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Regression analysis Module 15: Advanced interactions (The attached PDF file has better formatting.) Selecting the optimal model using sums of squares and degrees of freedom (F test) Tables 7.1 and 7.2 on page 139 are tested on the final exam. This posting explains the computations for the F test in these tables. The variables are: I = income, E = education, and T = type The regression sums of squares are Model | Terms | Sum of Squares | df | 1 | I, E, T, I × T, E × T | 24,794 | 8 | 2 | I, E, T, I × T | 24,556 | 6 | 3 | I, E, T, E × T | 23,842 | 6 | 4 | I, E, T | 23,666 | 4 | 5 | I, E | 23,074 | 2 | 6 | I, T, I × T | 23,488 | 5 | 7 | E, T, E × T | 22,710 | 5 |
For each model, The residual sum of squares is  . The regression sum of squares is  . The total sum of squares is  . The total sum of squares does not depend on the model; it is 28,347 in this illustration. Jacob: All three formulas for the sums of squares use only Y values, not X value or ß’s. Rachel: The regression sum of squares and the residual sum of squares use the fitted Y values, which depend on the X values. They vary by model. The degrees of freedom in Table 7.1 on page 139 are the number of explanatory variables in the model (k). The degrees of freedom are actually N-k-1. This illustration shows the degrees of freedom for the numerator of the F test, which is the difference in the number of variables in the full vs reduced models. N-1 is the same for all models, so it drops out of the difference. For the number of explanatory variables: I and E are one explanatory variable each. T, I × T, and E × T are two explanatory variables each. Table 7.2 shows the degrees of freedom and sum of squares in the numerator of the F test. Source | Models Contrasted | Sum of Squares | df | F | Income | 3 – 7 | 1,132 | 1 | 28.35 | Education | 2 – 6 | 1,068 | 1 | 26.75 | Type | 4 – 5 | 592 | 2 | 7.41 | Income × Type | 1 – 3 | 952 | 2 | 11.92 | Education × Type | 1 – 2 | 238 | 2 | 2.98 | Residuals | | 3,553 | 89 | | Total | | 28,347 | 97 | |
The total sum of squares is 28,347. The sample has 98 data points, so the total sum of squares has 98 – 1 = 97 degrees of freedom. The full model (Model 1) has a regression sum of squares of 24,794, so it has a residual sum of squares of 28,347 – 24,794 = 3,553. This residual sum of squares has 98 – 8 – 1 = 89 degrees of freedom. The denominator of the F ratio (for all tests) is 3,553 / 89 = 39.921. Illustration: To test the significance of income, we contrast models 3 and 7. The sum of squares is 23,842 for Model 3 and 22,710 for Model 7. The difference in the sum of squares is 23,842 – 22,710 = 1,132. Model 3 has 6 explanatory variables and Model 7 has 5 explanatory variables. The degrees of freedom in the numerator of the F test is 6 – 5 = 1. The numerator of the F ratio is 1,132 / 1 = 1,132. The F ratio is 1,132 / 39.921 = 28.356. Illustration: To test the significance of education × type, we contrast models 1 and 2. The sum of squares is 24,794 for Model 1 and 24,556 for Model 2. The difference in the sum of squares is 24,794 – 24,556 = 238. Model 1 has 8 explanatory variables and Model 2 has 6 explanatory variables. The degrees of freedom in the numerator of the F test is 8 – 6 = 2. The numerator of the F ratio is 238 / 2 = 119. The F ratio is 119 / 39.921 = 2.981. To find the p-values in Table 7.2, use a table of the F-distributions or statistical software, such as Excel. If an exam problem asks for a p-value, it will give a table.
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