rsquared

Rsquared, more commonly written as R-squared or R^2, is the coefficient of determination used in regression analysis to quantify how well a statistical model explains the variability of a dependent variable. It is defined as 1 minus the ratio of the sum of squared residuals to the total sum of squares: R^2 = 1 - SSE/SST. Here SSE is the sum over observations of the squared differences between observed values and the model's predicted values, and SST is the sum of squared differences between the observed values and their mean.

Interpretation: R^2 represents the proportion of the variance in the dependent variable that is explained by the model. Values near 0 indicate little explanatory power; values near 1 indicate a close fit. In simple linear regression with an intercept, R^2 also equals the square of the Pearson correlation between the observed outcomes and the predictor variable (or between observed values and the fitted values). In multiple regression, R^2 reflects the combined explanatory power of all predictors but does not reveal causation.

Variants and extensions: Adjusted R-squared modifies R^2 to account for the number of predictors relative to sample size, helping to avoid overfitting. It is defined as 1 - (1 - R^2) * (n - 1) / (n - p - 1), where n is sample size and p is number of predictors. In predictive modeling, out-of-sample or cross-validated R^2 assesses predictive performance on unseen data. For non-linear models or categorical outcomes, several pseudo-R^2 measures are used.

Limitations: R^2 does not indicate whether the model is appropriate, nor does a high value imply causation or good predictive performance on new data. It can be inflated by adding variables and is sensitive to outliers.