16.8 Dummy Variables
16.8.1 A Dummy variable
An indicator variable, or dummy variable, is an input variable that represents qualitative data, such as gender, race, etc. Typically, dummy variables are sometimes referred to as binary variables because they usually take just two values, \(1\) or \(0\), with \(1\) generally representing the presence of a characteristic and \(0\) representing the absence.
For example, if we had a dummy variable called male, \(1\) would indicate that the individual is male and \(0\) would indicate that the individual is female (or non-male to be more precise). Using a \(1\) and \(0\) to represent the presence or absence of a particular characteristic is arbitrary, but as we will see, it is quite convenient. Generally, we can define a dummy variable \(D\) as
\[ \begin{aligned} D = \begin{cases} 1 & \text{if characteristic is present} \\ 0 & \text{if characteristic is not present} \end{cases} \end{aligned} \]
16.8.2 In the Model
Perhaps the most common use of dummy variables in a linear regression model is to modify the intercept parameter. Adding an indicator variable \(D\) to the linear model, along with a new parameter \(\beta_2\) gives us
\[ y_i = \beta_{0} + \beta_{1}x + \beta_2 D + \epsilon_i \] Suppose we had a regression model where the variable \(D = \text{DEGREE}\) represented whether or not an individual obtained a college degree, with \(1\) indicating that he or she did, and \(0\) otherwise. In this model, our dependent variable, \(y\), is income, along with another independent variable, \(x= \text{EXPERIENCE}\), indicating how much work experience in years that individual has. Such a model can be represented as follows:
\[ \text{INCOME}_i = \beta_{0} + \beta_{1}\text{EXPERIENCE} + \beta_2 \text{DEGREE} + \epsilon_i \]
that is, for those with a college degree, the intercept of the model is \((\hat \beta_{0} + \hat \beta_2)\), while the intercept of the model for those without a college degree is simply \(\hat \beta_{0}\).
\[ \begin{align} \hat{\text{INCOME}} = \begin{cases} (\hat \beta_{0} + \hat \beta_2) + \hat \beta_{1}\text{EXPERIENCE}, & \text{If DEGREE}=1 \\ \hat \beta_{0} + \hat \beta_{1}\text{EXPERIENCE}, & \text{If DEGREE} = 0 \end{cases} \end{align} \]
16.8.3 An Example
We have information about the 2008-09 nine-month academic salary for Assistant Professors, Associate Professors and Professors in a college in the U.S. The data were collected as part of the on-going effort of the college’s administration to monitor salary differences between male and female faculty members. The available data set has 397 observations on the following 6 variables:
| label | description |
|---|---|
| rank | a factor with levels AssocProf, AsstProf and Prof |
| discipline | a factor with levels A (“theoretical” departments) or B (“applied” departments) |
| yrs.since.phd | years since PhD |
| yrs.service | years of service |
| sex | a factor with levels Female and Male |
| salary | nine-month salary, in dollars |
| Source: carData package in R |
| rank | discipline | yrs.since.phd | yrs.service | sex | salary |
|---|---|---|---|---|---|
| Prof | B | 27 | 14 | Male | 147349 |
| AsstProf | B | 2 | 0 | Male | 78000 |
| AssocProf | B | 48 | 53 | Male | 90000 |
| Prof | A | 33 | 31 | Male | 111350 |
| Prof | A | 39 | 36 | Male | 117515 |
| Prof | A | 49 | 43 | Male | 72300 |
| Prof | A | 16 | 11 | Male | 88175 |
| Prof | A | 23 | 23 | Male | 108200 |
| Prof | A | 25 | 15 | Male | 95329 |
| Prof | A | 30 | 26 | Male | 136660 |
| Note: First 10 observations |
16.8.3.1 Categorical variables with two levels
Suppose that, we wish to investigate differences in salaries between males and females.
Based on the gender variable, we can create a new dummy variable, \(D\), that takes the value:
\[ \begin{align} D = \begin{cases} 1 & \text{if a person is male} \\ 0 & \text{if a person is female} \end{cases} \end{align} \] and use this variable as a predictor in the regression model $ Y_i=_0 + _1D + _i $, leading to:
\[ \begin{align} \begin{cases} \beta_{0} + \beta_1, & \text{if person is male: } D=1 \\ \beta_{0} , & \text{if person is female: } D=0 \end{cases} \end{align} \] The coefficients can be interpreted as follow:
\[ \begin{align} \begin{cases} \beta_{0} + \beta_1 & \text{s the mean salary among males}\\ \beta_{0} & \text{is the mean salary among females}\\ \beta_1 & \text{is the mean difference in salary between males and females} \end{cases} \end{align} \]
| salary 000 | ||||
|---|---|---|---|---|
| Regressor | \(\hat \beta\) | std. Error | \(t_{stat}\) | \(p\)-value |
| (Intercept) | 101.00 | 4.81 | 21.00 | <0.001 |
| sex [Male] | 14.09 | 5.06 | 2.78 | 0.0057 |
| Observations | 397 | |||
| R2 / R2 adjusted | 0.019 / 0.017 | |||
From the output above (where salary is divided by 1000) the mean salary for female is estimated to be \(101.002\), whereas males are estimated a total of \(101.002 + 14.088 = 115.090\). The \(p\)-value for the dummy variable \(\text{sexMale}\) is significant, suggesting that there is a statistical evidence of a difference in mean salary between the genders.
The decision to code males as \(1\) and females as \(0\) (baseline) is arbitrary, and has no effect on the regression computation, but does alter the interpretation of the coefficients.
Let’s see the output when \(D=1\) if the person is female:
| salary 000 | ||||
|---|---|---|---|---|
| Regressor | \(\hat \beta\) | std. Error | \(t_{stat}\) | \(p\)-value |
| (Intercept) | 115.09 | 1.59 | 72.50 | <0.001 |
| sex [Female] | -14.09 | 5.06 | -2.78 | 0.0057 |
| Observations | 397 | |||
| R2 / R2 adjusted | 0.019 / 0.017 | |||
Now the estimates for \(\beta_0\) and \(\beta_1\) are \(115.090\) and \(-14.088\), respectively, leading once again to a prediction of mean salary of \(115.090\) for males and a estimation of \(115.090 - 14.088 = 101.002\) for females.
| Observations | 397 |
| Dependent variable | salary000 |
| Type | OLS linear regression |
| F(2,394) | 26.821 |
| R² | 0.120 |
| Adj. R² | 0.115 |
| Est. | 2.5% | 97.5% | t val. | p | |
|---|---|---|---|---|---|
| (Intercept) | 101.429 | 96.451 | 106.407 | 40.060 | 0.000 |
| sex_Female | -9.072 | -18.630 | 0.486 | -1.866 | 0.063 |
| yrs.service | 0.748 | 0.529 | 0.967 | 6.711 | 0.000 |
| Standard errors: OLS |
16.8.3.2 Categorical variables with more than two levels
WORK IN PROGRESS
References: