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?

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

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

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

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.