Question

Suppose that the variables \(y, x_{1}\), and \(x_{2}\) are related by the regression model $$ y=1.8+.1 x_{1}+.8 x_{2}+e $$ a. Construct a graph (similar to that of Figure \(14.5)\) showing the relationship between mean \(y\) and \(x_{2}\) for fixed values 10,20 , and 30 of \(x_{1}\). b. Construct a graph depicting the relationship between mean \(y\) and \(x_{1}\) for fixed values 50,55, and 60 of \(x_{2}\). c. What aspect of the graphs in Parts (a) and (b) can be attributed to the lack of an interaction between \(x_{1}\) and \(x_{2}\) ? d. Suppose the interaction term \(.03 x_{3}\) where \(x_{3}=x_{1} x_{2}\) is added to the regression model equation. Using this new model, construct the graphs described in Parts (a) and (b). How do they differ from those obtained in Parts (a) and (b)?

Short answer

For fixed values of \(x_{1}\) or \(x_{2}\), there are linear relationships between \(y\) and \(x_{1}\) or \(y\) and \(x_{2}\). The lack of interaction between \(x_{1}\) and \(x_{2}\) is reflected in parallel lines in the graphs. Adding an interaction term will change the slope of these lines, showing a change in the effect of one independent variable on the dependent variable \(y\) that depends on the value of the other independent variable.

Step-by-step solution

Graph without interaction term for fixed values of \(x_1\)

The regression equation is \(y=1.8+.1 x_{1}+.8 x_{2}+e\). To construct the graph, substitute different fixed values of \(x_{1}\) (10, 20, 30) into the regression equation. The equation becomes \(y = 1.8 + 1*x_{1} + 0.8*x_{2} + e\). Keeping \(x_{2}\) as the variable, plot y against \(x_{2}\)}, for each value of \(x_1\).

Graph without interaction term for fixed values of \(x_2\)

Again, use the regression equation \(y=1.8+.1 x_{1}+.8 x_{2}+e\). This time, substitute different fixed values of \(x_{2}\) (50, 55, 60) into the regression equation. The equation becomes \(y = 1.8 + .1 * x_{1} + .8 * x_{2} + e\). Keeping \(x_{1}\) as the variable, plot y against \(x_{1}\)}, for each value of \(x_2\).

Interpret the graphs for interaction between \(x_{1}\) and \(x_{2}\)

If there is no interaction between \(x_{1}\) and \(x_{2}\), this means the effect of \(x_{1}\) on y doesn't depend on \(x_{2}\) and vice versa. This is reflected in the graphs as parallel lines: the mean of \(y\) changes with \(x_{1}\) or \(x_{2}\) but the slopes of the lines are constant - they don't depend on the specific value of either \(x_{1}\) or \(x_{2}\)