coefficient of determination

coefficient of determination, in statistics, R² (or r²), a measure that assesses the ability of a model to predict or explain an outcome in the linear regression setting. More specifically, R² indicates the proportion of the variance in the dependent variable (Y) that is predicted or explained by linear regression and the predictor variable (X, also known as the independent variable).

In general, a high R² value indicates that the model is a good fit for the data, although interpretations of fit depend on the context of analysis. An R² of 0.35, for example, indicates that 35 percent of the variation in the outcome has been explained just by predicting the outcome using the covariates included in the model. That percentage might be a very high portion of variation to predict in a field such as the social sciences; in other fields, such as the physical sciences, one would expect R² to be much closer to 100 percent. The theoretical minimum R² is 0. However, since linear regression is based on the best possible fit, R² will always be greater than zero, even when the predictor and outcome variables bear no relationship to one another.

R² increases when a new predictor variable is added to the model, even if the new predictor is not associated with the outcome. To account for that effect, the adjusted R² (typically denoted with a bar over the R in R²) incorporates the same information as the usual R² but then also penalizes for the number of predictor variables included in the model. As a result, R² increases as new predictors are added to a multiple linear regression model, but the adjusted R² increases only if the increase in R² is greater than one would expect from chance alone. In such a model, the adjusted R² is the most realistic estimate of the proportion of the variation that is predicted by the covariates included in the model.

When only one predictor is included in the model, the coefficient of determination is mathematically related to the Pearson’s correlation coefficient, r. Squaring the correlation coefficient results in the value of the coefficient of determination. The coefficient of determination can also be found with the following formula: R² = MSS/TSS = (TSS − RSS)/TSS, where MSS is the model sum of squares (also known as ESS, or explained sum of squares), which is the sum of the squares of the prediction from the linear regression minus the mean for that variable; TSS is the total sum of squares associated with the outcome variable, which is the sum of the squares of the measurements minus their mean; and RSS is the residual sum of squares, which is the sum of the squares of the measurements minus the prediction from the linear regression.

The coefficient of determination shows only association. As with linear regression, it is impossible to use R² to determine whether one variable causes the other. In addition, the coefficient of determination shows only the magnitude of the association, not whether that association is statistically significant.