The Target Parameter of Adjusted R-Squared in Fixed-Design Experiments

R-squared (R²) and adjusted R-squared (R²_Adj) are sometimes viewed as statistics detached from any target parameter, and sometimes as estimators for the population multiple correlation. The latter interpretation is meaningful only if the explanatory variables are random. This article proposes an alternative perspective for the case where the x’s are fixed. A new parameter is defined, in a similar fashion to the construction of R², but relying on the true parameters rather than their estimates. (The parameter definition includes also the fixed x values.) This parameter is referred to as the “parametric” coefficient of determination, and denoted by ρ²_*. The proposed ρ²_* remains stable when irrelevant variables are removed (or added), unlike the unadjusted R², which always goes up when variables, either relevant or not, are added to the model (and goes down when they are removed). The value of the traditional R²_Adj may go up or down with added (or removed) variables, either relevant or not. It is shown that the unadjusted R² overestimates ρ²_*, while the traditional R²_Adj underestimates it. It is also shown that for simple linear regression the magnitude of the bias of R²_Adj can be as high as the bias of the unadjusted R² (while their signs are opposite). Asymptotic convergence in probability of R²_Adj to ρ²_* is demonstrated. The effects of model parameters on the bias of R² and R²_Adj are characterized analytically and numerically. An alternative bi-adjusted estimator is presented and evaluated.