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

The absence of an interaction term in the original regression model resulted in graphs showing uniform slopes regardless of the fixed variable. However, the introduction of an interaction term in the regression model led to changes in the slopes of lines in the new graphs, as the effect of one variable on \(y\) now depends on the level of the other variable.

Step-by-step solution

Graph of mean \(y\) and \(x_{2}\) for fixed \(x_{1}\) values

Since the values of \(x_{1}\) are fixed, this becomes a simple linear regression with slope .8. For fixed \(x_{1}=10\), \(y = 1.8 + 10*.1 + .8*x_{2} = 2.8 + .8*x_{2}\). Similarly, with \(x_{1}=20\) and \(x_{1}=30\), the relationship becomes \(y = 3.8 + .8*x_{2}\) and \(y = 4.8 + .8*x_{2}\) respectively. Plot these three equations on a graph, where \(x_{2}\) is on the x-axis and \(y\) is on the y-axis.

Graph of mean \(y\) and \(x_{1}\) for fixed \(x_{2}\) values

Now let's fix \(x_{2}\) and let \(x_{1}\) vary. As before, this is a regression line with slope .1. For \(x_{2}=50, 55, 60\), the following relationships are obtained: \(y = 1.8 + .1*x_{1} + 50*.8 = 41.8 + .1*x_{1}\), \(y = 45.8 + .1*x_{1}\) and \(y = 49.8 + .1*x_{1}\). Plot these three equations on a graph where \(x_{1}\) is on the x-axis and \(y\) is on the y-axis.

Understand lack of interaction

The graphs in part (a) and (b) all show straight lines with equal slopes. This uniformity is due to the lack of an interaction term. Because there is no interaction term, the effect of one variable on \(y\) is independent of the level of the other variable.

Introduce interaction term

Now that an interaction term is introduced, the model should change. With the new equation \(y = 1.8 + .1*x_{1} + .8*x_{2} + .03*x_{3}\) where \(x_{3} = x_{1}*x_{2}\), the slope of the regression line should now change depending on the values of \(x_{1}\) and \(x_{2}\). This will result in different slopes for different fixed values.

Construct new graphs

With the new model, we plot new graphs just like in step 1 and step 2. However, now the lines will have different slopes due to the influence of both \(x_{1}\) and \(x_{2}\) on one another through the interaction term. These new equations will look like \(y = 2.8 + (0.8 + 0.03*10)*x_{2}\), \(y = 3.8 + (0.8 + 0.03*20)*x_{2}\), \(y = 4.8 + (0.8 + 0.03*30)*x_{2}\) for fixed \(x_{1}\) values, and \(y = 41.8 + (0.1 + 0.03*50)*x_{1}\), \(y = 45.8 + (0.1 + 0.03*55)*x_{1}\), \(y = 49.8 + (0.1 + 0.03*60)*x_{1}\) for fixed \(x_{2}\) values.

Key concepts

Simple Linear Regression

In the realm of statistical analysis, simple linear regression is a fundamental tool used to understand the relationship between two variables. Imagine you're studying how study time affects exam scores, or how temperature impacts ice cream sales. In these cases, simple linear regression helps to explain the impact of the independent variable (like study time or temperature) on the dependent variable (like exam scores or sales).

The general form of a simple linear regression equation is \(y = \beta_0 + \beta_1x + \epsilon\), where \(y\) is the dependent variable, \(x\) is the independent variable, \(\beta_0\) is the y-intercept, \(\beta_1\) is the slope of the line, and \(\epsilon\) represents the random error.

Imagine you want to predict a student's exam score based on their study time. If a student doesn't study at all, the intercept, \(\beta_0\), gives the expected exam score; and for each additional hour of study, the slope, \(\beta_1\), indicates how much the exam score is expected to increase. In this simple linear regression model, each variable has a direct and clear-cut relationship with the other.

Interaction Term

Sometimes in regression analysis, the effect of one independent variable on the dependent variable may depend on the level of another independent variable. This is where an interaction term comes into play.

An interaction term is created by multiplying two independent variables together and including this product as a new variable in the regression model. It allows us to see if there's a combined effect of the two independent variables on the dependent one. For example, considering the study time and the difficulty of the material (another independent variable), the combined effect on exam score might be different from studying more of either harder or easier material alone. The interaction can reveal this nuanced relationship.

In mathematical terms, if our original model was \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon\), an interaction term would look like \(\beta_3(x_1x_2)\), making the new model \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3(x_1x_2) + \epsilon\). This new term, \(\beta_3(x_1x_2)\), represents how the relationship between \(x_1\) and the outcome \(y\) changes as \(x_2\) changes, and vice versa.

Fixed Values Analysis

When dealing with multiple regression analysis, fixed values analysis is employed to understand the distinct effect of one independent variable on the dependent variable while holding another independent variable constant. This type of analysis is also known as 'holding other factors constant' or 'ceteris paribus' in economics.

In the context of the provided problem, we analyze the relationship between \(y\) and \(x_2\) by fixing \(x_1\) at certain values. For instance, if we fix \(x_1\) at 10, any variation in \(y\) can be attributed to changes in \(x_2\) since the effect of \(x_1\) is constant. Visualizing this relationship often involves plotting graphs for each fixed value of \(x_1\), allowing us to discern patterns or trends in how \(x_2\) affects \(y\).

Similarly, the same approach is applied to examine how \(x_1\) influences \(y\) while fixing \(x_2\) at particular values. This process can highlight the independent impact of each variable and is crucial for understanding the dynamics of the variables in a multiple regression model.

Regression Model Graph

Regression model graphs are visual representations that help us understand the relationship between variables illustrated by a regression model. In the case of our problem, the regression model graphs show how mean \(y\) varies with \(x_1\) or \(x_2\), while holding the other variable at fixed values.

For simple linear regression without an interaction term, the graph will display a series of straight lines, each representing the regression line at a particular value of the fixed variable. These lines are parallel, indicating that the effect of the changing variable on \(y\) is the same irrespective of the level of the fixed variable. This reveals the absence of interaction between \(x_1\) and \(x_2\) in our model.

However, when we introduce an interaction term, the parallel lines may no longer hold true. Instead, the slopes of the lines change because the effect of one independent variable on \(y\) now varies depending on the level of the other variable. For example, as \(x_1\) increases, the slope of the relationship between \(y\) and \(x_2\) alters if there’s an interaction term that relates \(x_1\) and \(x_2\). If plotted on a graph, each line would have a different inclination, reflecting the effect of the interaction term on the dependent variable.