Question

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?

Short answer

a. A first-order model with one quantitative independent variable can be written as $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$.

b. A model including both quantitative and qualitative variables with 3 levels can be written as $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}$.

c. A model including both qualitative and quantitative variables with interactions can be written as $\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}{}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{\beta }{}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}$.

d. The response lines of the model in part c will only be parallel if no interaction amongst the variables is observed in the model. If there’s any interaction amongst the variables, then the response lines will be intersecting each other.

e. The model in part c will have only one response line when there’s no interaction in the model and the y-intercepts and slope values for all the variables are the same.

Step-by-step solution

Quantitative independent variable model

A first-order model with one quantitative independent variable can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$.

Quantitative and qualitative variable model

A model including both quantitative and qualitative variables with 3 levels can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}$.

Quantitative and qualitative variable model with interactions

A model including both qualitative and quantitative variables with interactions can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{x}{}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{\beta }{}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}$.

Graphical interpretation

The response lines of the model in part c will only be parallel if no interaction amongst the variables is observed in the model. If there’s any interaction amongst the variables, then the response lines will be intersecting each other.

Graphical interpretation

The model in part c will have only one response line when there’s no interaction in the model and the y-intercepts and slope values for all the variables are the same.