![Okuns law quarterly differences](https://upload.wikimedia.org/wikipedia/commons/7/77/Okuns_law_quarterly_differences.svg)
In
statistics, the coefficient of determination, denoted ''R''
2 or ''r''
2 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 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.
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 only an
intercept is included, then ''r''
2 is simply the square of the sample
correlation coefficient
A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
(i.e., ''r'') between the observed outcomes and the observed predictor values.
If additional
regressors are included, ''R''
2 is the square of the
coefficient of multiple correlation
In statistics, the coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions th ...
. In both such cases, the coefficient of determination normally ranges from 0 to 1.
There are cases where ''R''
2 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''
2 may still be negative, for example when linear regression is conducted without including an intercept, or when a non-linear function is used to fit the data. 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
The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents ...
in
regression analysis
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
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 Symmetric mean absolute percentage error (SMAPE or sMAPE) is an accuracy measure based on percentage (or relative) errors. It is usually defined as follows:
: \text = \frac \sum_^n \frac
where ''A't'' is the actual value and ''F't'' is the ...
on the test datasets in the article.
When evaluating the goodness-of-fit 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 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).
Definitions
![Coefficient of Determination](https://upload.wikimedia.org/wikipedia/commons/8/86/Coefficient_of_Determination.svg)
A
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of the ...
has ''n'' values marked ''y''
1,...,''y''
''n'' (collectively known as ''y''
''i'' or as a vector ''y'' =
1,...,''y''''n''">'y''1,...,''y''''n''sup>''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 with two
sums of squares formulas:
* The sum of squares of residuals, also called the
residual sum of squares
In statistics, the residual sum of squares (RSS), also known as the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepan ...
:
* The
total sum of squares
In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, y_i, i\leq n, it is defined as the sum over all squared dif ...
(proportional to the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
of the data):
The most general definition of the coefficient of determination is
In the best case, the modeled values exactly match the observed values, which results in
and
. A baseline model, which always predicts
, will have
. Models that have worse predictions than this baseline will have a negative
.
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 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
In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression (SSR – not to be confused with the residual sum of squares (RSS) or sum of squares of errors), is a quantity ...
, is defined as
:
In some cases, as in
simple linear regression, the
total sum of squares
In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, y_i, i\leq n, it is defined as the sum over all squared dif ...
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 In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be ...
(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
In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one ...
with an estimated intercept term, ''R''
2 equals the square of the
Pearson correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the 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 coefficient between two
explanatory variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
s, 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
A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
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''). According to Everitt, 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 measure of the
goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
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 occur when the model fits the data worse than the worst possible
least-squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
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 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 least-squares regression: that is, by minimizing ''SS''
res. In this case, ''R''
2 increases as the number of variables in the model is increased (''R''
2 is
monotone increasing
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...
with the number of variables included—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 likely 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
Weighted least squares (WLS), also known as weighted linear regression, is a generalization of ordinary least squares and linear regression in which knowledge of the variance of observations is incorporated into the regression.
WLS is also a speci ...
or
generalized least squares
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, ordinar ...
, 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 multiple linear model
Consider a linear model with
more than a single explanatory variable, of the form
:
where, for the ''i''th case,
is the response variable,
are ''p'' regressors, and
is a mean zero
error
An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'.
In statistics ...
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
, 1and represents the proportion of variability in ''Y''
''i'' that may be attributed to some linear combination of the regressors (
explanatory variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or deman ...
s) in ''X''.
''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 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, ''R''
2 is the square of the
Pearson product-moment correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the 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
In statistics, the coefficient of multiple correlation is a measure of how well a given variable can be predicted using a linear function of a set of other variables. It is the correlation between the variable's values and the best predictions th ...
.
Inflation of ''R''2
In
least squares regression using typical data, ''R''
2 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 ''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
F-test
An ''F''-test is any statistical test in which the test statistic has an ''F''-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model ...
can be performed on the
residual sum of squares
In statistics, the residual sum of squares (RSS), also known as the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepan ...
, similar to the F-tests in
Granger causality
The Granger causality test is a statistical hypothesis test for determining whether one time series is useful in forecasting another, first proposed in 1969. Ordinarily, regressions reflect "mere" correlations, but Clive Granger argued that cau ...
, 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 ''i''th 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 non-decreasing 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 non-zero 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;
*
omitted-variable bias
In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to those that were included.
More specifically, OV ...
exists;
* the correct
regression was used;
* the most appropriate set of independent variables has been chosen;
* there is
collinearity
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
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
or
) is an attempt to account for the phenomenon of the ''R''
2 automatically increasing when extra explanatory variables are added to the model. There are many different ways of adjusting (
). 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.
The adjusted ''R''
2 is defined as
:
where df
''res'' is the
degrees 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 size ''n'' and the number of variables ''p'' in the model, df
''res'' =''n-p''. df
''tot'' is given in the same way, but with ''p'' being unity for the mean, i.e. df
''tot'' = ''n-1''.
Inserting the degrees of freedom and using the definition of ''R^2'', it can be rewritten as:
:
where ''p'' is the total number of explanatory variables in the model, and ''n'' is the sample size.
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.
Adjusted ''R''
2 can be interpreted as a less biased estimator of the population ''R''
2, whereas the observed sample ''R''
2 is a positively biased estimate of the population value.
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
In machine learning and statistics, feature selection, also known as variable selection, attribute selection or variable subset selection, is the process of selecting a subset of relevant features (variables, predictors) for use in model construc ...
stage of model building.
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
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...
versions:
and
.
Despite using unbiased estimators for the population variances of the error and the dependent variable, adjusted ''R''
2 is not an unbiased estimator of the population ''R''
2,
which results by using the population variances of the errors and the dependent variable instead of estimating them.
Ingram Olkin
Ingram Olkin (July 23, 1924 – April 28, 2016) was a professor emeritus and chair of statistics and education at Stanford University and the Stanford Graduate School of Education. He is known for developing statistical analysis for evalua ...
and
John W. Pratt derived the
Minimum-variance unbiased estimator In statistics a minimum-variance unbiased estimator (MVUE) or uniformly minimum-variance unbiased estimator (UMVUE) is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.
For pra ...
for the population ''R''
2, which is known as Olkin-Pratt estimator. Comparisons of different approaches for adjusting ''R''
2 concluded that in most situations either an approximate version of the Olkin-Pratt estimator
or the exact Olkin-Pratt estimator should be preferred over (Ezekiel) adjusted ''R''
2.
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. 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
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
tables for them. The calculation for the partial ''R''
2 is
:
which is analogous to the usual coefficient of determination:
:
Generalizing and decomposing ''R''2
As explained above, model selection heuristics such as the Adjusted
criterion and the
F-test
An ''F''-test is any statistical test in which the test statistic has an ''F''-distribution under the null hypothesis. It is most often used when comparing statistical models that have been fitted to a data set, in order to identify the model ...
examine whether the total
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
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.
![Geometric R squared](https://upload.wikimedia.org/wikipedia/commons/c/c9/Geometric_R_squared_.svg)
Alternatively, one can decompose a generalized version of
to quantify the relevance of deviating from a hypothesis.
As Hoornweg (2018) shows, several shrinkage estimators – such as
Bayesian linear regression
Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well ...
,
ridge regression
Ridge regression is a method of estimating the coefficients of multiple- regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Als ...
, and the (adaptive)
lasso
A lasso ( or ), also called lariat, riata, or reata (all from Castilian, la reata 're-tied rope'), is a loop of rope designed as a restraint to be thrown around a target and tightened when pulled. It is a well-known tool of the Spanish an ...
– 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 Z-scores 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 in-sample accuracy improves by 75% if the data-optimized
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
('R-outer'). 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.
Click on the
lasso
A lasso ( or ), also called lariat, riata, or reata (all from Castilian, la reata 're-tied rope'), is a loop of rope designed as a restraint to be thrown around a target and tightened when pulled. It is a well-known tool of the Spanish an ...
for an example.
''R''2 in logistic regression
In the case of
logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression a ...
, usually fit by
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...
, there are several choices of
''pseudo-R''2.
One is the generalized ''R''
2 originally proposed by Cox & Snell, and independently by Magee:
:
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
In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after ...
.
Nagelkerke 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''
2max.
Comparison with norm of residuals
Occasionally, the
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
of residuals is used for indicating goodness of fit. This term is calculated as the square-root of the
sum of squares of residuals
In statistics, the residual sum of squares (RSS), also known as the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepan ...
:
:
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.
Another single-parameter indicator of fit is the
RMSE
The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents ...
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.
[OriginLab webpage, http://www.originlab.com/doc/Origin-Help/LR-Algorithm. Retrieved February 9, 2016.]
History
The creation of the coefficient of determination has been attributed to the geneticist
Sewall Wright
Sewall Green Wright FRS(For) Honorary FRSE (December 21, 1889March 3, 1988) was an American geneticist known for his influential work on evolutionary theory and also for his work on path analysis. He was a founder of population genetics alongsi ...
and was first published in 1921.
See also
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Anscombe's quartet
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Fraction of variance unexplained
In statistics, the fraction of variance unexplained (FVU) in the context of a regression task is the fraction of variance of the regressand (dependent variable) ''Y'' which cannot be explained, i.e., which is not correctly predicted, by the e ...
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Goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
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Nash–Sutcliffe model efficiency coefficient (
hydrological applications)
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Pearson product-moment correlation coefficient
In statistics, the Pearson correlation coefficient (PCC, pronounced ) ― also known as Pearson's ''r'', the Pearson product-moment correlation coefficient (PPMCC), the bivariate correlation, or colloquially simply as the correlation coefficient ...
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Proportional reduction in loss
Proportional reduction in loss (PRL) is a general framework for developing and evaluating measures of the reliability of particular ways of making observations which are possibly subject to errors of all types. Such measures quantify how much havin ...
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Regression model validation
In statistics, regression validation is the process of deciding whether the numerical results quantifying hypothesized relationships between variables, obtained from regression analysis, are acceptable as descriptions of the data. The validation ...
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Root mean square deviation
The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between values (sample or population values) predicted by a model or an estimator and the values observed. The RMSD represents ...
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Stepwise regression
Notes
Further reading
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{{DEFAULTSORT:Coefficient Of Determination
Regression diagnostics
Statistical ratios
Least squares