Impact of data structure on the estimators R-square and adjusted R-square in linear regression.

The effects of the data structure on the quality of the estimator R-square and adjusted R-square in linear multiple regression was evaluated by Monte Carlo simulation. A total of 216 data were generated through which the number of variables, the theoretical value of R-square value, the colinearity between the explanatory variables and the index of coefficient decrease which measures the importance of the explanatory variables in the model were controlled. The results confirmed that R-square is a biased estimator. If much of users are conscious that the estimator R^2 is biased, they consider that the estimator R-square_a is unbiased, or at least that its bias is negligible. This study however revealed that, in unfavourable situations (small theoretical values R-square_0 and small sample size compared to the number of explanatory variables) bias can be significant when the model is established by using selection procedure. The user’s attention is drawn to the risks incurred in the use of this parameter.