Can R2 be positive or negative?
In the context of linear regression, R2 is a measure of how well the regression model fits the observed data points. It represents the proportion of the variance in the dependent variable that can be explained by the independent variables.
Typically, R2 ranges between 0 and 1, where a value of 0 indicates that the independent variables do not explain any of the variance in the dependent variable, and a value of 1 indicates that the independent variables explain all of the variance. However, it is possible for R2 to take on negative values in certain situations.
When can R2 be negative?
In linear regression, R2 can be negative only when the intercept (or perhaps the slope) is constrained. This situation may arise when the model is misspecified or when there are constraints imposed on the regression coefficients. For example, if the intercept is constrained to be positive, it is possible for the model to perform poorly and result in a negative R2. However, it is important to note that in most practical applications, R2 is expected to be positive.
Why is R2 typically positive?
In a well-fitted linear regression model, R2 is expected to be positive. This is because R2 is related to the square of the correlation coefficient (r) between the dependent and independent variables. The correlation coefficient ranges between -1 and 1, where a value of -1 indicates a perfect negative linear relationship, a value of 1 indicates a perfect positive linear relationship, and a value of 0 indicates no linear relationship.
Since R2 is the square of r, it will always be positive or zero. A positive R2 indicates that the independent variables are explaining a proportion of the variance in the dependent variable, while a value of zero suggests that the independent variables have no explanatory power.
It is important to interpret R2 in conjunction with other model evaluation metrics and consider the specific context and objectives of the analysis. While R2 provides a useful summary measure of model fit, it should not be solely relied upon to draw conclusions about the adequacy of the regression model. Other considerations, such as the significance of the regression coefficients, residual analysis, and the theoretical basis of the model, should also be taken into account.
While R2 is typically positive in linear regression, it is possible for it to be negative under certain constrained conditions. However, in most practical scenarios, R2 is expected to be positive, indicating the proportion of variance explained by the independent variables.