Editing Coefficient of determination (section)

{{Short description|Indicator for how well data points fit a line or curve}}
{{Distinguish|Coefficient of variation}}
{{expand German|date=September 2019|Bestimmtheitsmaß}}
[[File:Okuns law quarterly differences.svg|300px|thumb|[[Ordinary least squares]] regression of [[Okun's law]]. Since the regression line does not miss any of the points by very much, the ''R''<sup>2</sup> of the regression is relatively high.]]

In [[statistics]], the '''coefficient of determination''', denoted ''R''<sup>2</sup> or ''r''<sup>2</sup> and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
It is a [[statistic]] used in the context of [[statistical model]]s whose main purpose is either the [[Prediction#Statistics|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.<ref>{{cite book|last1=Steel|first1=R. G. D.|last2=Torrie|first2=J. H.|year=1960|title=Principles and Procedures of Statistics with Special Reference to the Biological Sciences|publisher=[[McGraw Hill]]}}</ref><ref>{{cite book |last1=Glantz |first1=Stanton A. |last2=Slinker |first2=B. K. |year=1990 |title=Primer of Applied Regression and Analysis of Variance |publisher=McGraw-Hill |isbn=978-0-07-023407-9}}</ref><ref>{{cite book |last1=Draper |first1=N. R. |last2=Smith |first2=H. |year=1998 |title=Applied Regression Analysis |publisher=Wiley-Interscience |isbn=978-0-471-17082-2}}</ref>

There are several definitions of ''R''<sup>2</sup> that are only sometimes equivalent. In [[simple linear regression]] (which includes an [[regression intercept|intercept]]), ''r''<sup>2</sup> is simply the square of the sample [[Pearson product-moment correlation coefficient|''correlation coefficient'']] (''r''), between the observed outcomes and the observed predictor values.<ref name=Devore>{{cite book |last1 = Devore|first1 = Jay L.|title = Probability and Statistics for Engineering and the Sciences| edition=8th |publisher = Cengage Learning |location = Boston, MA | year = 2011 |isbn =978-0-538-73352-6 |pages=508–510}}</ref> If additional [[regressor]]s are included, ''R''<sup>2</sup> 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 ''R''<sup>2</sup> 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, ''R''<sup>2</sup> may still be negative, for example when linear regression is conducted without including an intercept,<ref>{{cite book |last=Barten |first=Anton P. |author-link=Anton Barten |chapter=The Coeffecient of Determination for Regression without a Constant Term |editor-first=Risto |editor-last=Heijmans |editor2-first=Heinz |editor2-last=Neudecker |title=The Practice of Econometrics |location=Dordrecht |publisher=Kluwer |year=1987 |isbn=90-247-3502-5 |pages=181–189 }}</ref> or when a non-linear function is used to fit the data.<ref>{{cite journal |doi=10.1016/S0304-4076(96)01818-0 |title=An R-squared measure of goodness of fit for some common nonlinear regression models |year=1997 |last1=Colin Cameron |first1=A. |last2=Windmeijer |first2=Frank A.G. |journal=Journal of Econometrics |volume=77 |issue=2 |pages=1790–2 }}</ref> 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 [[mean absolute error|MAE]], [[MAPE]], [[mean square error|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 certain test datasets.<ref>{{cite journal |
doi=10.7717/peerj-cs.623|
title= The coefficient of determination R-squared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation|
year=2021 |
last1= Chicco |
first1=Davide |
last2= Warrens |
first2=Matthijs J. |
last3=Jurman|
first3=Giuseppe|
journal= PeerJ Computer Science |
volume=7 |
issue=e623 |
pages=e623|
pmid= 34307865|
pmc= 8279135|
doi-access=free}}</ref>

When evaluating the goodness-of-fit of simulated (''Y''<sub>pred</sub>) versus measured (''Y''<sub>obs</sub>) values, it is not appropriate to base this on the ''R''<sup>2</sup> of the linear regression (i.e., ''Y''<sub>obs</sub>= ''m''·''Y''<sub>pred</sub> + b).{{cn|date=August 2021}} The ''R''<sup>2</sup> quantifies the degree of any linear correlation between ''Y''<sub>obs</sub> and ''Y''<sub>pred</sub>, while for the goodness-of-fit evaluation only one specific linear correlation should be taken into consideration: ''Y''<sub>obs</sub> = 1·''Y''<sub>pred</sub> + 0 (i.e., the 1:1 line).<ref>{{cite journal |doi=10.1029/1998WR900018 |title= Evaluating the use of "goodness-of-fit" measures in hydrologic and hydroclimatic model validation |year=1999 |last1= Legates |first1=D.R. |last2= McCabe |first2=G.J. |journal= Water Resour. Res. |volume=35 |issue=1 |pages=233–241 |bibcode= 1999WRR....35..233L |s2cid= 128417849 |doi-access= }}</ref><ref>{{cite journal |doi=10.1016/j.jhydrol.2012.12.004|title= Performance evaluation of hydrological models: statistical significance for reducing subjectivity in goodness-of-fit assessments |year=2013 |last1= Ritter |first1=A. |last2= Muñoz-Carpena |first2=R. |journal= Journal of Hydrology |volume=480 |issue=1 |pages=33–45 |bibcode= 2013JHyd..480...33R }}</ref>