14.44 Anxiety, depression, and multiple regression: We conducted a second regression analysis on the data from the previous exercise. In addition to depression at year 1, we included a second predictor variable to predict anxiety at year 3. We also included anxiety at year 1. (We might expect that the best predictor of anxiety at a later point in time is one’s anxiety at an earlier point in time.) Here is the output for that analysis.

a. From this software output, write the regression equation.

b. As the first predictor variable, depression at year 1, increases by 1 point, what happens to the predicted score on anxiety at year 3?

c. As the second predictor variable, anxiety at year 1, increases by 1 point, what happens to the predicted score on anxiety at year 3?

d. Compare the predictive utility of depression at year 1 using the regression equation in the previous exercise and using the regression equation you just wrote in part

(a) of this exercise. In which regression equation is depression at year 1 a better predictor? Given that we’re using the same sample, is depression at year 1 actually better at predicting anxiety at year 3 in one regression equation versus the other? Why do you think there’s a difference?

e. The following table is the correlation matrix for the three variables. As you can see, all three are highly correlated with one another. If we look at the intersection of each pair of variables, the number next to “Pearson correlation” is the correlation coefficient. For example, the correlation between “Anxiety year 1” and “Depression year 1” is 0.549.

Which two variables show the strongest correlation? How might this explain the fact that depression at year 1 seems to be a better predictor when it’s the only predictor variable than when anxiety at year 1 also is included? What does this tell us about the importance of including third variables in the regression analyses when possible?

f. Let’s say you want to add a fourth predictor variable. You have to choose among three possible predictor variables: (1) a variable highly correlated with both predictor variables and the outcome variable, (2) a variable highly correlated with the outcome variable but not correlated with either predictor variable, and (3) a variable not correlated with either of the predictor variables or with the outcome variable. Which of the three variables is likely to make the multiple regression equation better? That is, which is likely to increase the proportionate reduction in error? Explain.