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Chapter 12: 86P (page 765)

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.

Short Answer

Expert verified

A second-order model with one quantitative variable and one qualitative variable with 4 levels can be written as Ey=尾0+尾1x1+尾2x12+尾3x2+尾4x3+尾5x4.

Step by step solution

01

Variable conditions

There are two independent variables: one quantitative variable (say x1) with a model in second-order and one qualitative variable with 4 levels (for k levels, (k-1) no of variables will be introduced in the model; namely x2,x3, and x4). There are no interactions observed between the two independent variables.

02

Model for E(y)

A second-order model with one quantitative variable and one qualitative variable with 4 levels can be written asEy=尾0+尾1x1+尾2x12+尾3x2+尾4x3+尾5x4

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Most popular questions from this chapter

Question: Do blondes raise more funds? Refer to the Economic Letters (Vol. 100, 2008) study of whether the color of a female solicitor鈥檚 hair impacts the level of capital raised, Exercise 12.75 (p. 756). Recall that 955 households were contacted by a female solicitor to raise funds for hazard mitigation research. In addition to the household鈥檚 level of contribution (in dollars) and the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female), the researcher also recorded the beauty rating of the solicitor (measured quantitatively, on a 10-point scale).

  1. Write a first-order model (with no interaction) for mean contribution level, E(y), as a function of a solicitor鈥檚 hair color and her beauty rating.
  2. Refer to the model, part a. For each hair color, express the change in contribution level for each 1-point increase in a solicitor鈥檚 beauty rating in terms of the model parameters.
  3. Write an interaction model for mean contribution level, E(y), as a function of a solicitor鈥檚 hair color and her beauty rating.
  4. Refer to the model, part c. For each hair color, express the change in contribution level for each 1-point increase in a solicitor鈥檚 beauty rating in terms of the model parameters.
  5. Refer to the model; part c. Illustrate the interaction with a graph.

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: Write a second-order model relating the mean of y, E(y), to

a. one quantitative independent variable

b. two quantitative independent variables

c. three quantitative independent variables [Hint: Include allpossible two- way cross-product terms and squared terms.]

Question: Refer to Exercise 12.82.

a. Write a complete second-order model that relates E(y) to the quantitative variable.

b. Add the main effect terms for the qualitative variable (at three levels) to the model of part a.

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 curves of the model have the same shape but different y-intercepts?

e. Under what circumstances will the response curves of the model be parallel lines?

f. Under what circumstances will the response curves of the model be identical?

Question: Consider the model:

y=尾0+尾1x1+尾2x2+尾3x3+蔚

where x1 is a quantitative variable and x2 and x3 are dummy variables describing a qualitative variable at three levels using the coding scheme

role="math" localid="1649846492724" x2=1鈥�鈥�鈥�iflevel鈥�20鈥�鈥�otherwise鈥�鈥�鈥�鈥�鈥�鈥�鈥�x3=1鈥�鈥�鈥�if鈥�level鈥�30鈥�鈥�鈥�otherwise

The resulting least squares prediction equation is y^=44.8+2.2x1+9.4x2+15.6x3

a. What is the response line (equation) for E(y) when x2 = x3 = 0? When x2 = 1 and x3 = 0? When x2 = 0 and x3 = 1?

b. What is the least squares prediction equation associated with level 1? Level 2? Level 3? Plot these on the same graph.
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