The regression discontinuity (RD) design is a quasi-experimental approach that can be used to estimate a causal effect in instances where the intervention is assigned based on whether an observed continuous “assignment” variable exceeds some arbitrary threshold.
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First introduced by Thistlewaite and Campbell, the RDD is predicated on the assumption that units (e.g., people, cities, states) with values of the assignment variable just below the threshold (who do not receive the intervention) are good controls for those just above the threshold (who do receive the intervention). In these instances, provided that the units cannot systematically sort around the cutoff, the selection mechanism is completely known and perfect measured, effectively randomizing intervention assignment to the units.
In the potential outcomes framework, a causal effect is the difference in potential outcomes between two different interventions simultaneously applied to the same unit (e.g., individual, school, city). Unfortunately, this definition requires that the same unit be observed simultaneously in two different states, a contradiction that Holland designated the Fundamental Problem of Causal Inference. In response to this problem, Rubin proposed a statistical solution that a population can be used to estimate the average causal effect of an intervention, relative to a control, across different units with the assumption that no unobserved variables confound the relationship between the exposure and outcome (i.e., exchangeability of units). The randomization of treatment assignments became the ‘gold standard’ method to create an expectation of exchangeability in units between two arms of any experiment.
Experiments can randomize intervention assignment to overcome the exchangeability assumptions, whereas non-experimental studies require several additional strong and untestable assumptions to measure causal intervention effects (see, (10)). Instances exist, however, when some arbitrary cutoff assigns an intervention to some people and does not affect others, which in expectation can create exchangeability of units around the threshold.
The rationale for the RD design is that there is no systematic difference in those who are just over the threshold (e.g., over age 21 years) and those who are just below (e.g., less than 21 years). Therefore, the ability to legally purchase alcohol, in this example, is based only on month of birth; essentially treatment is randomly assigned to some and not others.
(1) the forcing variable (Xi);
(2) threshold value (Z);
(3) outcome variable (Yi);
(4) functional form of the relationship between the exposure and outcome.
The rationale for the RD design is that the probability of treatment assignment is discontinuous and deterministic at the threshold. Equation 1 shows that treatment will be assigned (Ti= 1) when xi is greater than or equal to z, whereas no treatment will be assigned (Ti= 0) when xi is less than z. Validity of this analysis assumes that those who receive treatment should be more on one side of the threshold value than the other, i.e., there is limited cross-over in treatment assignment around the threshold.
The functional form between the forcing variable (Xi) and outcome (Yi) is the final component necessary for a RD analysis. The relationship between the 2 variables should rely on a priori theory, but data-driven approaches have been developed to supplement theory (12, 13). This point requires re-emphasis, data-driven methods to estimate the functional form should be used to complement the a priori relationship between the 2 variables, not determine the relationship. The simplest form between the forcing and outcome variable is a linear relationship (Equation 2). Similar to a typical linear regression model, Equation 2 shows that the outcome variable (Yi) is estimated by an intercept (α), slope (β), and error term (εi). Unlike a typical linear regression model, the treatment covariate (Ti) is a deterministic function of the forcing variable (Xi).
In addition, functional form can be independently estimated for observations at either side of the cutoff with local-polynomial nonparametric regression models. Insofar as RD designs use the functional form of the forcing and outcome variable to estimate the casual effect at the cutoff, the bandwidth (i.e., number of observations on each side of the threshold used to inform the local-polynomial nonparametric regression model) can greatly influence the estimated causal effect. Three different methods have largely informed bandwidth selection for these models, one proposed by Calonico, Cattaneo, and Titiunik (14), another by Imbens and Kalyanaraman (15), and a cross-validation method. Each method has been validated and shown robust to data irregularities, suggesting that an a priori hypothesized form better informs the best bandwidth. The determination of the functional form is the final component to estimate the causal intervention effect. In the next section, I will review internal and external threats to the validity of RD designs.
Overrides to intervention assignment at the threshold can occur through either unintentional or intentional (systematic) processes—each of these mechanisms will have different implications for the effect estimates. Importantly, if the misassignment is the product of some stochastic process (i.e., due to chance alone), then the effect estimates can be robust to these violations of the threshold assumption. However, cases not assigned to treatment as a result of a more systematic process should be identified and removed prior to analysis. In instances when these cases cannot be removed in advance, the analysis can continue, classifying these data based on the unit’s eligibility score instead of the treatment received, which yields unbiased estimates of the effect of assignment to treatment, instead of the treatment effect itself. However, misassignment that results from units’ that precisely manipulate the value of their forcing variable to receive, or not receive, treatment is a source of non-ignorable bias. For example, individual consumers of health insurance aware that the income threshold for single persons to qualify for Medicaid is $16,105 could manipulate their earned income (e.g., not report casual income) to fall below the threshold and receive benefits. Such deliberate falsification to the assignment variable at even a nominal level has no methodological cure.
the selection mechanism is completely known and perfectly measured, removing selection as a threat to bias, as is testing or instrumentation because both groups come from the same data. Irrespective of likelihood of confounding by an exogenous factor, two validity tests can be performed as sensitivity tests. First, covariates can be examined in relation to the forcing variable for discontinuity at the threshold. That is, the RD model can be re-estimated with the forcing variable as the dependent variable and covariates as the forcing variable. Second, potential discontinuity can be measured at additional points to rule out other explanations. Imbens and Lemieux (12) suggested that the data be limited to one side of the discontinuity, taking the median of that particular side as the revised threshold value in that section, and test whether there is discontinuity at that part of the model. The presence of discontinuity in either of these sensitivity analyses may suggest a threat to internal validity.