Chapter 12, Problem 169UP

To determine

Write a model for one quantitative variable that is at 4 different levels.

Answer to Problem 169UP

The model for one quantitative variable that is at 4 different levels is,

$\begin{array}{l}E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3\\ \text{where,}\\ x_1=\{\begin{array}{l}1\text{if }y\text{ is observed level}2\\ 0\text{otherwise}\end{array}\\ x_2=\{\begin{array}{l}1\text{if }y\text{ is observed level}3\\ 0\text{otherwise}\end{array}\\ x_3=\{\begin{array}{l}1\text{if }y\text{ is observed level}4\\ 0\text{otherwise}\end{array}\end{array}$

Explanation of Solution

Calculation:

In general for a model relating to $E\left(y\right)$ one qualitative independent variable with k variables, the model would be,

$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\mathrm{...}+{\beta }_{k-1}{x}_{k-1}$

Then the variable $x_i$ is the dummy variable for $i+1$ level. That is,

$x_i=\{\begin{array}{l}1\text{if }y\text{ is observed level}i+1\left(\text{level}A\right)\\ 0\text{otherwise}\left(\text{level}B\right)\end{array}$

The model for one quantitative variable that is at 4 different levels is,

$\underset{\_}{E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3}$

Where,

$\begin{array}{l}\underset{\_}{x_1=\{\begin{array}{l}1\text{if }y\text{ is observed level}2\\ 0\text{otherwise}\end{array}}\\ \underset{\_}{x_2=\{\begin{array}{l}1\text{if }y\text{ is observed level}3\\ 0\text{otherwise}\end{array}}\\ \underset{\_}{x_3=\{\begin{array}{l}1\text{if }y\text{ is observed level}4\\ 0\text{otherwise}\end{array}}\end{array}$

For this model, the values of ${\beta }_i^{\text{'}s}$ are defined as,

$\begin{array}{l}{\beta }_0={\mu }_A\\ {\beta }_1={\mu }_B-{\mu }_A\\ {\beta }_2={\mu }_C-{\mu }_A\\ {\beta }_3={\mu }_D-{\mu }_A\end{array}$

Also, ${\mu }_i$ denotes the mean of y for the level I for the qualitative variable and are defined as,

$\begin{array}{l}{\mu }_A={\beta }_0\\ {\mu }_B={\beta }_0+{\beta }_1\\ {\mu }_C={\beta }_0+{\beta }_2\\ {\mu }_D={\beta }_0+{\beta }_3\end{array}$