Suppose that in the Framingham study [Haynes et al., 1978]...

Suppose that in the Framingham study [Haynes et al., 1978] we want to examine the relationship between type $\mathrm{A}$ behavior and anger (as given by the four anger variables). We would like to be sure that the relationship does not occur because of joint relationships with the other variables; that is, we want to adjust for all the variables other than type A (variable 1) and the anger variables $11,12,13$, and 17. (a) What quantity would you use to look at this? (b) If the value (squared) is 0.019 , what is the value of the $F$-statistic to test for significance? The degrees of freedom?

Suppose that in the Framingham study [Haynes et al., 1978] we want to examine the relationship between type $\mathrm{A}$ behavior and anger (as given by the four anger variables). We would like to be sure that the relationship does not occur because of joint relationships with the other variables; that is, we want to adjust for all the variables other than type A (variable 1) and the anger variables $11,12,13$, and 17.
(a) What quantity would you use to look at this?
(b) If the value (squared) is 0.019 , what is the value of the $F$-statistic to test for significance? The degrees of freedom?

Key Concepts

F-Test for Nested Models

When a block of variables is added to a model (in a situation with nested models), one can test if these variables account for a statistically significant amount of variability. The test statistic is formed by dividing the mean square due to the extra predictors (obtained from the extra sum?of?squares) by the mean square error from the full model. In formula form it is F = [ (r²/df?) / ((1 – r²)/df?) ], where r² is the squared partial correlation, df? is the number of predictors (the numerator degrees?of?freedom) added, and df? is the error degrees?of?freedom in the full model.

Degrees of Freedom in the Extra SS Test

When you perform an extra sum?of?squares (or partial R²) F?test, the number of degrees of freedom in the numerator is the number of predictors being added (in this case the anger variables, or the block under study) and the denominator degrees of freedom is n minus the total number of predictors in the full model (including those already adjusted for as well as the ones added). In the given example, if one is testing the 4 anger variables (and type A is held apart or is part of the block, as specified), the numerator degrees of freedom is either 4 (or 5) and the denominator degrees of freedom is n minus the number of predictors in the full model.

Partial R² (Squared Partial Correlations)

This is the proportion of variability in the response that is uniquely attributable to a set of predictors after removing the effects of all other variables. In the context of adjusting for possible confounding factors, one compares the full model (which includes the predictors of interest) with a reduced model (which excludes them) and then calculates the extra sum?of?squares. The squared partial correlation (often denoted by the partial R²) is exactly this extra proportion of variation explained by the additional predictors.

Extra Sum-of-Squares Principle

This concept refers to the method by which one assesses the contribution of one or more variables to a regression model. The idea is to compare the sum-of-squares error for a model with and without the new variables. The difference, properly scaled, measures the unique contribution of the variables added. This approach leads to a formal F?test of whether the additional variables significantly improve the fit of the model.

Please give Ace some feedback