is the coefficient of determination in regression models. It shows the proportion of variance explained by the model in relation to the total variance.
- (or ) = Model / explained sum of squares (amount of variation in the dependent variable that is explained by the model)
- = Unexplained sum of squares (amount of variation not explained by the model)
- = Total sum of squares
Interpretation of R² values
The value of R² corresponds to the amount of variation that can be „explained“ by the model (i.e. the amount of variation being accounted for as the model can‘t prove causalities).
A value of 0 means no fit at all, while a value of 1 means perfect fit. For example means that 95% of the variation can be accounted for by the model.
Relevance of the R² value
- R² can be used to compare models based on the same dataset which use the same dependent variable
- Very low or high values can serve as a warning sign for a badly specified model. When it is high, some of the independent variables could be too strongly correlated or identical to the dependent variable.
- When estimating models with individual survey data, low values (< 10%) are normal because individual behaviours and attitudes are hard to predict.
Common misinterpretations of R²
- R² can‘t show how strong the relationship between X and Y is. This is instead indicated by the slope parameter
- It also doesn‘t show how well the model matches the true population
- It can‘t show the „correctness“ of the empirical model
- A model with a higher R²- value is not inherently better
Adjusted R²
In multivariate regression models the introduction of additional independent variables, leads higher R² values even though the variation stays the same.
As a solution, the Adjusted R² value includes the number of independent variables. The idea is, that new variables have to make up a penalty by making up in explained variance for the R² value to raise. This can, in theory, lead to negative R² values.
- = number of independent variables
- = number of observations