Thus far, it has been implicitly assumed that all sample units comply with their treatment assignments, that is, that all treatments and no controls receive intervention services. Under this “sharp” RD design, the probability of receiving the treatment changes from zero to one at the cutoff value.
The “fuzzy” RD design (Trochim 1984; Hahn et al. 2001) allows for noncompliers— treatment group nonparticipants and control group crossovers. Under this design, the jump in the probability of receiving the treatment at the cutoff is less than one. As an example, Van der Klaauw (2002) examined the effects of financial aid offers on college attendance, where the “score” variable was based on the applicant’s SAT scores and grades, and cutoffs were based on rules used by colleges to award aid. Applicants in higher scoring groups were more likely to receive financial aid offers than applicants in lower scoring groups. However, some higher scoring applicants did not receive financial aid offers (treatment group nonparticipants) and some lower scoring applicants did receive offers (crossovers). Thus, this is a fuzzy RD design, because application information (such as essays and extracurricular activities) that was not measured in the score also played a role in financial aid award decisions.
Under the fuzzy RD design, modifications to the estimation methods discussed above are necessary to obtain impacts that adjust for noncompliers. This situation is analogous to the distinction between intention-to-treat (ITT) and treatment-on-the-treated (TOT) estimators under RA designs (Angrist et al. 1996).
The modifications can be understood by first classifying units at the cutoff score into four mutually exclusive compliance categories: compliers, never-takers, always-takers, and defiers (Angrist et al. 1996). Let RTi denote the “potential” service receipt indicator variable in the treatment condition, and RCi denote the potential service receipt indicator variable in the control condition. Compliers (CL) are those would receive intervention services only if they were assigned to the treatment group (RTi=1 and RCi=0). Nevertakers (N) are those who would never receive treatment services (RTi=0 and RCi=0) and always-takers are those would always receive treatment services (RTi=1 and RCi=1). Finally, defiers are those who would receive the treatment only in the control condition (RTi=0 and RCi=1).
The ATEK parameter for the pooled sample can be expressed as a weighted average of the ATEK parameters for each of the unobserved compliance groups:
where pg is the percentage of the study population in compliance group g(Σpg =1), and ATEK_g is the associated impact parameter. The ATEK_CL parameter under the fuzzy RD design can then be identified under two key assumptions (Hahn et al. 2001; Imbens and Lemieux 2008). The first is that there are no defiers—the monotonicity assumption. This implies that pD =0 and pCL =(PT -PC), where PT is the treatment group participation rate (service receipt rate) and PC is the control group crossover rate. The second key assumption is that the distributions of potential outcomes are independent of treatment assignments for the never-takers and always-takers—the exclusion restriction. This assumption implies that never-takers and always-takers receive identical services regardless of the treatment condition to which they are assigned. This restriction implies that ATEK_N =ATEK_A =0.
Under these two assumptions, the following impact parameter can be identified under the fuzzy RD design:
This parameter represents the average causal effect of the treatment for compliers at the cutoff score.
A consistent estimator for the ATEK_CL parameter can be obtained by dividing consistent estimators for the numerator and denominator in (14), which can both be obtained using RD regression methods. The ATEK parameter can be estimated using equation (8). An estimator for (pT -pC) can be obtained as the parameter estimate on TiRD from a regression of observed treatment receipt status, ri, on TiRD and a smooth function of Scorei.
This ATEK_CL ratio estimator can also be obtained using instrumental variables (IV) techniques (Hahn et al. 2001) that are similar to the methods developed by Bloom (1984) and Angrist et al.(1996) to adjust for noncompliers in RA evaluations. For example, consider the following two-stage-least-squares procedure: (1) Calculate predicted values, r̂i, from the RD regression model discussed above for estimating (pT -pC); (2) Estimate equation (8) using r̂i in place of TiRD. The second-stage coefficient estimate on r̂i yields the ATEK_CL estimator.
In principle, the variance of this ATEK_CL estimator must account for the estimation error in both the ATEK and (pT -pC) parameters.7 As an approximation, however, I treat (pT -pC) as fixed and use equation (12) to obtain the following asymptotic variance expression for the ATEK_CL estimator:
The corresponding variance expression for the TOT estimator under the RA design is:
where (qT -qC) represents the treatment-control difference in service receipt rates for the full study population (not just those at the cutoff).
Consequently, the design effect for the fuzzy RD design is:
This design effect reduces to (11) if service receipt rates for units right around the cutoff mimic service receipt rates for units across the full score distribution.