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Chapter 14: Problem 11

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

Expert verified
By adding the interaction term to the regression model, the relationship between the dependent and independent variables is allowed to change depending on the level of the other independent variable. Graphically, this is reflected in the cessation of parallelism between the lines for different fixed values of \(x_{1}\) and \(x_{2}\).

Step by step solution

01

Construct the graph for the relationship between mean \(y\) and \(x_{2}\) for fixed values of \(x_{1}\)

By substituting the given fixed values \(x_{1} = 10, 20, 30\) into the regression equation, you obtain three corresponding lines which indicate mean \(y\) against \(x_{2}\) holding \(x_{1}\) constant. This graph depicts the changes in the mean value of \(y\) as \(x_{2}\) varies for each fixed value of \(x_{1}\).
02

Construct the graph for the relationship between mean \(y\) and \(x_{1}\) for fixed values of \(x_{2}\)

Similarly, for each \(x_{2} = 50, 55, 60\), substitute these fixed values into the regression equation to obtain three lines demonstrating the respective mean \(y\) against \(x_{1}\) when \(x_{2}\) is held constant. This graph represents the changes in the mean value of \(y\) as \(x_{1}\) varies for each fixed value of \(x_{2}\).
03

Analyzing the graphs

The aspect that can be attributed to the lack of interaction between \(x_{1}\) and \(x_{2}\) is that the lines on the graph are parallel. This is because in the absence of interaction term, the effect of a unit change in \(x_{1}\) or \(x_{2}\) on \(y\) does not depend on the level of the other variable.
04

Adding the interaction term and update the regression model

Here, interaction term \(.03 x_{3}\) where \(x_{3}=x_{1} x_{2}\) is added to the model, which updates the regression model to \(y=1.8+.1 x_{1}+.8 x_{2}+.03 x_{3}+e\). This new model allows the effect of a unit change in one independent variable to depend on the level of another independent variable.
05

Constructing new graphs under the updated model

Substitute the mentioned fixed values of \(x_{1}\) and \(x_{2}\) into the updated regression model. The new graphs for mean \(y\) against \(x_{2}\) and \(x_{1}\) can be constructed using these updated settings. But in these graphs, the slopes of the lines will no longer be parallel indicating the interaction effect between \(x_{1}\) and \(x_{2}\).

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Key Concepts

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

Understanding Interaction Term
In multiple regression analysis, an **interaction term** is a valuable concept. It allows us to explore how the relationship between independent variables affects the dependent variable, depending on their combined influence. An interaction term is often a product of two independent variables. For example, in the exercise, we added the term \(.03 x_{3}\) where \(x_{3} = x_{1} x_{2}\) to the regression model. This transformed the model to focus not just on individual impacts of \(x_1\) and \(x_2\), but also how they interact together.

When an interaction term is included, it indicates that the effect of one independent variable on the dependent variable may change depending on the value of another independent variable. In simple terms, it means the combined effect is different from the sum of their individual effects. By visualizing this through graphs, you can easily see how interaction terms make the slopes of the regression lines non-parallel because the effect of one variable depends on another variable's level.
Understanding Regression Model
A **regression model** helps us determine the relationship between different variables. When dealing with real-world data, we often want to understand how changes in certain variables affect a particular outcome. For this, we use regression models as formulas that express these relationships mathematically.

In a multiple regression model, such as the one in the exercise, we study more than one independent variable influencing the dependent variable. Originally, our model was: \[ y = 1.8 + 0.1x_1 + 0.8x_2 + e \] where each term serves a role:
  • The **constant** \(1.8\) is the baseline level of the dependent variable, \(y\), when all independent variables \(x_1\) and \(x_2\) are zero.
  • The **coefficients** \(0.1\) for \(x_1\) and \(0.8\) for \(x_2\) explain how much \(y\) is expected to increase with a one-unit increase in \(x_1\) or \(x_2\), respectively, assuming all other variables remain constant.
  • The **error term** \(e\) captures random variation or other unexplained influences on \(y\).
With the new model, the addition of the interaction term \(.03 x_3\) reflects the more complex, real-world scenario where the combined changes of \(x_1\) and \(x_2\) influence \(y\) simultaneously.
Dependent and Independent Variables Clarified
In regression analysis, figuring out the roles of **dependent and independent variables** is crucial because it shapes how we build our models. The dependent variable, often denoted as \(y\), is the primary factor we're trying to predict or explain through the model. It's what changes in response to shifts in the independent variables.

On the other hand, **independent variables** (such as \(x_1\) and \(x_2\) in our exercise) are the factors that provide the input or influence that help determine changes in \(y\). These are also called predictor variables because they help to predict the outcome, or dependent variable.
  • The independent variables are manipulated within the model to see how they impact the dependent variable.
  • Understanding their roles assists in making efficient, accurate models for predicting outcomes and also for exploring the dynamic between different factors.
The whole goal is to make the dependent variable as predictable as possible by optimally using the information from the independent variables. In the exercise, by examining how \(x_1\) and \(x_2\) independently and jointly (with the interaction term) affect \(y\), we get a more comprehensive understanding of the relationships present in the data.

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Most popular questions from this chapter

Suppose that a multiple regression data set consists of \(n=15\) observations. For what values of \(k\), the number of model predictors, would the corresponding model with \(R^{2}=.90\) be judged useful at significance level .05? Does such a large \(R^{2}\) value necessarily imply a useful model? Explain.

The ability of ecologists to identify regions of greatest species richness could have an impact on the preservation of genetic diversity, a major objective of the World Conservation Strategy. The article "Prediction of Rarities from Habitat Variables: Coastal Plain Plants on Nova Scotian Lakeshores" (Ecology [1992]: 1852 - 1859) used a sample of \(n=37\) lakes to obtain the estimated regression equation \(\begin{aligned} \hat{y}=& 3.89+.033 x_{1}+.024 x_{2}+.023 x_{3} \\ &+.008 x_{4}-.13 x_{5}-.72 x_{6} \end{aligned}\) where \(y=\) species richness, \(x_{1}=\) watershed area, \(x_{2}=\) shore width, \(x_{3}=\) drainage \((\%), x_{i}=\) water color (total color units), \(x_{5}=\) sand \((\%)\), and \(x_{6}=\) alkalinity. The coefficient of multiple determination was reported as \(R^{2}=.83 .\) Use a test with significance level \(.01\) to decide whether the chosen model is useful.

This exercise requires the use of a computer package. The authors of the article "Absolute Versus per Unit Body Length Speed of Prey as an Estimator of Vulnerability to Predation" (Animal Behaviour [1999]: 347 - 352) found that the speed of a prey (twips/s) and the length of a prey (twips \(\times 100\) ) are good predictors of the time (s) required to catch the prey. (A twip is a measure of distance used by programmers.) Data were collected in an experiment in which subjects were asked to "catch" an animal of prey moving across his or her computer screen by clicking on it with the mouse. The investigators varied the length of the prey and the speed with which the prey moved across the screen. The following data are consistent with summary values and a graph given in the article. Each value represents the average catch time over all subjects. The order of the various speed-length combinations was randomized for each subject. $$ \begin{array}{ccc} \text { Prey Length } & \text { Prey Speed } & \text { Catch Time } \\ \hline 7 & 20 & 1.10 \\ 6 & 20 & 1.20 \\ 5 & 20 & 1.23 \\ 4 & 20 & 1.40 \\ 3 & 20 & 1.50 \\ 3 & 40 & 1.40 \\ 4 & 40 & 1.36 \\ 6 & 40 & 1.30 \\ 7 & 40 & 1.28 \\ 7 & 80 & 1.40 \\ 6 & 60 & 1.38 \\ 5 & 80 & 1.40 \\ 7 & 100 & 1.43 \\ 6 & 100 & 1.43 \\ 7 & 120 & 1.70 \\ 5 & 80 & 1.50 \\ 3 & 80 & 1.40 \\ 6 & 100 & 1.50 \\ 3 & 120 & 1.90 \\ \hline \end{array} $$ a. Fit a multiple regression model for predicting catch time using prey length and speed as predictors. b. Predict the catch time for an animal of prey whose length is 6 and whose speed is 50 . c. Is the multiple regression model useful for predicting catch time? Test the relevant hypotheses using \(\alpha=.05\). d. The authors of the article suggest that a simple linear regression model with the single predictor \(x=\frac{\text { length }}{\text { speed }}\) might be a better model for predicting catch time. Calculate the \(x\) values and use them to fit this linear regression model. e. Which of the two models considered (the multiple regression model from Part (a) or the simple linear regression model from Part (d)) would you recommend for predicting catch time? Justify your choice.

The article "Impacts of On-Campus and OffCampus Work on First-Year Cognitive Outcomes" (Journal of College Student Development [1994]: 364- 370) reported on a study in which \(y=\) spring math comprehension score was regressed against \(x_{1}=\) previous fall test score, \(x_{2}=\) previous fall academic motivation, \(x_{3}=\) age, \(x_{6}=\) number of credit hours, \(x_{5}=\) residence \(\left(1\right.\) if on campus, 0 otherwise), \(x_{6}=\) hours worked on campus, and \(x_{7}=\) hours worked off campus. The sample size was \(n=210\), and \(R^{2}=.543\). Test to see whether there is a useful linear relationship between \(y\) and at least one of the predictors.

The article "The Value and the Limitations of High-Speed Turbo-Exhausters for the Removal of Tar-Fog from Carburetted Water-Gas" (Society of Chemical Industry Journal [1946]: \(166-168\) ) presented data on \(y=\) tar content \(\left(\right.\) grains \(\left./ 100 \mathrm{ft}^{3}\right)\) of a gas stream as a function of \(x_{1}=\) rotor speed (rev/minute) and \(x_{2}=\) gas inlet temperature \(\left({ }^{\circ} \mathrm{F}\right)\). A regression model using \(x_{1}, x_{2}, x_{3}=x_{2}^{2}\) and \(x_{i}=x_{1} x_{2}\) was suggested: $$ \begin{aligned} \text { mean } y \text { value }=& 86.8-.123 x_{1}+5.09 x_{2}-.0709 x_{3} \\ &+.001 x_{i} \end{aligned} $$ a. According to this model, what is the mean \(y\) value if \(x_{1}=3200\) and \(x_{2}=57 ?\) b. For this particular model, does it make sense to interpret the value of a \(\beta_{2}\) as the average change in tar content associated with a 1 -degree increase in gas inlet temperature when rotor speed is held constant? Explain.
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