Question

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?

Short answer

a. A complete second-order model equation for y relating to the quantitative variable can be written as $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}$.

b. A complete second-order model equation for y relating to the quantitative variable and qualitative variables having 3 levels can be written as $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{3}}$.

c. A complete second-order model equation for y relating to the quantitative variable and qualitative variables having 3 levels and interaction terms can be written as $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}$.

d. The response curves will have the same shape when the slope parameter of all the variables are the same but the line intercepts have different values due to the changes in the 1-unit changes in the value of y due to the value of qualitative variables.

e. The response curves of the model will be parallel lines when the slope parameters for quantitative and qualitative variables are the same while the y-intercept values are different.

f. The response curves of the model will be identical when the slope parameters and the y-intercept values are the same for all the response lines, meaning the quantitative and qualitative variables have the identical slope and y-intercept values.

Step-by-step solution

Second-order model equation

A complete second-order model equation for y relating to the quantitative variable can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}$

Second-order model equation with added dummy variables

A complete second-order model equation for y relating to the quantitative variable and qualitative variables having 3 levels can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{3}}$

Second-order model equation with added dummy variables and interaction terms

A complete second-order model equation for y relating to the quantitative variable and qualitative variables having 3 levels and interaction terms can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{{\mathbf{x}}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}$

Graphical interpretation

The response curves will have the same shape when the slope parameter of all the variables are the same but the line intercepts have different values due to the changes in the 1-unit changes in the value of y due to the value of qualitative variables.

Graphical interpretation

The response curves of the model will be parallel lines when the slope parameters for quantitative and qualitative variables are the same while the y-intercept values are different.

Graphical interpretation

The response curves of the model will be identical when the slope parameters and the y-intercept values are the same for all the response lines, meaning the quantitative and qualitative variables have the identical slope and y-intercept values.