What is the difference between the R-square and the adjusted R-square? Do we need to
use the adjusted R-square when the R-square indicates a model fit reasonably well in a
data set?
The R-square (R^2) and the adjusted R-square (R^2_adjusted) are both statistical measures used to evaluate the goodness-of-fit of a regression model. However, they have distinct purposes and interpretations.
R-square represents the proportion of variation in the dependent variable (response variable) that is explained by the independent variables (predictor variables) in the regression model. It ranges from 0 to 1, with a higher value indicating a better fit. Specifically, R-square is calculated as the ratio of the sum of squares of the regression (SSR) to the total sum of squares (SST), where SSR represents the explained variation and SST represents the total variation:
R-square = SSR / SST
Adjusted R-square, on the other hand, adjusts the R-square for the number of predictor variables and the sample size. It penalizes the inclusion of irrelevant variables in the model and prevents the R-square value from being artificially inflated due to overfitting. As more predictors are added to the model, the R-square tends to increase even if the additional predictors do not contribute significantly to explaining the variation in the dependent variable.
Adjusted R-square is calculated as:
R-square_adjusted = 1 - (1 - R-square) * (n - 1) / (n - p - 1)
Where n is the sample size and p is the number of predictor variables.
So, to answer your question, while R-square indicates how well a model fits the data by explaining the variation in the dependent variable, adjusted R-square is a more conservative measure that takes into account the complexity of the model and adjusts for the number of predictors. It is generally recommended to use the adjusted R-square when comparing models or when evaluating the usefulness of adding additional predictors. If the R-square value is high and the adjusted R-square is close to the R-square, it suggests that the model is a good fit, given the number of predictors and the sample size.