What is R-Squared?
The R-squared statistic quantifies the predictive accuracy of a statistical model. It shows the proportion of variance in the outcome variable that is explained by the predictions. It is also known as the coefficient of determination, R², r², and r-square. This article will go over the key properties of R², how it is computed and its limitations.
Key properties of R²
R² typically has a value in the range of 0 through to 1. A value of 1 indicates that predictions are identical to the observed values; it is not possible to have a value of R² of more than 1. A value of 0 indicates that there is no linear relationship between the observed and predicted values, where “linear” in this context means that it is still possible that there is a non-linear relationship between the observed and predicted values. Finally, a value of 0.5 means that half of the variance in the outcome variable is explained by the model. Sometimes the R² is presented as a percentage (e.g., 50%).
How is the R-squared statistic computed?
There are many equivalent ways of computing R². Perhaps the simplest is:
R² = Explained sum-of-squares / Total sum-of-squares
I've illustrated this in the table below. The first column, called Observed, shows the nine observed values (i.e., of the outcome variable). The second column contains the observed values minus their average value of 1.95. The third column squares these values. The sum of these squared values is called the Total sum-of-squares (TSS).
The fourth column shows the predicted values (in this case from a linear regression). The Explained-sum-of-squares are computed using the predicted values in the same way as was done with the observed values. The ratio of these numbers, 0.18909/3.27 = 0.05783 = R².
An alternative way of computing R² is as the square of Pearson’s product-moment correlation. In most conventional situations these two calculations will produce the same values. They can differ when the model being used is not sensible (e.g., a model where the predictions are less accurate than chance) or when being computed for data that has not been used when fitting the model.
How to use R²
R² has two main uses. One is to provide a basic summary of how well a model fits the data. If R² is only 0.1, then in an absolute sense the R² is only explaining a tenth of what can be explained. Similarly, an R² of .99 is explaining almost all that can be explained.
The other main application of R² is to compare models. All else being equal, a model with a higher R² is a better model.
Limitations of R²
A common misunderstanding of R² is that there is a threshold. For example, that a model needs to have an R² of more than 0.9 to be good. This is rarely true: for example, a model which predicts that future share prices may be able to earn billions of dollars in profits for a hedge fund even if the R² is only 0.01.
R² can is also problematic when comparing models. In the case of regression, for example, if you add an extra predictor the R² will almost always increase. Therefore, while it is common for researchers to have a look at R² when comparing models, more sophisticated methods (e.g., statistical tests, information criteria) should be used most of the time.
Variants of R²
There are a number of variants of R². The most well-known is the Adjusted R² Statistic, which is designed to make it possible to compare R² across models with different numbers of predictors. Various pseudo-R² statistics have been developed for models for categorical outcome variables.