91Ó°ÊÓ

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)?

Step by step solution

01

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 \(x_1\).

02

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 \(x_2\).

03

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 \(x_{2}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interaction Term

An interaction term in a regression model is an additional variable created by multiplying two or more predictors. This term captures the effect that occurs when the influence of one independent variable on the dependent variable depends on the level of another independent variable. For example, in our revised equation with the interaction term \.03 x_3\ where \(x_3 = x_1 x_2\), the interaction term allows us to see how the combined effect of \(x_1\) and \(x_2\) influences \(y\). Without the interaction term, one could only see the isolated effects of \(x_1\) and \(x_2\). This means:

• If there's no interaction, the variables affect \(y\) independently.
• With the interaction, the effect of \(x_1\) can depend on \(x_2\) and vice versa.

Adding an interaction term often changes the interpretation of a model, as it brings a richer, more nuanced understanding of how variables interplay in affecting the outcome.

Regression Model

A regression model is a statistical way to understand relationships between a dependent variable and one or more independent variables. It helps predict outcomes and understand underlying trends by fitting a mathematical equation to the observed data. In this exercise, the regression model is given by:\[y = 1.8 + 0.1x_1 + 0.8x_2 + e\]Here, \(y\) is the dependent variable influenced by \(x_1\) and \(x_2\), with \(e\) representing a random error term. The regression coefficients (i.e., \(.1\) and \(.8\)) show how much \(y\) changes when \(x_1\) or \(x_2\) increases by one unit. A positive coefficient indicates that as the independent variable increases, the dependent variable also increases. This model assumes linearity, meaning the relationships are represented by straight lines.

Mean Response

The mean response is the average predicted value of the dependent variable in a regression model for given levels of the predictors. It represents the expected value of \(y\) based on fixed values of \(x_1\) and \(x_2\). In the given model, substituting different fixed values of one of the predictors while keeping the other fixed provides the mean response, which is then used to plot the relationship between \(y\) and each predictor individually.

• The mean response changes as the predictor values change.
• Parallel lines on the graph indicate consistent mean responses, regardless of other fixed values.

Graphically, viewing the mean response helps in understanding the isolated impact of each predictor separately on the dependent variable.

Fixed Values

In the context of regression, fixed values refer to specific, constant levels of an independent variable used to analyze the effect on the dependent variable. In problems like this, researchers often fix values for one variable to observe how changes in another variable affect \(y\).Consider:

• In Part (a), \(x_1\) is set to 10, 20, and 30 to study \(y\) vs. \(x_2\).
• In Part (b), the roles reverse with \(x_2\) fixed at 50, 55, and 60.

The practice of fixing values helps isolate the influence of one variable, making it easier to understand its unique contribution. This approach highlights whether or not interaction effects are present by observing the parallelism of lines on a graph.