When Is Regression Discontinuity Appropriate?

Can an Eligibility Threshold Create a Credible Causal Comparison Without Randomization?

Suppose a university offers an AI-supported tutoring program to students who score below 60 on a diagnostic assessment. Students scoring 60 or above continue receiving existing academic support.

You want to determine whether the additional tutoring improves subsequent achievement. However, the university will not randomly assign students because the program follows an established eligibility policy.

Comparing every student below the threshold with every student above it would be problematic. Students with substantially different diagnostic scores may differ in prior knowledge, academic preparation, and other characteristics associated with achievement.

But what about students scoring 59 and 60? Their diagnostic performance is very similar, yet the assignment rule places them on opposite sides of the eligibility threshold.

Regression discontinuity exploits this kind of assignment mechanism. Its usefulness depends on whether the threshold creates a credible local comparison and whether other factors could explain an apparent outcome discontinuity.

Use Regression Discontinuity When Treatment Changes at a Defensible Cutoff

When Regression Discontinuity Provides a Credible Research Design

What makes a study a regression discontinuity design?

A regression discontinuity design, commonly abbreviated RDD or RD, evaluates the effect of an intervention by examining an outcome discontinuity at a threshold used to determine treatment assignment or eligibility.

The assignment variable is called the running variable, forcing variable, or assignment variable. It may be a test score, age, income, institutional performance measure, or another numerical quantity.

For example, a university might assign additional academic support to students whose diagnostic scores fall below a specified cutoff. Students immediately above and below that threshold may provide a useful comparison because their running-variable values are close.

The central identifying argument is that, without the intervention, the relevant potential outcome functions would be continuous at the cutoff.

If the intervention produces a discontinuous change in the outcome at that threshold, the jump may be attributed to treatment under the design's assumptions.

Regression discontinuity is one of several approaches that can support causal inference without random assignment.

The cutoff must actually influence treatment assignment

A threshold is useful when it plays a genuine role in determining intervention eligibility or receipt.

Suppose a diagnostic score below 60 makes a student eligible for AI tutoring. The threshold changes treatment eligibility according to a known rule.

By contrast, dividing students into high-achieving and low-achieving groups at a score of 60 merely for analysis does not create regression discontinuity if treatment assignment was unrelated to that threshold.

Researchers should establish how the cutoff was defined, when it was established, whether it was consistently applied, and what happened to students on either side.

The suitability of an assignment cutoff depends on more than its numerical value. The assignment mechanism must support the intended causal comparison.

Why observations near the cutoff are especially informative

Regression discontinuity focuses on units with running-variable values close to the threshold.

Students scoring 59 and 60 may be more comparable in relevant pretreatment characteristics than students scoring 25 and 95.

However, this does not mean that students immediately around the threshold are literally identical or randomly assigned.

The causal argument relies on assumptions about the continuity of potential outcomes and the absence of problematic sorting or manipulation at the cutoff.

Researchers should also recognize that the observed running variable may be discrete. If scores take only a small number of possible values, there may be insufficient support arbitrarily close to the cutoff for conventional continuity-based estimation.

Therefore, local comparability is a design argument that must be supported, not simply assumed because two scores differ by one point.

Sharp and fuzzy regression discontinuity answer different assignment situations

Two major forms of regression discontinuity are commonly distinguished.

In a sharp design, all students below a cutoff might receive tutoring and all students above it might not.

In a fuzzy design, some eligible students may decline tutoring, while some ineligible students obtain access through exceptions.

The fuzzy design uses the discontinuity in treatment probability to identify an effect under additional assumptions, commonly including instrumental-variable conditions and an appropriate monotonicity assumption for a local complier interpretation.

Researchers should not classify a study as sharp merely because an official eligibility rule exists. Actual treatment receipt must be examined.

The continuity assumption is central

The central regression discontinuity assumption is that relevant potential outcome functions would be continuous at the cutoff in the absence of the treatment discontinuity.

In practical terms, units just above and below the threshold should not experience an abrupt change in their untreated expected outcomes for reasons unrelated to the intervention.

For example, if students below 60 receive AI tutoring but also become eligible for a separate financial assistance program at exactly the same cutoff, an achievement discontinuity may reflect both interventions.

Similarly, if students can strategically manipulate their diagnostic scores to qualify for tutoring, the populations immediately around the cutoff may differ systematically.

Continuity cannot be established by a single statistical test. Researchers must examine the assignment process and plausible alternative mechanisms.

Why precise manipulation of the running variable is a concern

Regression discontinuity becomes less credible when individuals or administrators can precisely influence which side of the cutoff a unit occupies.

For example, if teachers can adjust diagnostic scores after learning that scores below 60 qualify students for additional tutoring, students immediately below the threshold may have been selected through discretionary decisions.

This can undermine the argument that potential outcomes change smoothly across the threshold.

Researchers should investigate whether the running variable was measured before eligibility decisions, whether scoring procedures were standardized, and whether exceptions or discretionary adjustments occurred.

Density tests and examination of pretreatment characteristics can help identify suspicious patterns, but passing these diagnostics does not prove that manipulation was absent.

Why the running variable should be measured consistently

The credibility of the design depends on accurate and consistent measurement of the assignment variable.

Suppose diagnostic scores are generated using different assessment forms across institutions. If the forms are not comparable, the same numerical cutoff may not represent equivalent levels of prior achievement.

Researchers should verify the scoring procedure, timing, reliability, and treatment of missing or revised scores.

Measurement error can also complicate treatment assignment and estimation, particularly when the observed score differs from the quantity actually used for eligibility.

In educational settings, rounding, score heaping, and limited score categories deserve attention because they affect the support available near the cutoff.

Why a discontinuity in treatment must be established

Before interpreting an outcome discontinuity, researchers should examine whether the cutoff actually changes treatment assignment or receipt.

For a sharp design, treatment should follow the cutoff rule as specified.

For a fuzzy design, there must be a meaningful discontinuity in the probability of treatment at the cutoff.

If students below and above the threshold have nearly identical probabilities of receiving tutoring, the cutoff may provide little useful identifying variation.

A weak treatment discontinuity can produce imprecise estimates and make fuzzy regression discontinuity inference difficult.

The strength and interpretation of the first-stage discontinuity should therefore be reported.

How the local treatment effect is estimated

In a conventional sharp regression discontinuity design, researchers estimate the outcome regression function on each side of the cutoff and examine the difference between its limiting values at the threshold.

In a fuzzy design, the outcome discontinuity is commonly divided by the discontinuity in treatment probability to estimate an appropriate local treatment effect under additional assumptions.

Neither calculation is sufficient by itself to establish causality. The interpretation depends on the assignment mechanism, continuity, and estimation assumptions.

Bandwidth selection affects the analysis

The bandwidth determines how far from the cutoff observations are included in the local estimation procedure.

A narrow bandwidth emphasizes observations close to the threshold but may leave relatively few observations and produce imprecise estimates.

A wider bandwidth includes more observations but requires stronger approximation assumptions about the outcome regression functions across a broader range.

Researchers commonly use local polynomial estimation with data-driven bandwidth selection and robust bias-corrected inference.

However, the chosen bandwidth should be justified and sensitivity to reasonable alternatives examined.

Researchers should not select a bandwidth simply because it produces a statistically significant result.

Why high-order global polynomials can be problematic

Fitting one high-degree polynomial across a wide range of the running variable may produce unstable estimates near the cutoff.

The resulting discontinuity can be sensitive to observations far from the threshold and to the chosen polynomial degree.

Methodological guidance generally favors appropriately specified local polynomial methods over automatically fitting high-order global polynomials.

The functional form, bandwidth, kernel, and inferential procedure should be selected according to the design and justified analytically.

Visual inspection can help identify implausible model behavior, but it should not substitute for formal estimation.

How to examine the data graphically

A regression discontinuity plot typically places the running variable on the horizontal axis and the outcome on the vertical axis.

The cutoff is marked clearly, and outcome patterns are displayed on both sides.

Binned means and fitted curves may help readers see whether a discontinuity appears near the threshold.

However, bin widths, axis scales, and smoothing choices can influence the visual impression.

Researchers should report the statistical estimate and uncertainty rather than interpreting a visible jump as sufficient evidence of causality.

Why covariate balance is useful but not definitive

Pretreatment characteristics should generally evolve smoothly through the cutoff under a credible continuity-based design.

Researchers may examine whether age, prior achievement, socioeconomic characteristics, or other predetermined variables display unexpected discontinuities.

Substantial jumps can indicate sorting, multiple assignment rules, or other threats to identification.

However, a nonsignificant balance test does not prove continuity. Tests may lack power, particularly with limited observations near the cutoff.

Covariate checks should complement investigation of the assignment mechanism.

Why the estimated effect is usually local

Regression discontinuity generally identifies an effect at the assignment threshold.

For example, a tutoring program's estimated effect for students scoring near 60 may not represent its effect for students scoring 20 or 90.

Students far from the cutoff may have different learning needs, baseline preparation, or responsiveness to the intervention.

Researchers should distinguish internal validity near the threshold from generalizability to other parts of the population.

Extrapolating beyond the cutoff requires additional assumptions or evidence.

When regression discontinuity is preferable to difference-in-differences

Regression discontinuity may be attractive when a credible threshold-based assignment mechanism exists and sufficient observations are available near the cutoff.

Difference-in-differences instead relies on a comparison of outcome changes over time and assumptions about untreated counterfactual trends.

For example, a tutoring program allocated by a diagnostic cutoff may naturally support regression discontinuity, while a university-wide policy introduced at one institution but not another may be more suitable for difference-in-differences.

The designs identify different effects and rely on different assumptions.

Researchers should select the strategy that best matches the actual assignment mechanism rather than choosing according to statistical familiarity.

When regression discontinuity is inappropriate

The design may be unsuitable when no genuine assignment cutoff exists, when the running variable can be manipulated in ways that undermine continuity, or when multiple interventions change at the same threshold and cannot be distinguished.

It may also be difficult to implement when there are too few observations near the cutoff, the running variable has very limited support, or treatment probability does not change meaningfully at the threshold.

In these circumstances, researchers should consider whether another quasi-experimental design better reflects the intervention assignment process.

Evaluating AI-Supported Tutoring Assigned Through a Diagnostic Score

Interpretation: The cutoff provides a potentially credible local comparison because intervention assignment changes at a known diagnostic threshold.

However, the estimate is causal only if the assignment and continuity assumptions are defensible. If another intervention also begins below 60, the observed jump may reflect a combined policy effect rather than AI tutoring alone.

The design's strength comes from the threshold-based assignment mechanism, not simply from comparing students with different scores.

Common Mistakes When Selecting Regression Discontinuity

Use the Assignment Rule to Determine Whether Regression Discontinuity Is Defensible

Begin by verifying that the intervention is allocated through a genuine numerical threshold. Establish the running variable, cutoff, assignment rule, and actual treatment receipt.

Next, investigate whether units can manipulate the assignment variable and whether other interventions change at the same threshold.

Finally, determine whether sufficient observations are available near the cutoff for an appropriate local analysis.

The appropriate design follows the actual assignment mechanism. Regression discontinuity is compelling when the cutoff provides credible identifying variation, not merely when a dataset contains a continuous score and a convenient threshold.

Before Using Regression Discontinuity

Frequently Asked Questions About Regression Discontinuity

Regression Discontinuity Is Appropriate When the Cutoff Supports a Credible Local Comparison

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