Abstract
The coefficient of determination R^2 and the coefficient of determination adjusted for the degrees of freedom R*^2 are common statistics to evaluate the goodness of fit in a linear regression model. The coefficient of determination doubly adjusted for the degrees of freedom R**^2 is also used in applied research in the field of the quality control. It should be noted that R*^2 and R**^2 are proposed to select an adequate model and that the statistic which has good performance for the model selection cannot always perform well for the point estimation of the goodness of fit. In this paper, we consider the small sample properties of R^2, R*^2, R**^2, max{0, R*^2} and max{0, R**^2} as point estimators of the population coefficient of determination. We have the following results: (1) all types of estimators are unreliable when the sample size is small: (2) if we have to use the estimator, max{0, R*^2} is most desirable: (3)R^2 has the severe upward bias: (4) R**^2 and max{0, R**^2} have the severe downward biases, and their MSE's are larger than others, which imply that R**^2 and max {0, R**^2} are not suitable estimators for the population coefficient of determination.
