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## 5.3 Regression when X is a Binary Variable

Instead of using a continuous regressor $X$, we might be interested in running the regression

$Y_i = \beta_0 + \beta_1 D_i + u_i \tag{5.2}$

where $D_i$ is a binary variable, a so-called dummy variable. For example, we may define $D_i$ as follows:

$D_i = \begin{cases} 1 \ \ \text{if STR in i^{th} school district < 20} \\ 0 \ \ \text{if STR in i^{th} school district \geq 20} \\ \end{cases} \tag{5.3}$

The regression model now is

$TestScore_i = \beta_0 + \beta_1 D_i + u_i. \tag{5.4}$

Let us see how these data look like in a scatter plot:

# Create the dummy variable as defined above
CASchools$D <- CASchools$STR < 20

# Plot the data
plot(CASchools$D, CASchools$score,            # provide the data to be plotted
pch = 20,                                # use filled circles as plot symbols
cex = 0.5,                               # set size of plot symbols to 0.5
col = "Steelblue",                       # set the symbols' color to "Steelblue"
xlab = expression(D[i]),                 # Set title and axis names
ylab = "Test Score",
main = "Dummy Regression") With $D$ as the regressor, it is not useful to think of $\beta_1$ as a slope parameter since $D_i \in \{0,1\}$, i.e., we only observe two discrete values instead of a continuum of regressor values. There is no continuous line depicting the conditional expectation function $E(TestScore_i | D_i)$ since this function is solely defined for $x$-positions $0$ and $1$.

Therefore, the interpretation of the coefficients in this regression model is as follows:

• $E(Y_i | D_i = 0) = \beta_0$, so $\beta_0$ is the expected test score in districts where $D_i=0$ where $STR$ is above $20$.

• $E(Y_i | D_i = 1) = \beta_0 + \beta_1$ or, using the result above, $\beta_1 = E(Y_i | D_i = 1) - E(Y_i | D_i = 0)$. Thus, $\beta_1$ is the difference in group-specific expectations, i.e., the difference in expected test score between districts with $STR < 20$ and those with $STR \geq 20$.

We will now use R to estimate the dummy regression model as defined by the equations (5.2) and (5.3) .

# estimate the dummy regression model
dummy_model <- lm(score ~ D, data = CASchools)
summary(dummy_model)
#>
#> Call:
#> lm(formula = score ~ D, data = CASchools)
#>
#> Residuals:
#>     Min      1Q  Median      3Q     Max
#> -50.496 -14.029  -0.346  12.884  49.504
#>
#> Coefficients:
#>             Estimate Std. Error t value Pr(>|t|)
#> (Intercept)  650.077      1.393 466.666  < 2e-16 ***
#> DTRUE          7.169      1.847   3.882  0.00012 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 18.74 on 418 degrees of freedom
#> Multiple R-squared:  0.0348,	Adjusted R-squared:  0.0325
#> F-statistic: 15.07 on 1 and 418 DF,  p-value: 0.0001202
summary() reports the $p$-value of the test that the coefficient on (Intercept) is zero to to be < 2e-16. This scientific notation states that the $p$-value is smaller than $\frac{2}{10^{16}}$, so a very small number. The reason for this is that computers cannot handle arbitrary small numbers. In fact, $\frac{2}{10^{16}}$ is the smallest possble number R can work with.

The vector CASchools$D has the type logical (to see this, use typeof(CASchools$D)) which is shown in the output of summary(dummy_model): the label DTRUE states that all entries TRUE are coded as 1 and all entries FALSE are coded as 0. Thus, the interpretation of the coefficient DTRUE is as stated above for $\beta_1$.

One can see that the expected test score in districts with $STR < 20$ ($D_i = 1$) is predicted to be $650.1 + 7.17 = 657.27$ while districts with $STR \geq 20$ ($D_i = 0$) are expected to have an average test score of only $650.1$.

Group specific predictions can be added to the plot by execution of the following code chunk.

# add group specific predictions to the plot
points(x = CASchools\$D,
y = predict(dummy_model),
col = "red",
pch = 20)

Here we use the function predict() to obtain estimates of the group specific means. The red dots represent these sample group averages. Accordingly, $\hat{\beta}_1 = 7.17$ can be seen as the difference in group averages.

summary(dummy_model) also answers the question whether there is a statistically significant difference in group means. This in turn would support the hypothesis that students perform differently when they are taught in small classes. We can assess this by a two-tailed test of the hypothesis $H_0: \beta_1 = 0$. Conveniently, the $t$-statistic and the corresponding $p$-value for this test are computed by summary().

Since t value $= 3.88 > 1.96$ we reject the null hypothesis at the $5\%$ level of significance. The same conclusion results when using the $p$-value, which reports significance up to the $0.00012\%$ level.

As done with linear_model, we may alternatively use the function confint() to compute a $95\%$ confidence interval for the true difference in means and see if the hypothesized value is an element of this confidence set.

# confidence intervals for coefficients in the dummy regression model
confint(dummy_model)
#>                  2.5 %    97.5 %
#> (Intercept) 647.338594 652.81500
#> DTRUE         3.539562  10.79931

We reject the hypothesis that there is no difference between group means at the $5\%$ significance level since $\beta_{1,0} = 0$ lies outside of $[3.54, 10.8]$, the $95\%$ confidence interval for the coefficient on $D$.