remaining 51% of the variability is still unaccounted for. 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 R^{2} is equivalent to
where n is the number of observations (cases) on the variables.
In this form R^{2} is expressed as the ratio of the explained variance (variance of the model's predictions, which is SS_{reg} / n) to the total variance (sample variance of the dependent variable, which is SS_{tot} / n).
This partition of the sum of squares holds for instance when the model values ƒ_{i} have been obtained by linear regression. A milder sufficient condition reads as follows: The model has the form
where the q_{i} are arbitrary values that may or may not depend on i or on other free parameters (the common choice q_{i} = x_{i} is just one special case), and the coefficient estimates and 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
In linear least squares multiple regression with an estimated intercept term, R^{2} equals the square of the Pearson correlation coefficient between the observed and modeled (predicted) 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 and explanatory variable
It should not be confused with the correlation between two coefficient estimates, defined as
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 R^{2} 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 (by creating a revised predictor of the form α + βƒ_{i}).^{[citation needed]} According to Everitt (p. 78),^{[10]} this usage is specifically the definition of the term "coefficient of determination": the square of the correlation between two (general) variables.
Interpretation
R^{2} is a statistic that will give some information about the goodness of fit of a model. In regression, the R^{2} coefficient of determination is a statistical measure of how well the regression predictions approximate the real data points. An R^{2} of 1 indicates that the regression predictions perfectly fit the data.
Values of R^{2} outside the range 0 to 1 can occur when the model fits the data worse than a horizontal hyperplane. This would occur when the wrong model was chosen, or nonsensical constraints were applied by mistake. If equation 1 of Kvålseth^{[11]} is used (this is the equation used most often), R^{2} can be less than zero. If equation 2 of Kvålseth is used, R^{2} can be greater than one.
In all instances where R^{2} is used, the predictors are calculated by ordinary leastsquares regression: that is, by minimizing SS_{res}. In this case R^{2} increases as we increase the number of variables in the model (R^{2} is monotone increasing with the number of variables included—i.e., it will never decrease). This illustrates a drawback to one possible use of R^{2}, where one might keep adding variables (Kitchen sink regression) to increase the R^{2} 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 R^{2} 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 R^{2}. The explanation of this statistic is almost the same as R^{2} but it penalizes the statistic as extra variables are included in the model. For cases other than fitting by ordinary least squares, the R^{2} statistic can be calculated as above and may still be a useful measure. If fitting is by weighted least squares or generalized least squares, alternative versions of R^{2} can be calculated appropriate to those statistical frameworks, while the "raw" R^{2} may still be useful if it is more easily interpreted. Values for R^{2} can be calculated for any type of predictive model, which need not have a statistical basis.
In a nonsimple linear model
Consider a linear model with more than a single explanatory variable, 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 R^{2} is a measure of the global fit of the model. Specifically, R^{2} is an element of [0, 1] and represents the proportion of variability in Y_{i} that may be attributed to some linear combination of the regressors (explanatory variables) in X.^{[12]}
R^{2} is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, R^{2} = 1 indicates that the fitted model explains all variability in , while R^{2} = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope = 0, intercept = ) between the response variable and regressors). An interior value such as R^{2} = 0.7 may be interpreted as follows: "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 R^{2}, 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 nonzero 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, R^{2} is the square of the Pearson productmoment correlation coefficient relating the regressor and the response variable. More generally, R^{2} is the square of the correlation between the constructed predictor and the response variable. With more than one regressor, the R^{2} can be referred to as the coefficient of multiple determination.
Inflation of R^{2}
In least squares regression, R^{2} is weakly increasing with increases in the number of regressors in the model. Because increases in the number of regressors increase the value of R^{2}, R^{2} 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 Ftest can be performed on the residual sum of squares, similar to the Ftests in Granger causality, though this is not always appropriate. As a reminder of this, some authors denote R^{2} by R_{q}^{2}, 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
where X_{i} 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 X_{i}.
The optimal value of the objective is weakly smaller as more explanatory variables are added and hence additional columns of (the explanatory data matrix whose ith row is X_{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 depends only on y, the nondecreasing property of R^{2} follows directly from the definition above.
The intuitive reason that using an additional explanatory variable cannot lower the R^{2} is this: Minimizing is equivalent to maximizing R^{2}. When the extra variable is included, the data always have the option of giving it an estimated coefficient of zero, leaving the predicted values and the R^{2} unchanged. The only way that the optimization problem will give a nonzero coefficient is if doing so improves the R^{2}.
Caveats
R^{2} does not indicate whether:
 the independent variables are a cause of the changes in the dependent variable;
 omittedvariable 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
Adjusted R^{2}
The use of an adjusted R^{2} (one common notation is , pronounced "R bar squared"; another is ) is an attempt to take account of the phenomenon of the R^{2} automatically and spuriously increasing when extra explanatory variables are added to the model. It is a modification due to Henri Theil of R^{2} that adjusts for the number of explanatory terms in a model relative to the number of data points.^{[13]} The adjusted R^{2} can be negative, and its value will always be less than or equal to that of R^{2}. Unlike R^{2}, the adjusted R^{2} increases only when the increase in R^{2} (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 R^{2} computed each time, the level at which adjusted R^{2} 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 R^{2} is defined as
where p is the total number of explanatory variables in the model (not including the constant term), and n is the sample size.
Adjusted R^{2} can also be written as
where df_{t} is the degrees of freedom n– 1 of the estimate of the population variance of the dependent variable, and df_{e} is the degrees of freedom n – p – 1 of the estimate of the underlying population error variance.
The principle behind the adjusted R^{2} statistic can be seen by rewriting the ordinary R^{2} as
where and are 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 R^{2} can be interpreted as an unbiased (or less biased) estimator of the population R^{2}, whereas the observed sample R^{2} is a positively biased estimate of the population value.^{[14]} Adjusted R^{2} 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.^{[14]}
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.^{[15]}^{[16]}^{[17]} 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 R^{2} is relatively straightforward after estimating two models and generating the ANOVA tables for them. The calculation for the partial R^{2} is
which is analogous to the usual coefficient of determination:
Generalizing and decomposing ^{[18]}
As explained above, model selection heuristics such as the Adjusted criterion and the Ftest examine whether the total sufficiently increases to determine whether 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 will hardly increase even if the new regressor is of relevance. As a result, the abovementioned heuristics will ignore relevant regressors when crosscorrelations are high.
Alternatively, one can decompose a generalized version of to quantify the relevance of deviating from a hypothesis.^{[18]} As Hoornweg (2018) shows, several shrinkage estimators  such as Bayesian linear regression, ridge regression, and the (adaptive) lasso  make use of this decomposition of when they gradually shrink parameters from the unrestricted OLS solutions towards the hypothesized values. Let us first define the linear regression model as
It is assumed that the matrix is standardized with Zscores and that the column vector is centered to have a mean of zero. Let the column vector refer to the hypothesized regression parameters and let the column vector denote the estimated parameters. We can then define
An of 75% means that the insample accuracy improves by 75% if the dataoptimized solutions are used instead of the hypothesized values. In the special case that is a vector of zeros, we obtain the traditional again.
The individual effect on of deviating from a hypothesis can be computed with ('Router'). This times matrix is given by
where . The diagonal elements of exactly add up to . If regressors are uncorrelated and is a vector of zeros, then the diagonal element of simply corresponds to the value between and . When regressors and are correlated, might increase at the cost of a decrease in . As a result, the diagonal elements of 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 to quantify the relevance of deviating from a hypothesized value.^{[18]} Click on the lasso for an example.
in logistic regression
In the case of logistic regression, usually fit by maximum likelihood, there are several choices of pseudoR^{2}.
One is the generalized R^{2} originally proposed by Cox & Snell,^{[19]} and independently by Magee:^{[20]}
where is the likelihood of the model with only the intercept, 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:
where D is the test statistic of the likelihood ratio test.
Nagelkerke^{[21]} noted that it had the following properties:
 It is 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, R^{2} is between 0 and : thus, Nagelkerke suggested the possibility to define a scaled R^{2} as R^{2}/R^{2}_{max}.^{[22]}
Comparison with norm of residuals
Occasionally, the norm of residuals is used for indicating goodness of fit. This term is calculated as the squareroot of the sum of squared residuals:
Both R^{2} and the norm of residuals have their relative merits. For least squares analysis R^{2} 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 R^{2} is the term acts to normalize the value. If the y_{i} values are all multiplied by a constant, the norm of residuals will also change by that constant but R^{2} will stay the same. As a basic example, for the linear least squares fit to the set of data:
R^{2} = 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 R^{2} remains the same, but norm of residuals = 302.
Other single parameter indicators of fit include the standard deviation of the residuals, or the RMSE of the residuals. These would have values of 0.151 and 0.174 respectively for the above example given that the fit was linear with an unforced intercept.^{[23]}
History
The creation of the coefficient of determination has been attributed to the geneticist Sewall Wright and was first published in 1921.^{[24]}
See also
 Fraction of variance unexplained
 Goodness of fit
 Nash–Sutcliffe model efficiency coefficient (hydrological applications)
 Pearson productmoment correlation coefficient
 Proportional reduction in loss
 Regression model validation
 Root mean square deviation
 ttest of
Notes
 ^ Steel, R. G. D.; Torrie, J. H. (1960). Principles and Procedures of Statistics with Special Reference to the Biological Sciences. McGraw Hill.
 ^ Glantz, Stanton A.; Slinker, B. K. (1990). Primer of Applied Regression and Analysis of Variance. McGrawHill. ISBN 9780070234079.
 ^ Draper, N. R.; Smith, H. (1998). Applied Regression Analysis. WileyInterscience. ISBN 9780471170822.
 ^ Devore, Jay L. (2011). Probability and Statistics for Engineering and the Sciences (8th ed.). Boston, MA: Cengage Learning. pp. 508–510. ISBN 9780538733526.
 ^ Barten, Anton P. (1987). "The Coeffecient of Determination for Regression without a Constant Term". In Heijmans, Risto; Neudecker, Heinz (eds.). The Practice of Econometrics. Dordrecht: Kluwer. pp. 181–189. ISBN 9024735025.
 ^ Colin Cameron, A.; Windmeijer, Frank A.G. (1997). "An Rsquared measure of goodness of fit for some common nonlinear regression models". Journal of Econometrics. 77 (2): 1790–2. doi:10.1016/S03044076(96)018180.
 ^ Imdadullah, Muhammad. "Coefficient of Determination". itfeature.com.
 ^ Legates, D.R.; McCabe, G.J. (1999). "Evaluating the use of "goodnessoffit" measures in hydrologic and hydroclimatic model validation". Water Resour. Res. 35 (1): 233–241. doi:10.1029/1998WR900018.
 ^ Ritter, A.; MuñozCarpena, R. (2013). "Performance evaluation of hydrological models: statistical significance for reducing subjectivity in goodnessoffit assessments". Journal of Hydrology. 480 (1): 33–45. doi:10.1016/j.jhydrol.2012.12.004.
 ^ Everitt, B. S. (2002). Cambridge Dictionary of Statistics (2nd ed.). CUP. ISBN 9780521810999.
 ^ Kvalseth, Tarald O. (1985). "Cautionary Note about R2". The American Statistician. 39 (4): 279–285. doi:10.2307/2683704. JSTOR 2683704.
 ^ Computing Adjusted R2 for Polynomial Regressions
 ^ Theil, Henri (1961). Economic Forecasts and Policy. Holland, Amsterdam: North.^{[page needed]}
 ^ ^{a} ^{b} Shieh, Gwowen (20080401). "Improved shrinkage estimation of squared multiple correlation coefficient and squared crossvalidity coefficient". Organizational Research Methods. 11 (2): 387–407. doi:10.1177/1094428106292901. ISSN 10944281.
 ^ Richard AndersonSprecher, "Model Comparisons and R^{2}", The American Statistician, Volume 48, Issue 2, 1994, pp. 113–117.
 ^ (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.
 ^ "R implementation of coefficient of partial determination"
 ^ ^{a} ^{b} ^{c} Hoornweg, Victor (2018). "Part II: On Keeping Parameters Fixed". Science: Under Submission. Hoornweg Press. ISBN 9789082918809.
 ^ Cox, D. D.; Snell, E. J. (1989). The Analysis of Binary Data (2nd ed.). Chapman and Hall.
 ^ Magee, L. (1990). "R^{2} measures based on Wald and likelihood ratio joint significance tests". The American Statistician. 44. pp. 250–3. doi:10.1080/00031305.1990.10475731.
 ^ Nagelkerke, Nico J. D. (1992). Maximum Likelihood Estimation of Functional Relationships, PaysBas. Lecture Notes in Statistics. 69. ISBN 9780387977218.
 ^ 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.
 ^ OriginLab webpage, http://www.originlab.com/doc/OriginHelp/LRAlgorithm. Retrieved February 9, 2016.
 ^ Wright, Sewell (January 1921). "Correlation and causation". Journal of Agricultural Research. 20: 557–585.
References
 Gujarati, Damodar N.; Porter, Dawn C. (2009). Basic Econometrics (Fifth ed.). New York: McGrawHill/Irwin. pp. 73–78. ISBN 9780073375779.
 Hughes, Ann; Grawoig, Dennis (1971). Statistics: A Foundation for Analysis. Reading: AddisonWesley. pp. 344–348. ISBN 0201030217.
 Kmenta, Jan (1986). Elements of Econometrics (Second ed.). New York: Macmillan. pp. 240–243. ISBN 9780023650703.