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Chapter 12: Q111E (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

E(Y)=β0+β1xi

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

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 – 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 β parameter isβi^sβi^

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 – 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

Consider relating E(y) to two quantitative independent variables x1 and x2.

  1. Write a first-order model for E(y).

  2. Write a complete second-order model for E(y).

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)=β0+β1x1+β2x2+β3x3 , 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.

Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 11.27 (p. 657). Recall that opening weekend box office revenue data (in millions of dollars) were collected for a sample of 24 recent movies. In addition to each movie’s tweet rate, i.e., the average number of tweets referring to the movie per hour 1 week prior to the movie’s release, the researchers also computed the ratio of positive to negative tweets (called the PN-ratio).

a) Give the equation of a first-order model relating revenue (y)to both tweet rate(x1)and PN-ratio(x2).

b) Which b in the model, part a, represents the change in revenue(y)for every 1-tweet increase in the tweet rate(x1), holding PN-ratio(x2)constant?

c) Which b in the model, part a, represents the change in revenue (y)for every 1-unit increase in the PN-ratio(x2), holding tweet rate(x1)constant?

d) The following coefficients were reported:R2=0.945andRa2=0.940. Give a practical interpretation for bothR2andRa2.

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a. Write a complete, first-order model for E(y) as a function of the six independent variables.

b. Consider a test of whether the leadership score of either the purser or the head flight attendant (or both) is statistically useful for predicting team goal attainment. Give the null and alternative hypotheses as well as the reduced model for this test.

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d. The p-value of the subset F-test for comparing the two models for successful cabin crews was reported in the article as p 6 .05. Formally test the null hypothesis using α = .05. What do you conclude?

e. The two models were also fit to the data for the n = 24 unsuccessful cabin crews with the following results: R2 = .14 for reduced model, R2 = .15 for complete model. On the basis of this information only, give your opinion regarding the null hypothesis for unsuccessful cabin crews.

f. The p-value of the subset F-test for comparing the two models for unsuccessful cabin crews was reported in the article as p < .10. Formally test the null hypothesis using α = .05. What do you conclude?

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d. A t-test on theβ-value for the interaction term in the model,x1x2, resulted in a p-value of 0.02. Practically interpret this result, usingα=0.05.
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