Just as in the simple linear regression model, we can decompose the total variation in the dependent variation into two components: explained variation and unexplained variation.

R² = (Total variation - Unexplained variation)/Total variation

Recall that R² = 1 - SSE/SST, where SST = ∑(y_t - y-bar)2, and SSE = ∑(e-hat_ti)². The SST remains constant when another explanatory variable is added to a model because such an addition has no effect on the sum ∑(y_t - y-bar)². On the other hand, adding an additional explanatory variable to the model causes SSE to decline provided the estimated coefficient of the new variable is not exactly 0 and thus causes the value of R² to increase. To this extent, then, 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.

Another measure of goodness-of-fit takes into account the number of explanatory variables included in an equation. The measure, called the adjusted R square and denoted R-bar², is calculated as shown below:

Adjusted R² = 1 - [SSE/(n - k - 1)]/[SST/(n - 1)]
= 1 - [(n-1)/(n - k - 1)] (1 - R²)

• 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.

Continue with our example discussed earlier:

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.