Regression Discontinuity Design

Regression discontinuity design (abbreviated RDD) is a quasi-experimental causal inference architecture in epidemiology, biostatistics, and health policy evaluation that estimates the causal effect of an intervention when treatment assignment is determined by whether a continuous baseline running variable (also termed the forcing variable or assignment variable, X) falls above or below a fixed numerical cutoff threshold (c). Introduced by educational psychologists Donald L. Thistlethwaite and Donald T. Campbell in 1960 and formalized within the potential-outcomes and instrumental-variables frameworks by Guido W. Imbens, Thomas Lemieux, and Jinyong Hahn (2001, 2008), regression discontinuity exploits clinical guidelines and public health eligibility rules—such as initiating antiretroviral therapy when CD4⁺ T-lymphocyte counts fall below 350 cells/μL, prescribing statins when 10-year atherosclerotic cardiovascular risk exceeds 10%, admitting neonates to intensive care when birth weight is below 1,500 g, or granting Medicare health insurance at age 65.0 years. Because baseline biological measurement noise renders individuals immediately above the cutoff (X = c + ε) exchangeable with individuals immediately below the cutoff (X = c − ε), comparing outcome limits on either side of the threshold neutralizes confounding by indication without randomized allocation.

In observational pharmacoepidemiology and clinical care, treatments are purposefully prescribed to patients with worse prognosis: antihypertensive drugs are given to patients with elevated systolic blood pressure, and intensive care is reserved for critically ill patients. Consequently, across the full distribution of patients, the running variable X is a strong confounder (T ← X → Y). Standard multivariable regression or propensity-score matching struggles when clinical guidelines create near-deterministic treatment rules because there is minimal positivity (common support) between treated and untreated patients at the same value of X.

Regression discontinuity turns this violation of global positivity into a local quasi-experiment. While a patient with a baseline CD4⁺ count of 150 cells/μL is clinically incomparable to a patient with 600 cells/μL, a single flow-cytometry blood draw exhibits stochastic diurnal and analytical variability (often ±25 to 40 cells/μL). Whether a patient whose true underlying mean CD4⁺ count is 350 cells/μL happens to record a lab value of 348 cells/μL (eligible for immediate antiretroviral initiation) or 352 cells/μL (deferred under historical guidelines) is effectively random in a small neighborhood [c − h, c + h] around the threshold c = 350.

Sharp RDD (Deterministic Treatment Rule)

In a sharp regression discontinuity design, treatment status T_i ∈ {0, 1} is a deterministic step function of the running variable X_i at cutoff c:

T_i = 1 if X_i ≥ c, and T_i = 0 if X_i < c

(or vice versa when treatment is assigned for X_i < c). The probability of receiving treatment jumps discontinuously from 0 to 1 at X = c. Provided that the conditional expectations of the potential outcomes E[Y_i(1) | X_i = x] and E[Y_i(0) | X_i = x] are continuous functions of x across the threshold x = c, the Local Average Treatment Effect at the Cutoff (τ_SRD) is identified by the vertical jump in the conditional expectation of the observed outcome Y:

τ_SRD = E[Y_i(1) − Y_i(0) | X_i = c] = lim_x→c⁺ E[Y_i | X_i = x] − lim_x→c⁻ E[Y_i | X_i = x]

Fuzzy RDD (Probabilistic Guideline Adherence as an Instrumental Variable)

In real-world clinical practice, physicians rarely follow numerical thresholds with 100% determinism: some patients slightly below a blood-pressure or LDL-cholesterol threshold receive treatment due to family history, while some patients slightly above the threshold decline medication. When crossing the threshold c produces a discontinuous jump in the probability of treatment that is strictly less than 1 (0 < lim_x→c⁺ Pr(T_i = 1 | X_i = x) − lim_x→c⁻ Pr(T_i = 1 | X_i = x) < 1), the architecture is a fuzzy regression discontinuity design.

As demonstrated by Hahn, Todd, and Van der Klaauw (2001), a fuzzy RDD is mathematically equivalent to a local instrumental variable design (analogous in algebraic ratio structure to Mendelian randomization), where the threshold indicator Z_i = 1(X_i ≥ c) serves as the instrument. Dividing the intention-to-treat outcome jump at the cutoff (the numerator) by the compliance jump in treatment uptake at the cutoff (the denominator) yields the Wald estimator for the Local Average Treatment Effect among Compliers at the Cutoff (τ_FRD):

τ_FRD = [ lim_x→c⁺ E[Y_i | X_i = x] − lim_x→c⁻ E[Y_i | X_i = x] ] / [ lim_x→c⁺ E[T_i | X_i = x] − lim_x→c⁻ E[T_i | X_i = x] ]

This requires the standard instrumental-variable monotonicity assumption (crossing the cutoff does not cause any patient to switch away from the recommended treatment) and local exclusion restriction (crossing the threshold affects the clinical outcome Y only through treatment T, and not through a simultaneous bundled intervention triggered at the exact same cutoff c).

Avoiding High-Order Global Polynomials

To estimate the one-sided limits at X = c, modern biostatistical practice avoids fitting high-order global polynomials (such as cubic or quartic curves across the entire range of X), which Andrew Gelman and Guido Imbens (2019) proved assign erratic weights to observations far from the cutoff and produce overly narrow confidence intervals. Instead, investigators restrict estimation to observations within a symmetric bandwidth h around the cutoff (X_i ∈ [c − h, c + h]) and fit local linear regression (weighted by a triangular kernel K((X_i − c) / h) that assigns maximum weight at X_i = c):

Y_i = α + τ Z_i + β₋ (X_i − c)(1 − Z_i) + β₊ (X_i − c)*Z_i* + ε_i

Allowing distinct slopes β₋ below the cutoff and β₊ above the cutoff prevents biological nonlinearity in X from masquerading as a discontinuous treatment jump τ. Optimal data-driven bandwidths h_MSE that balance squared bias (which grows with wider h) against variance (which grows with narrower h) and bias-corrected robust confidence intervals were derived by Sebastian Calonico, Matias D. Cattaneo, and Max H. Farrell (2014, implemented in rdrobust).

Because the continuity assumption cannot be directly observed for counterfactual outcomes, every regression discontinuity study performs three mandatory empirical falsification checks:

1. McCrary (2008) Density Test for Running-Variable Manipulation ("Heaping"): If clinicians or patients intentionally manipulate or selectively round recorded measurements of X to qualify for treatment (for example, rounding birth weight to 1,490 g to secure neonatal intensive care reimbursement, or repeating blood tests until a patient qualifies for a drug), observations will bunch (heap) immediately on the eligible side of c. A discontinuous jump in the probability density f(x) at x = c invalidates local exchangeability.
2. Covariate Balance Continuity Tests (Placebo Outcomes): Baseline predetermined covariates W (such as patient age, sex, smoking history, or baseline comorbidities) must exhibit zero discontinuity at X = c, serving as negative control outcomes.
3. Placebo Cutoff Falsification Tests: Re-running the local linear estimator at artificial cutoffs c′ where no clinical guideline change occurs (e.g., CD4⁺ = 420 or 280 cells/μL) should yield null effect estimates (τ̂ ≈ 0).

Can causal effects from a regression discontinuity study be generalized far away from the cutoff?

Not without strong parametric extrapolation assumptions. Because RDD identifies the treatment effect specifically at the threshold X = c (and for compliers in a fuzzy design), a survival benefit demonstrated for initiating antiretroviral therapy at CD4⁺ = 350 cells/μL does not automatically measure the benefit of initiating therapy at CD4⁺ = 700 cells/μL or 50 cells/μL. However, when guidelines shift the threshold over time across multiple cutoffs, multi-cutoff RDD designs can trace out the treatment effect curve across a broader clinical spectrum.

How does measurement error in the running variable affect RDD?

Unlike classical regression dilution bias in ordinary least squares, random biological or assay measurement error embedded in the clinician's observed assignment variable X is actually beneficial for sharp RDD identification: as long as physicians assign treatment based on the noisy observed lab value X rather than the unobserved true latent value, the noise locally randomizes patients with similar latent health across the cutoff c. Conversely, if the investigator only observes a coarsened or retrospectively noisy proxy of the exact variable the clinician used, the jump at c is smoothed out and attenuated.

1. Thistlethwaite, D. L., & Campbell, D. T. (1960). Regression-discontinuity analysis: An alternative to the ex post facto experiment. Journal of Educational Psychology, 51(6), 309–317. https://doi.org/10.1037/h0044319
2. Imbens, G. W., & Lemieux, T. (2008). Regression discontinuity designs: A guide to practice. Journal of Econometrics, 142(2), 615–635. https://doi.org/10.1016/j.jeconom.2007.05.001
3. Bor, J., Moscoe, E., Mutevedzi, P., Newell, M. L., & Bärnighausen, T. (2014). Regression discontinuity designs in epidemiology: causal inference without randomized trials. Epidemiology, 25(5), 729–737. https://doi.org/10.1097/EDE.0000000000000138
4. Calonico, S., Cattaneo, M. D., & Titiunik, R. (2014). Robust nonparametric confidence intervals for regression-discontinuity designs. Econometrica, 82(6), 2295–2326. https://doi.org/10.3982/ECTA11757