Question

Question:Consider the first-order model equation in three quantitative independent variables $\mathbf{E}\mathbf{\left(}\mathbf{Y}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{-}\mathbf{3}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{5}{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_{\mathbf{3}}$

1. Graph the relationship between Y and ${\mathbf{x}}_{\mathbf{3}}$for ${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$ and ${\mathbf{x}}_2\mathbf{=}1$
2. Repeat part a for ${\mathbf{x}}_1\mathbf{=}1$and ${\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{2}$
3. How do the graphed lines in parts a and b relate to each other? What is the slope of each line?
4. If a linear model is first-order in three independent variables, what type of geometric relationship will you obtain when is graphed as a function of one of the independent variables for various combinations of the other independent variables?

Short answer

(A) Graph

(B) Graph

(C) The two lines are parallel to each other where the slope of the lines is -1.

(D) The geometric relationship between $\mathbf{E}\mathbf{\left(}\mathbf{Y}\mathbf{\right)}$ and any independent variable for various combinations of other variables will be a linear relationship.

Step-by-step solution

Step-by-Step SolutionStep 1: Graph

Given, $\mathbf{E}\mathbf{\left(}\mathbf{Y}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{-}\mathbf{3}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{5}{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_3$for ${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{2}$and ${\mathbf{x}}_2\mathbf{=}1$

$\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{-}\mathbf{3}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{-}{\mathbf{x}}_{\mathbf{3}}$

$\mathbf{y}\mathbf{=}\mathbf{1}\mathbf{-}{\mathbf{x}}_{\mathbf{3}}$

Now to plot this equation, we make a table

Y	1	0
X3	0	1

Graph

Given,

$\mathbf{E}\mathbf{\left(}\mathbf{Y}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{-}\mathbf{3}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}\mathbf{5}{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_3$for ${\mathbf{x}}_1\mathbf{=}1$ and ${\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{2}$

$$\begin{gathered} \mathbf{y}\mathbf{=}\mathbf{2}\mathbf{-}\mathbf{3}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{+}\mathbf{5}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{-}{\mathbf{x}}_{\mathbf{3}} \\ \mathbf{y}\mathbf{=}\mathbf{-}\mathbf{11}\mathbf{-}{\mathbf{x}}_{\mathbf{3}} \end{gathered}$$

Now to plot this equation, we make a table

Y	-11	0
X3	0	-11

Relationship between the two graphs

From the two graphs, it is visible that the two lines are parallel to each other. The slope of for both the lines is -1 with the intercept value changing for part (a) and part (b). In part (a) the intercept value was 1 while in part (b) the intercept value is -11

Geometric relationship between   and any independent variable

The geometric relationship between and any independent variable for various combinations of other variables will be a linear relationship.

Any linear model which is a first-order model in three independent variables will also have a linear geometric relationship betweenand any independent variable for various combinations of other variables.

The first order linear model means that it’s a linear regression model where the variables’ maximum power in the model is 1.