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Chapter 12: Q112E (page 781)

Question: There are six independent variables, x1, x2, x3, x4, x5, and x6, that might be useful in predicting a response y. A total of n = 50 observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to be useful. The software fits all possible one-variable models of the form

where xi is the ith independent variable, i = 1, 2, 鈥�, 6. The information in the table is provided from the computer printout.

E(Y)=尾0+尾1xi

a. Which independent variable is declared the best one variable predictor of y? Explain.

b. Would this variable be included in the model at this stage? Explain.

c. Describe the next phase that a stepwise procedure would execute.

Short Answer

Expert verified

Answer

a. For x1, the t-value is the highest (3.80), x1 is the best one variable predictor of y.

b. Since x1 is the best one-variable predictor of y, the variable will be included in the model at this stage.

c. The stepwise program now begins to search through the remaining (k 鈥� 1) independent variables for the best two-variable model of the form. Therefore, here the program begins searching remaining 5 independent variables for the best two-variable model.

Step by step solution

01

One-variable best predictor

T-test values for testing of single 尾 parameter is尾i^s尾i^

Here for x1, x2,x3,x4,x5,and x6t-values are 3.80, -90, 2.98, 1.21, -6.02, and 0.857 respectively. Since for x1, the t-value is the highest, x1 is the best one variable predictor of y.

02

Stepwise procedure

Since x1 is the best one-variable predictor of y, the variable will be included in the model at this stage.

03

Stepwise procedure

The stepwise program now begins to search through the remaining (k 鈥� 1) independent variables for the best two-variable model of the form. Therefore, here the program begins searching remaining 5 independent variables for the best two-variable model.

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Most popular questions from this chapter

Do blondes raise more funds? During fundraising, does the physical appearance of the solicitor impact the level of capital raised? An economist at the University of Nevada- Reno designed an experiment to answer this question and published the results in Economic Letters (Vol. 100, 2008). Each in a sample of 955 households was contacted by a female solicitor and asked to contribute to the Center for Natural Hazards Mitigation Research. The level of contribution (in dollars) was recorded as well as the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female).

a) Consider a model for the mean level of contribution, E(y), that allows for different means depending on the hair color of the solicitor. Create the appropriate number of dummy variables for hair color. (Use minority female as the base level.)

b) Write the equation of the model, part a, incorporating the dummy variables.

c) In terms of the b鈥檚 in the model, what is the mean level of contribution for households contacted by a blond Caucasian solicitor?

d) In terms of the b鈥檚 in the model, what is the difference between the mean level of contribution for households contacted by a blond solicitor and those contacted by a minority female?

e) One theory posits that blond solicitors will achieve the highest mean contribution level, but that there will be no difference between the mean contribution levels attained by brunette Caucasian and minority females. If this theory is true, give the expected signs of the鈥檚 in the model.

f) The researcher found the b estimate for the dummy variable for blond Caucasian to be positive and significantly different from 0 (p-value < 0.01). The尾estimate for the dummy variable for brunette Caucasian was also positive, but not significantly different from 0 (p-value < 0.10). Do these results support the theory, part e?

Write a model that relates E(y) to two independent variables鈥攐ne quantitative and one qualitative at four levels. Construct a model that allows the associated response curves to be second-order but does not allow for interaction between the two independent variables.

The first-order model E(y)=尾0+尾1x1was fit to n = 19 data points. A residual plot for the model is provided below. Is the need for a quadratic term in the model evident from the residual plot? Explain.

Question: Chemical plant contamination. Refer to Exercise 12.18 (p. 725) and the U.S. Army Corps of Engineers study. You fit the first-order model,E(Y)=尾0+尾1x1+尾2x2+尾3x3 , to the data, where y = DDT level (parts per million),X1= number of miles upstream,X2= length (centimeters), andX3= weight (grams). Use the Excel/XLSTAT printout below to predict, with 90% confidence, the DDT level of a fish caught 300 miles upstream with a length of 40 centimeters and a weight of 1,000 grams. Interpret the result.

Consider a multiple regression model for a response y, with one quantitative independent variable x1 and one qualitative variable at three levels.

a. Write a first-order model that relates the mean response E(y) to the quantitative independent variable.

b. Add the main effect terms for the qualitative independent variable to the model of part a. Specify the coding scheme you use.

c. Add terms to the model of part b to allow for interaction between the quantitative and qualitative independent variables.

d. Under what circumstances will the response lines of the model in part c be parallel?

e. Under what circumstances will the model in part c have only one response line?
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