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Chapter 14: Problem 32

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.

Short Answer

Expert verified
The number of predictors \(k\) for which the model is useful can be found by trying different values of \(k\) that satisfy the inequality given in Step 2. A high \(R^{2}\) of 0.90 does not necessarily imply a useful model and could be due to overfitting if \(k\) is close to \(n\).

Step by step solution

01

Calculate the F-critical value

First, determine the critical value of F at the 0.05 significance level and degrees of freedom \(df1 = k\) and \(df2 = n-k-1\). Let's assign this value to \(F_{critical}\). Because we do not know \(k\) yet, we cannot calculate this value numerically. However, it is critical to know that this value is needed.
02

Determine the number of predictors for which the model is useful

Next, set up the inequality \((R^{2} / k) / ((1 - R^{2}) / (n - k - 1)) > F_{critical}\) and solve for \(k\). You'll have to try different values of \(k\) to see which ones will satisfy this inequality. Remember, \(k\) must be less than the total number of observations, \(n\).
03

Interpret the value of \(R^{2}\)

A high \(R^{2}\) value of 0.90 means that 90% of the variability in the response variable is explained by the predictors in the model. However, this does not necessarily imply a useful model. It could be due to overfitting, especially if the number of predictors \(k\) is close to the number of observations \(n\). Always consider other factors such as the context of the problem, the size and representativeness of the sample, and the plausibility of the causative relationships suggested by the model.

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Most popular questions from this chapter

A manufacturer of wood stoves collected data on \(y=\) particulate matter concentration and \(x_{1}=\) flue temperature for three different air intake settings (low, medium, and high). a. Write a model equation that includes indicator variables to incorporate intake setting, and interpret each of the \(\beta\) coefficients. b. What additional predictors would be needed to incorporate interaction between temperature and intake setting?

Consider the dependent variable \(y=\) fuel efficiency of a car (mpg). a. Suppose that you want to incorporate size class of car, with four categories (subcompact, compact, midsize, and large), into a regression model that also includes \(x_{1}=\) age of car and \(x_{2}=\) engine size. Define the necessary indicator variables, and write out the complete model equation. b. Suppose that you want to incorporate interaction between age and size class. What additional predictors would be needed to accomplish this?

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 Influence of Temperature and Sunshine on the Alpha-Adid Contents of Hops" (Agricultural Meteorology [1974]: 375-382) used a multiple regression model to relate \(y=\) yield of hops to \(x_{1}=\) average temperature \(\left({ }^{\circ} \mathrm{C}\right)\) between date of coming into hop and date of picking and \(x_{2}=\) average percentage of sunshine during the same period. The model equation proposed is $$ y=415.11-6.60 x_{1}-4.50 x_{2}+e $$ a. Suppose that this equation does indeed describe the true relationship. What mean yield corresponds to an average temperature of 20 and an average sunshine percentage of 40 ? b. What is the mean yield when the average temperature and average percentage of sunshine are \(18.9\) and 43, respectively? c. Interpret the values of the population regression coefficients.

Data from a sample of \(n=150\) quail eggs were used to fit a multiple regression model relating $$ \begin{aligned} y &=\text { eggshell surface area }\left(\mathrm{mm}^{2}\right) \\ x_{1} &=\text { egg weight }(\mathrm{g}) \\ x_{2} &=\text { egg width }(\mathrm{mm}) \\ x_{3} &=\text { egg length }(\mathrm{mm}) \end{aligned} $$ ("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 \(\quad 10.561+1.535 x_{1}-0.178 x_{2}-0.045 x_{3}\) and \(R^{2}=.996 .\) 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.
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