STA 506 2.0 Linear Regression Analysis

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# STA 506 2.0 Linear Regression Analysis
## Lecture 12-ii: Indicator Variables
### Dr Thiyanga S. Talagala
### 2020-11-21

---

## Introduction

- All the independent variables we have considered up to this point have been measured on a continuous scale.

- Regression analysis can be generalized to incorporate qualitative variables.

- Examples of qualitative variables:

- Gender (male, female)
    - Smoking status (smoker, nonsmoker)
    - Employment status (full-time, part-time, unemployed)
    - BMI (underweight, normal, overweight, obese)

- Categorical variables can be incorporated into regression through **indicator variables**.

- Sometimes indicator variables are called **dummy variable.**

---

## Creating dummy variables

```
   IQ Gender  BMI
1  10   Male 20.2
2  20   Male 20.5
3 100   Male 18.5
4  98   Male 25.0
5 100 Female 24.9
6  11 Female 31.0
7  50 Female 18.5
8  70 Female 20.0
```

Indicator variable for `Gender`

`$$D_i = \left\{
    \begin{array}\\
        1 & \mbox{if male} \\
        0 & \mbox{if female} 
    \end{array}
\right.$$`

The choice of 0 and 1 to identify the levels of a qualitative variable is arbitrary.

---

## Regression equation

.pull-left[

```
   IQ Gender  BMI D
1  10   Male 20.2 1
2  20   Male 20.5 1
3 100   Male 18.5 1
4  98   Male 25.0 1
5 100 Female 24.9 0
6  11 Female 31.0 0
7  50 Female 18.5 0
8  70 Female 20.0 0
```
]

.pull-right[

Indicator variable for `Gender`

`$$D_i = \left\{
    \begin{array}\\
        1 & \mbox{if male} \\
        0 & \mbox{if female} 
    \end{array}
\right.$$`

]

Regression equation

`$$y_i = \beta_0 + \beta_1 x_i + \beta_2D_i + \epsilon_i,$$`
where `$x$` represents the variable `BMI`.

---

## Regression equation (cont.)

`$$y_i = \beta_0 + \beta_1 x_i + \beta_2D_i + \epsilon_i,$$`

where `$x$` represents the variable `BMI`.

### Regression equation for **males**, `$D_i = 1$`

`$$y_i = \beta_0 + \beta_1 x_i + \beta_2 + \epsilon_i,$$`

`$$y_i = (\beta_0 +  \beta_2) + \beta_1 x_i + \epsilon_i,$$`

- Thus the relationship between **IQ score** and **BMI** for **males** is a straight line with intercept `$\beta_0+\beta_2$` and slope `$\beta_1$`.

### Regression equation for **females**, `$D_i = 0$`

`$$y_i = \beta_0 + \beta_1 x_i + \epsilon_i,$$`

- Thus the relationship between **IQ score** and **BMI** for **females** is a straight line with intercept `$\beta_0$` and slope `$\beta_1$`.

---
## Regression equation (cont.)

.pull-left[
Regression equation for **males**, `$D_i = 1$`

`$$y_i = (\beta_0 +  \beta_2) + \beta_1 x_i + \epsilon_i,$$`

Regression equation for **females**, `$D_i = 0$`

`$$y_i = \beta_0 + \beta_1 x_i + \epsilon_i,$$`
]

.pull-right[
![height=1](dummy.PNG)
]
---

.pull-left[

![](dummy.PNG)

]

.pull-right[

- Two parallel lines (a common slope and different intercepts).

- `$\beta_2$` expresses differences in heights between the two regression lines.

### Interpretations

- `$\beta_1$` - Change in the mean response, `$\mu_Y$`, for each additional unit increase in the BMI when other variables held at constant.

- `$\beta_2$` is a measure of how much higher (or lower) the mean response of the male group is than that of the female group.

( `$\beta_2$` is a measure of the difference in mean IQ score resulting from changing from male group to female group)
]

---
## Qualitative variable with more than 2 levels

In general, a qualitative variable with `$k$` levels is represented by `$k-1$` indicator variables, each taking the values 0 and 1.

.pull-left[

```
   IQ         BMI headcir D1 D2
1  10      Normal    50.2  0  1
2  20      Normal    50.5  0  1
3 100       Obese    58.5  0  0
4  98       Obese    55.0  0  0
5 100 Underweight    54.9  1  0
6  11 Underweight    40.0  1  0
7  50 Underweight    48.5  1  0
8  70 Underweight    50.0  1  0
```
]

.pull-right[

`$D_1$` | `$D_2$` | Description |
---|---|---| ---|
1|0|observation is from underweight |
0|1|observation is from normal |
0|0|observation is from Obese |
]

.pull-left[
`$$D_{1i} = \left\{
    \begin{array}\\
        1 & \mbox{if underweight} \\
        0 & \mbox{if otherwise} 
    \end{array}
\right.$$`

`$$D_{2i} = \left\{
    \begin{array}\\
        1 & \mbox{if normal} \\
        0 & \mbox{if otherwise} 
    \end{array}
\right.$$`

]

---
## Your turn

Write the regression equations for the three levels.

```
   IQ headcir D1 D2
1  10    50.2  0  1
2  20    50.5  0  1
3 100    58.5  0  0
4  98    55.0  0  0
5 100    54.9  1  0
6  11    40.0  1  0
7  50    48.5  1  0
8  70    50.0  1  0
```

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Acknowledgement

Introduction to Linear Regression Analysis, Douglas C. Montgomery, Elizabeth A. Peck, G. Geoffrey Vining

[Dr. Thiyanga S. Talagala](https://thiyanga.netlify.app/)