- Chapter 1: Introduction
- Chapter 2: Measuring Statistical Power
- Chapter 3: Considered Designs
- Chapter 4: Aggregated Designs: RD Design Theory and Design
- Chapter 5: Multilevel RD Designs
- Chapter 6: Selecting the Score Range for the Sample
- Chapter 7: Illustrative Precision Calculations
- Chapter 8: Summary and Conclusions
- References
- List of Tables
- List of Figures
- Appendix A
- Appendix B
- PDF & Related Info

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
*R _{Ci}* denote the potential service receipt indicator variable
in the control condition.

The *ATE _{K}* parameter for the pooled sample can be expressed as
a weighted average of the

where *p _{g}* is the percentage of the study population in compliance
group

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 *ATE _{K_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

This *ATE _{K_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,

In principle, the variance of this *ATE _{K_CL}* estimator must account
for the estimation error in both the

The corresponding variance expression for the TOT estimator under the RA design is:

where (*q _{T}* -

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.