The following ANOVA table is from a multiple regression analysis. $$\begin{array} { | c | c | c | c | c | c | } \hline \text { Source } & \mathrm { df } & \mathrm { SS } & \mathrm { MS } & F & p \ \hline \text { Repression } & 3 & 1500 & & & \ \hline \text { Error } & 26 & & & & \ \hline \text { Total } & & 2300 & & & \ \hline \end{array}$$ The adjusted R²value is __________. A)0.65 B)0.39 C)0.61 D)0.53 E)0.78

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Quizplus · Accepted Answer

The adjusted $R^2$ value is calculated using the formula $$R^2_{adj} = 1 - \left(\frac{MSE}{MST}\right)$$ where $MSE$ is the mean square error and $MST$ is the mean square total. First, we need to calculate $MSE$ and $MST$. Given that the total sum of squares (SST) is 2300 and the sum of squares due to regression (SSR) is 1500, the sum of squares due to error (SSE) is $SST - SSR = 2300 - 1500 = 800$. With 26 degrees of freedom for error, $MSE = \frac{SSE}{df_{error}} = \frac{800}{26}$. The total degrees of freedom is 29 (since $df_{total} = df_{regression} + df_{error} = 3 + 26$), so $MST = \frac{SST}{df_{total}} = \frac{2300}{29}$. However, to calculate $R^2_{adj}$, we actually use the formula $$R^2_{adj} = 1 - \left(\frac{SSE / (n - k - 1)}{SST / (n - 1)}\right)$$ where $n$ is the number of observations (30, since $df_{total} = 29$ and $df = n - 1$), and $k$ is the number of predictors (3). Plugging in the numbers, $$R^2_{adj} = 1 - \left(\frac{800 / (30 - 3 - 1)}{2300 / (30 - 1)}\right)$$ simplifies to $$R^2_{adj} = 1 - \left(\frac{800 / 26}{2300 / 29}\right)$$, which, after calculation, gives an $R^2_{adj}$ value that rounds to 0.61, indicating that approximately 61% of the variability in the dependent variable is accounted for by the model, after adjusting for the number of predictors.

The Following ANOVA Table Is from a Multiple Regression Analysis