PDF file location: http://www.murraylax.org/rtutorials/regression_dummy.pdf

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A dummy variable or binary variable is a variable that takes on a value of 0 or 1 as an indicator that the observation has some kind of characteristic. Common examples:

It is common to use dummy variables as explanatory variables in regression models, if binary categorical variables are likely to influence the outcome variable.

1. Example: Factors Affecting Monthly Earnings

Let us examine a data set that explores the relationship between total monthly earnings (MonthlyEarnings) and a number of variables on an interval scale (i.e. numeric quantities) that may influence monthly earnings including including each person’s IQ (IQ), a measure of knowledge of their job (Knowledge), years of education (YearsEdu), and years experience (YearsExperience), years at current job (Tenure).

The data set also includes dummy variables that may explain monthly earnings, including whether or not the person is black / African American (Black), whether or not the person lives in a Southern U.S. state (South), and whether or not the person lives in an urban area (Urban).

The code below downloads a CSV file that includes data on the above variables from 1980 for 935 individuals and assigns it to a data frame that we name wages.

wages <- read.csv("http://murraylax.org/datasets/wage2.csv");

The following call to lm() estimates a multiple regression predicting monthly earnings based on the eight explanatory variables given above, which includes three dummy variables. The next call to summary() displays some summary statistics for the estimated regression.

lmwages <- lm(MonthlyEarnings 
              ~ IQ + Knowledge + YearsEdu + YearsExperience + Tenure
              + Black + South + Urban,
              data = wages)
summary(lmwages)
## 
## Call:
## lm(formula = MonthlyEarnings ~ IQ + Knowledge + YearsEdu + YearsExperience + 
##     Tenure + Black + South + Urban, data = wages)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -874.42 -229.18  -40.25  181.26 2163.02 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -451.0098   121.3752  -3.716 0.000215 ***
## IQ                 2.5966     0.9963   2.606 0.009301 ** 
## Knowledge          6.5545     1.8142   3.613 0.000319 ***
## YearsEdu          47.6530     7.1378   6.676 4.22e-11 ***
## YearsExperience   12.4833     3.1746   3.932 9.04e-05 ***
## Tenure             6.2910     2.4049   2.616 0.009043 ** 
## Black           -110.6660    39.2222  -2.822 0.004882 ** 
## South            -50.8222    25.7903  -1.971 0.049068 *  
## Urban            155.4316    26.4621   5.874 5.94e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 356.7 on 926 degrees of freedom
## Multiple R-squared:  0.2285, Adjusted R-squared:  0.2219 
## F-statistic: 34.29 on 8 and 926 DF,  p-value: < 2.2e-16

The p-values in the right-most column reveal that all of the coefficients are statistically significantly different from zero at the 5% significance level. We have statistical evidence that all of these variables influence monthly earnings.

The coefficient on Black is equal to -110.67. This means that even after accounting for the effects of all the other explanatory variables in the model (includes educational attainment, experience, location, knowledge, and IQ), black / African American people earn on average $110.67 less per month than non-black people.

The coefficient on South is -50.82. Accounting for the impact of all the variables in the model, people that live in Southern United States earn on average $50.82 less per month than others.

The coefficient on Urban is 155.43. Accounting for the impact of all the variables in the model, people that live in urban areas earn $155.43 more per month, which probably reflects a higher cost of living.

We can compute confidence intervals for these effects with the following call to confint()

confint(lmwages, parm=c("Black", "South", "Urban"), level = 0.95)
##           2.5 %      97.5 %
## Black -187.6407 -33.6913263
## South -101.4365  -0.2079364
## Urban  103.4989 207.3642822

2. Dummy Interactions with Numeric Explanatory Variables

We found that black people have lower monthly earnings on average than non-black people. In our regression equation, this implies that the intercept is lower for black people than non-black people. We can also test whether a dummy variable affects the slope multiplying other variables.

For example, are there differences in the returns to education for black versus non-black people? To answer this, we include an interaction effect between Black and YearsEdu:

lmwages <- lm(MonthlyEarnings
              ~ IQ + Knowledge + YearsEdu + YearsExperience + Tenure
              + Black + South + Urban + Black*YearsEdu,
              data = wages)
summary(lmwages)
## 
## Call:
## lm(formula = MonthlyEarnings ~ IQ + Knowledge + YearsEdu + YearsExperience + 
##     Tenure + Black + South + Urban + Black * YearsEdu, data = wages)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -871.77 -223.35  -39.15  183.60 2166.96 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -484.8569   122.7181  -3.951 8.38e-05 ***
## IQ                 2.5965     0.9951   2.609 0.009224 ** 
## Knowledge          6.6834     1.8135   3.685 0.000242 ***
## YearsEdu          50.0652     7.2573   6.899 9.73e-12 ***
## YearsExperience   12.0943     3.1784   3.805 0.000151 ***
## Tenure             6.3322     2.4022   2.636 0.008528 ** 
## Black            328.4032   249.9481   1.314 0.189211    
## South            -48.6125    25.7902  -1.885 0.059753 .  
## Urban            155.1421    26.4318   5.870 6.09e-09 ***
## YearsEdu:Black   -35.0262    19.6929  -1.779 0.075630 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 356.3 on 925 degrees of freedom
## Multiple R-squared:  0.2312, Adjusted R-squared:  0.2237 
## F-statistic:  30.9 on 9 and 925 DF,  p-value: < 2.2e-16

We see here that when accounting for an interaction effect between race and education, the coefficient on the Black dummy variable becomes insignificant, but the coefficient on the interaction term is negative and significant at the 10% level. The coefficient on the interaction term equal to -35.03 means the slope on education is 35.03 less when \(Black=1\).

The coefficient on the interaction term is interpreted as the additional marginal effect of the numeric variable for the group associated with the dummy variable equal to 1. For this example:

3. Interacting Dummy Variables with Each Other

Let us interact two of the dummy variables to understand this interpretation and motivation. In the call to lm() below, we use our baseline model and interact South and Urban:

lmwages <- lm(MonthlyEarnings
              ~ IQ + Knowledge + YearsEdu + YearsExperience + Tenure
              + Black + South + Urban + South*Urban,
              data = wages)
summary(lmwages)
## 
## Call:
## lm(formula = MonthlyEarnings ~ IQ + Knowledge + YearsEdu + YearsExperience + 
##     Tenure + Black + South + Urban + South * Urban, data = wages)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -885.94 -228.09  -36.76  173.16 2153.62 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -516.8840   124.9159  -4.138 3.83e-05 ***
## IQ                 2.7472     0.9968   2.756 0.005964 ** 
## Knowledge          6.6968     1.8118   3.696 0.000232 ***
## YearsEdu          48.1580     7.1275   6.757 2.50e-11 ***
## YearsExperience   12.9375     3.1753   4.074 5.01e-05 ***
## Tenure             6.1817     2.4007   2.575 0.010178 *  
## Black           -109.0280    39.1521  -2.785 0.005467 ** 
## South             30.3594    45.5537   0.666 0.505288    
## Urban            200.1871    33.5683   5.964 3.51e-09 ***
## South:Urban     -116.3504    53.8671  -2.160 0.031033 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 356 on 925 degrees of freedom
## Multiple R-squared:  0.2324, Adjusted R-squared:  0.2249 
## F-statistic: 31.12 on 9 and 925 DF,  p-value: < 2.2e-16

To interpret the meaning of the coefficient on South, Urban, and South*Urban, we will ignore (hold constant) all the terms in the regression equation that do not include one of these variables.

3.1 Difference between Urban and Rural Workers in the North/East/West

Workers in the North / East / and West U.S. have \(South=0\). Here \(South=0\), \((South~x~Urban)=0\), so neither the coefficient on the interaction nor the coefficient on South come into play.

The coefficient for \(b_{Urban}\) implies that in the Non-Southern U.S., urban workers earn on average $200.19 more in monthly earnings than rural workers.

3.2 Difference between Urban and Rural Workers in the South

When focusing on workers in the South, \(South=1\) and the interaction term comes into play.

  • Impact for urban workers in the south = \(b_{South}(1) + b_{Urban}(1) + b_{Urban*South}(1)\)
  • Impact for rural workers in the south = \(b_{South}(1) + b_{Urban}(0) + b_{Urban*South}(0)\)
  • Difference = \(b_{Urban} + b_{Urban*South} = 200.19 - 116.35 =\) $83.84

In the Southern U.S. states, urban workers on average earn $83.84 more in monthly earnings than rural workers.

3.3 Difference between Southern and North/East/West Monthly Earnings for Urban Workers

  • Impact for Southern urban workers = \(b_{South}(1) + b_{Urban}(1) + b_{Urban*South}(1)\)
  • Impact for Non-Southern urban workers = \(b_{South}(0) + b_{Urban}(1) + b_{Urban*South}(0)\)
  • Difference = \(b_{South} + b_{Urban*South} = 30.36 - 116.35 =\) -$85.99

For urban workers, workers in the South earn $85.99 less in monthly earnings than workers outside the South.

3.4 Difference between Southern and North/East/West Monthly Earnings for Rural Workers

Rural workers have \(Urban=0\) and so the interaction term \(Urban~x~South=0\), so we can ignore both of those coefficients. The coefficient for \(b_{South}\) implies that Southern rural workers earn on average $30.36$ more per month than Non-Southern rural workers.

4 Three-Way Interactions and Higher!

What?! Things aren’t complicated enough for you?! Do at your own peril!

I have seen people include higher order interaction effects like South * Urban * Black * YearsEdu in their regressions. It has never been obvious to me that they understood what their results meant.