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Consider a regression analysis with three independent variables \(x_{1}, x_{2}\), and \(x_{3}\). Give the equation for the following regression models: a. The model that includes as predictors all independent variables but no quadratic or interaction terms; b. The model that includes as predictors all independent variables and all quadratic terms; c. All models that include as predictors all independent variables, no quadratic terms, and exactly one interaction term; d. The model that includes as predictors all independent variables, all quadratic terms, and all interaction terms (the full quadratic model).

Short Answer

Expert verified
a. y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3\)\nb. y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1^2 + \beta5x2^2 + \beta6x3^2\)\nc. \ Model 1: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1x2\)\ Model 2: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1x3\)\ Model 3: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x2x3\) \nd. y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1x2 + \beta5x2x3 + \beta6x1x3 + \beta7x1^2 + \beta8x2^2 + \beta9x3^2\)

Step by step solution

01

Model with only Independent Variables

The model that includes only the independent variables x1, x2, and x3, but no quadratic or interaction terms will be: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3\) where, \(\beta0, \beta1, \beta2, \beta3\) are the regression coefficients.
02

Model with Independent Variables and Quadratic Terms

The model that includes all the independent variables and all quadratic terms would now include the squares of x1, x2 and x3. Its equation will be: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1^2 + \beta5x2^2 + \beta6x3^2\)
03

Models with Independent Variables and One Interaction Term

There could be three models that include all independent variables with exactly one interaction term. Model 1: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1x2\) Model 2: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1x3\) Model 3: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x2x3\)
04

Full Quadratic Model with all Independent, Quadratic and Interaction Variables

The full quadratic model includes all independent variables, all quadratic terms, and interaction terms. Its equation will be: y = \(\beta0 + \beta1x1 + \beta2x2 + \beta3x3 + \beta4x1x2 + \beta5x2x3 + \beta6x1x3 + \beta7x1^2 + \beta8x2^2 + \beta9x3^2\)

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Key Concepts

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

Independent Variables
Independent variables, sometimes referred to as predictors or factors, are the backbone of regression analysis. They are the inputs to the model you are building. In the context of the original exercise, the independent variables are represented by \(x_1\), \(x_2\), and \(x_3\). These variables help predict or explain the changes in the dependent variable, often denoted as \(y\).

The equation with only independent variables would be simple and straightforward:
  • \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3\)
Where \(\beta_0, \beta_1, \beta_2,\) and \(\beta_3\) are coefficients that measure the effect of these independent variables on \(y\).

In this model, each independent variable has a linear relationship with the dependent variable. That means the change in \(y\) is directly proportional to changes in each of \(x_1, x_2,\) and \(x_3\).
Quadratic Terms
Quadratic terms are included in a regression model to capture non-linear relationships. These terms come into play when the effect of an independent variable on the dependent variable is not constant across all values, appearing instead as a curve.

Incorporating quadratic terms means squaring the independent variables, creating expressions like \(x_1^2, x_2^2,\) and \(x_3^2\). This results in the regression equation:
  • \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1^2 + \beta_5x_2^2 + \beta_6x_3^2\)
Here, \(\beta_4, \beta_5,\) and \(\beta_6\) are the coefficients for the quadratic terms.

Including these terms allows the model to account for curvature in the data, revealing a more complex relationship between the predictors and the dependent variable. This could be especially useful in fields like economics or biology, where such non-linear interactions are common.
Interaction Terms
Interaction terms are a crucial component when different independent variables influence each other's effect on the dependent variable. Instead of looking at the impact of single independent variables, interaction investigates how combinations affect the outcome variable.

For instance, adding an interaction term \(x_1x_2\) helps determine how \(x_1\) and \(x_2\) together influence changes in \(y\).

The regression equations with one interaction term might look like:
  • \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1x_2\)
  • \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1x_3\)
  • \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_2x_3\)
Such terms are invaluable for understanding scenarios where the effect of one variable is dependent on the level of another, providing greater depth in analyzing data interactions.
Full Quadratic Model
The full quadratic model is a comprehensive representation that includes all layers of complexity from the predictors. It combines independent variables, quadratic terms, and interaction terms in a single regression equation. This model helps capture non-linear and interactive effects simultaneously.

The equation looks like this:
  • \(y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \beta_3x_3 + \beta_4x_1x_2 + \beta_5x_2x_3 + \beta_6x_1x_3 + \beta_7x_1^2 + \beta_8x_2^2 + \beta_9x_3^2\)
This type of model provides flexibility, capturing:
  • Individual linear effects of predictors.
  • Curvature through quadratic terms.
  • Interactions between pairs of predictors.
This approach is ideal when seeking to understand a complex system influenced by multiple, possibly interrelated factors. However, keep in mind that with more terms, the risk of overfitting increases, so model selection techniques and validation are crucial to ensure robustness.

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Most popular questions from this chapter

The article “Readability of Liquid Crystal Displays: A Response Surface" (Human Factors [1983]: \(185-190\) ) used a multiple regression model with four independent variables, where \(y=\) error percentage for subjects reading a fourdigit liquid crystal display \(x_{1}=\) level of backlight (from 0 to \(\left.122 \mathrm{~cd} / \mathrm{m}\right)\) \(x_{2}=\) character subtense (from \(.025^{\circ}\) to \(\left.1.34^{\circ}\right)\) \(x_{3}=\) viewing angle \(\left(\right.\) from \(0^{\circ}\) to \(\left.60^{\circ}\right)\) \(x_{4}=\) level of ambient light (from 20 to \(1500 \mathrm{~lx}\) ) The model equation suggested in the article is \(y=1.52+.02 x_{1}-1.40 x_{2}+.02 x_{3}-.0006 x_{4}+e\) a. Assume that this is the correct equation. What is the mean value of \(y\) when \(x_{1}=10, x_{2}=.5, x_{3}=50\), and \(x_{4}=100 ?\) b. What mean error percentage is associated with a backlight level of 20 , character subtense of .5 , viewing angle of 10 , and ambient light level of 30 ? c. Interpret the values of \(\beta_{2}\) and \(\beta_{3}\).

Data from a sample of \(n=150\) quail eggs were used to fit a multiple regression model relating $$ y=\text { eggshell surface area }\left(\mathrm{mm}^{2}\right) $$ \(x_{1}=\) egg weight \((\mathrm{g})\) \(x_{2}=e g g\) width \((\mathrm{mm})\) $$ x_{3}=\text { egg length }(\mathrm{mm}) $$ (“Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors," International Journal of Poultry Science [2008]: 85-88). The resulting estimated regression function was $$ \begin{array}{l} 10.561+1.535 x_{1}-0.178 x_{2}-0.045 x_{3} \\ \text { and } R^{2}=.996 \end{array} $$ a. Carry out a model utility test to determine if this multiple regression model is useful. b. A simple linear regression model was also used to describe the relationship between \(y\) and \(x_{1}\), resulting in the estimated regression function \(6.254+1.387 x_{1}\). The \(P\) -value for the associated model utility test was reported to be less than .01 , and \(r^{2}=.994 .\) Is the linear model useful? Explain. c. Based on your answers to Parts (a) and (b), which of the two models would you recommend for predicting eggshell surface area? Explain the rationale for your choice.

The authors of the paper “Predicting Yolk Height, Yolk Width, Albumen Length, Eggshell Weight, Egg Shape Index, Eggshell Thickness, Egg Surface Area of Japanese Quails Using Various Egg Traits as Regressors" (International Journal of Poultry Science [2008]: 85-88) used a multiple regression model with two independent variables where $$ \begin{aligned} y &=\text { quail egg weight }(\mathrm{g}) \\ x_{1} &=\text { egg width }(\mathrm{mm}) \\ x_{2} &=\text { egg length }(\mathrm{mm}) \end{aligned} $$ The regression function suggested in the paper is \(-21.658+0.828 x_{1}+0.373 x_{2}\) a. What is the mean egg weight for quail eggs that have a width of \(20 \mathrm{~mm}\) and a length of \(50 \mathrm{~mm}\) ? b. Interpret the values of \(\beta_{1}\) and \(\beta_{2}\).

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, \(x_{4}=\) 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.

A number of investigations have focused on the problem of assessing loads that can be manually handled in a safe manner. The article "Anthropometric, Muscle Strength, and Spinal Mobility Characteristics as Predictors in the Rating of Acceptable Loads in Parcel Sorting" (Ergonomics [1992]: \(1033-1044\) ) proposed using a regression model to relate the dependent variable \(y=\) individual's rating of acceptable load \((\mathrm{kg})\) to \(k=3\) independent (predictor) variables: \(x_{1}=\) extent of left lateral bending \((\mathrm{cm})\) \(x_{2}=\) dynamic hand grip endurance (seconds) $$ x_{3}=\text { trunk extension ratio }(\mathrm{N} / \mathrm{kg}) $$ Suppose that the model equation is $$ \begin{array}{l} y=30+.90 x_{1}+.08 x_{2}-4.50 x_{3}+e \\ \text { and that } \sigma=5 \end{array} $$ a. What is the population regression function? b. What are the values of the population regression coefficients? c. Interpret the value of \(\beta_{1}\). d. Interpret the value of \(\beta_{3}\). e. What is the mean rating of acceptable load when extent of left lateral bending is \(25 \mathrm{~cm}\), dynamic hand grip endurance is 200 seconds, and trunk extension ratio is \(10 \mathrm{~N} / \mathrm{kg}\) ? f. If repeated observations on rating are made on different individuals, all of whom have the values of \(x_{1}\), \(x_{2},\) and \(x_{3}\) specified in Part (e), in the long run approximately what percentage of ratings will be between \(13.5 \mathrm{~kg}\) and \(33.5 \mathrm{~kg}\) ?

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