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

$$D_i=\{\begin{array}{cc}1& \text{if male}\\ 0& \text{if female}\end{array}$$

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

Regression equation

   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

Regression equation

$$y_i={\beta }_0+{\beta }_1{x}_i+{\beta }_2{D}_i+{ϵ}_i,$$ where $x$ represents the variable BMI.

Regression equation (cont.)

$$y_i={\beta }_0+{\beta }_1{x}_i+{\beta }_2{D}_i+{ϵ}_i,$$

where $x$ represents the variable BMI.

Regression equation for males, $D_i=1$

$$y_i={\beta }_0+{\beta }_1{x}_i+{\beta }_2+{ϵ}_i,$$

$$y_i=({\beta }_0+{\beta }_2)+{\beta }_1{x}_i+{ϵ}_i,$$

• 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, $D_i=0$

$$y_i={\beta }_0+{\beta }_1{x}_i+{ϵ}_i,$$

• 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.)

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.

   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

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