# Dummy Variable Trap

## What is the Dummy Variable Trap?

The Dummy Variable Trap occurs when two or more dummy variables created by one-hot encoding are highly correlated (multi-collinear). This means that one variable can be predicted from the others, making it difficult to interpret predicted coefficient variables in regression models. In other words, the individual effect of the dummy variables on the prediction model can not be interpreted well because of multicollinearity.

Using the one-hot encoding method, a new dummy variable is created for each categorical variable to represent the presence (1) or absence (0) of the categorical variable. For example, if tree species is a categorical variable made up of the values pine or oak, then tree species can be represented as a dummy variable by converting each variable to a one-hot vector. This means that a separate column is obtained for each category, where the first column represents if the tree is pine and the second column represents if the tree is oak. Each column will contain a 0 or 1 if the tree in question is of the column's species. These two columns are multi-collinear since if a tree is pine, then we know it's not oak and vice versa.

## Further explanation

To demonstrate the dummy variable trap, consider that we have a categorical variable of tree species, and assume that we have 7 trees:

$$\large x_{species} = [pine, oak, oak, pine, pine, pine, oak]$$

If the tree species variable is converted to dummy variables, the two vectors obtained:

$$\large x_{pine} = [1,0,0,1,1,1,0] \\[.5em] \large x_{oak} = [0,1,1,1,1,0,1]$$

Because a 1 in the pine column would mean a 0 in the oak column, we can say $\large x_{pine} = 1 – x_{oak}$. This results in two dummy variables that are multi-collinear, and so the dummy variable trap may occur in regression analysis.

To overcome the Dummy variable Trap, we drop one of the columns created when the categorical variable were converted to dummy variables by one-hot encoding. This can be done because the dummy variables include redundant information.

To see why this is the case, consider a multiple linear regression model for the given simple example as follows:

$$$$\large y = \beta_{0} + \beta_{1} {x_{pine}} + \beta_{2} {x_{oak}} + \epsilon$$$$

where $y$ is the response variable, $x_{pine}$ and $x_{oak}$ are the explanatory variables, $\beta_0$ is the intercept, $\beta_1$ and $\beta_2$ are the regression coefficients, and $\epsilon$ is the error term. Since these two dummy variables are multi-collinear — hence we know if a tree is pine, then it's not oak — we can substitute $x_{oak}$ by ($1 – x_{pine}$) in the multiple linear regression equation.

\large \begin{aligned} y &= \beta_{0} + \beta_{1} x_{pine} + ({1-x_{pine}}) \beta_{2} + \epsilon \\[.5em] &= (\beta_{0} + \beta_{2} ) + (\beta_{1} - \beta_{2}) x_{pine} + \epsilon \end{aligned}

As you can see, we were able to rewrite the regression equation using only $x_{pine}$, where the new coefficients to be predicted are $(\beta_{0} + \beta_{2})$ and $(\beta_{1} - \beta_{2})$. By dropping a dummy variable column, we can avoid this trap.

This example shows two categories, but this can be expanded to any number of categorical variables. In general, if we have $p$ number of categories, we will use $p-1$ dummy variables. Dropping one dummy variable to protect from the dummy variable trap.

## Python Example

The get_dummies() function in the Pandas library can be used to create dummy variables.

import pandas as pd



Create a simple categorical variable:

c1 = ['pine', 'oak', 'oak', 'pine', 'pine' ]



Then convert the categorical variable to dummy variables:

pd.get_dummies(c1)


Out:
oakpine
001
110
210
301
401

You can see that pandas has one-hot encoded our two tree species into two columns.

To drop the first dummy variable, we can specify the drop_first parameter in the get_dummies function:

pd.get_dummies(c1, drop_first=True)


Out:
pine
01
10
20
31
41

Now, we simply know whether a tree is pine or not pine.

For a more complex example, consider the following randomly created dataset:

# Defining a simple dataset
data ={
'gender' : ['female', 'male', 'male', 'male', 'male', 'male','female', 'male', 'male', 'female'],
'race' : ['white', 'hispanic', 'african', 'asian', 'asian', 'white', 'african', 'hispanic','white', 'african'],
'age' : [12,15,22,21,29,42,17, 25,14,47],
'income': [25,34,42,50,48,39,25,73,86,61]
}
df = pd.DataFrame (data, columns=data.keys())

print(df)


Out:
gender      race  age  income
0  female     white   12      25
1    male  hispanic   15      34
2    male   african   22      42
3    male     asian   21      50
4    male     asian   29      48
5    male     white   42      39
6  female   african   17      25
7    male  hispanic   25      73
8    male     white   14      86
9  female   african   47      61


pd.get_dummies(df, prefix=['gender', 'race'])


Out:
ageincomegender_femalegender_malerace_africanrace_asianrace_hispanicrace_white
01225100001
11534010010
22242011000
32150010100
42948010100
54239010001
61725101000
72573010010
81486010001
94761101000

The gender variable is converted to gender_female and gender_male, and the race variable is converted to the dummy variable race_african, race_asian, race_hispanic, and race_white. Again, to avoid the dummy variable trap, the last dummy variable is dropped by setting drop_first=True:

pd.get_dummies(df, prefix=['gender', 'race'], drop_first=True)


Out:
ageincomegender_malerace_asianrace_hispanicrace_white
012250001
115341010
222421000
321501100
429481100
542391001
617250000
725731010
814861001
947610000

The result shows that the gender_female and race_african dummy variables are dropped. If we have more than two categories, the dropped variable can be thought of as the absence of all other options, represented by zeros in every column. In this example, race_african = 1 - (race_asian + race_hispanic + race_white). In other words, if race_asian, race_hispanic, and race_white are all zero, the race of this record is assumed to be the dropped variable race_african.