R-squared (R²) measures how well an estimated regression fits the data. It is also known as the coefficient of determination. It is a measure of the goodness of fit of the regression line.

R² = Explained variation/Total variation = 1 - Unexplained variation/total variation

The higher the R², the better. R² values range from 0 to 1.

The value of R² depends on the number of explanatory variables included in the model. This causes a problem when we try to compare the goodness of fit of two models that have the same dependent variable but different number of explanatory variables.

Multiple regression software packages usually produce an adjusted R² as an alternative measure of goodness of fit. It does not automatically increase as independent variables are added to the model. Rather, it adjusts for the degrees of freedom by incorporating the number of independent variables.

• R² is always ≥ adjusted R².
• When a new independent variable is added, adjusted R² can decrease if adding that variable has only a small effect on R².
• In fact, adjusted R² can actually be negative if the correlation between the dependent variable and the independent variables is sufficient low.

We can use the information in an ANOVA table to determine R².

R² = 1 - SSE/SST = 1 - 20.8958/(574.7042 + 20.8958) = 0.9649. Adjusted R² = 1 - [(10 - 1)/(10 - 3)] (1 - 0.9649) = 0.9549.

In fact, R² and adjusted R² are often presented in an ANOVA table. Note that adjusted R² does not indicate whether a regression coefficient's predictions are true or biased. Residual plots and other statistics are required to determine whether or not the predictions are accurate.

Akaike's information criterion (AIC) and Schwarz's Bayesian information criteria (BIC) are also used to evaluate model fit and select the "best" model among a group with the same dependent variable. AIC is preferred if the purpose is prediction, BIC is preferred if goodness of fit is the goal, and lower values of both measures are better.