Regression Discontinuity Analysis

The Standard Model

To make these ideas more concrete, the following example will run through this text. In this hypothetical study, the assignment variable is a student’s score on a standardized test taken in 10th grade, the treatment is whether the student is enrolled in a standardized test prep class, and the focal outcome measure is “self-efficacy,” the student’s belief in the student’s ability to improve the student’s standardized test performance. The school provides the test prep class to students scoring in the lowest 30% on the 10th-grade test. The data from this hypothetical example are displayed in Figure 1.

Figure 1 Students’ self-efficacy scores increased significantly as a result of the test prep class

One key to understanding this type of analysis is noting that over and above the effect of the treatment variable, there is a relationship between the assignment variable and the outcome variable. In the given example, even if the test prep class is effective, it is quite possible that the students who attended the class will have lower self-efficacy scores on average than those who did not, simply because they started out at a lower level at the outset of the study. The question is whether the students who receive the class will have higher self-efficacy than would be predicted based on their 10th-grade test scores.

A regression discontinuity design is analyzed as follows: The outcome variable (Y, self-efficacy) is regressed on the treatment variable (X, attending the test prep class or not) and the assignment variable (Z, 10th-grade test score). One thus obtains the following regression equation:

$$Y=b_0+b_{1}X+b_{2}Z+e.$$

If the coefficient b1 is statistically significant, the data suggest that the treatment has an effect on the outcome variable. On the graph, this treatment effect will manifest as a vertical discrepancy between the two parallel regression lines. In the given example, the self-efficacy scores of the students in the test prep class were higher than would be expected based on their 10th-grade test scores. Because the treatment is dichotomous, the treatment effect is exactly equal to the coefficient b1. Students who attended the test prep class had self-efficacy scores 7.5 points higher as a result of taking the class.

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