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Chapter 12: 121E (page 802)

Consider fitting the multiple regression model

E(y)=尾0+尾1x1+尾2x2+尾3x3+尾4x4+尾5x5

A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain.

Short Answer

Expert verified

In this question, x4 and x2has a correlation of 0.93 and x4 and x5 has a correlation of 0.86. These correlation numbers are very high indicating a strong positive relationship between x4 and x2and x4 and x5 respectively. Thus, the problem of multicollinearity exists in the model.

Step by step solution

01

Multicollinearity check

Multicollinearity is checked by checking the correlation amongst the independent variables. If there is high correlation amongst any two independent variables, it is said that the problem of multicollinearity exists in the model.

02

Application of multicollinearity check

In this question, x4 and x2has a correlation of 0.93 and x4 and x5 has a correlation of 0.86. These correlation numbers are very high indicating a strong positive relationship between x4 and x2and x4 and x5 respectively. Thus, the problem of multicollinearity exists in the model.

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Most popular questions from this chapter

Consider fitting the multiple regression model

Ey=尾0+尾1x1+尾2x2+尾3x3+尾4x4+尾5x5

A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain.

Question: Tipping behaviour in restaurants. Can food servers increase their tips by complimenting the customers they are waiting on? To answer this question, researchers collected data on the customer tipping behaviour for a sample of 348 dining parties and reported their findings in the Journal of Applied Social Psychology (Vol. 40, 2010). Tip size (y, measured as a percentage of the total food bill) was modelled as a function of size of the dining party(x1)and whether or not the server complimented the customers鈥� choice of menu items (x2). One theory states that the effect of the size of the dining party on tip size is independent of whether or not the server compliments the customers鈥� menu choices. A second theory hypothesizes that the effect of size of the dining party on tip size is greater when the server compliments the customers鈥� menu choices as opposed to when the server refrains from complimenting menu choices.

a. Write a model for E(y) as a function of x1 and x2 that corresponds to Theory 1.

b. Write a model for E(y) as a function of x1and x2that corresponds to Theory 2.

c. The researchers summarized the results of their analysis with the following graph. Based on the graph, which of the two models would you expect to fit the data better? Explain.

Question: Bus Rapid Transit study. Bus Rapid Transit (BRT) is a rapidly growing trend in the provision of public transportation in America. The Center for Urban Transportation Research (CUTR) at the University of South Florida conducted a survey of BRT customers in Miami (Transportation Research Board Annual Meeting, January 2003). Data on the following variables (all measured on a 5-point scale, where 1 = very unsatisfied and 5 = very satisfied) were collected for a sample of over 500 bus riders: overall satisfaction with BRT (y), safety on bus (x1), seat availability (x2), dependability (x3), travel time (x4), cost (x5), information/maps (x6), convenience of routes (x7), traffic signals (x8), safety at bus stops (x9), hours of service (x10), and frequency of service (x11). CUTR analysts used stepwise regression to model overall satisfaction (y).

a. How many models are fit at step 1 of the stepwise regression?

b. How many models are fit at step 2 of the stepwise regression?

c. How many models are fit at step 11 of the stepwise regression?

d. The stepwise regression selected the following eight variables to include in the model (in order of selection): x11, x4, x2, x7, x10, x1, x9, and x3. Write the equation for E(y) that results from stepwise regression.

e. The model, part d, resulted in R2 = 0.677. Interpret this value.

f. Explain why the CUTR analysts should be cautious in concluding that the best model for E(y) has been found.

Question: Adverse effects of hot-water runoff. The Environmental Protection Agency (EPA) wants to determine whether the hot-water runoff from a particular power plant located near a large gulf is having an adverse effect on the marine life in the area. The goal is to acquire a prediction equation for the number of marine animals located at certain designated areas, or stations, in the gulf. Based on past experience, the EPA considered the following environmental factors as predictors for the number of animals at a particular station:

X1 = Temperature of water (TEMP)

X2 = Salinity of water (SAL)

X3 = Dissolved oxygen content of water (DO)

X4 = Turbidity index, a measure of the turbidity of the water (TI)

x5 = Depth of the water at the station (ST_DEPTH)

x6 = Total weight of sea grasses in sampled area (TGRSWT)

As a preliminary step in the construction of this model, the EPA used a stepwise regression procedure to identify the most important of these six variables. A total of 716 samples were taken at different stations in the gulf, producing the SPSS printout shown below. (The response measured was y, the logarithm of the number of marine animals found in the sampled area.)

a. According to the SPSS printout, which of the six independent variables should be used in the model? (Use 伪 = .10.)

b. Are we able to assume that the EPA has identified all the important independent variables for the prediction of y? Why?

c. Using the variables identified in part a, write the first-order model with interaction that may be used to predict y.

d. How would the EPA determine whether the model specified in part c is better than the first-order model?

e.Note the small value of R2. What action might the EPA take to improve the model?


Factors that impact an auditor鈥檚 judgment. A study was conducted to determine the effects of linguistic delivery style and client credibility on auditors鈥� judgments (Advances in Accounting and Behavioural Research, 2004). Two hundred auditors from Big 5 accounting firms were each asked to perform an analytical review of a fictitious client鈥檚 financial statement. The researchers gave the auditors different information on the client鈥檚 credibility and linguistic delivery style of the client鈥檚 explanation. Each auditor then provided an assessment of the likelihood that the client-provided explanation accounted for the fluctuation in the financial statement. The three variables of interest鈥攃redibility (x1), linguistic delivery style (x2) , and likelihood (y) 鈥攚ere all measured on a numerical scale. Regression analysis was used to fit the interaction model,y=尾0+尾1x1+尾2x2+尾3x1x2+蔚 . The results are summarized in the table at the bottom of page.

a) Interpret the phrase client credibility and linguistic delivery style interact in the words of the problem.

b) Give the null and alternative hypotheses for testing the overall adequacy of the model.

c) Conduct the test, part b, using the information in the table.

d) Give the null and alternative hypotheses for testing whether client credibility and linguistic delivery style interact.

e) Conduct the test, part d, using the information in the table.

f) The researchers estimated the slope of the likelihood鈥搇inguistic delivery style line at a low level of client credibility 1x1 = 222. Obtain this estimate and interpret it in the words of the problem.

g) The researchers also estimated the slope of the likelihood鈥搇inguistic delivery style line at a high level of client credibility 1x1 = 462. Obtain this estimate and interpret it in the words of the problem.
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