Discrete Running Variables, Heaping, and Donut Designs

Why the Running Variable’s Support Matters

A regression-discontinuity (RD) design compares outcomes just below and just above an assignment cutoff. Its central idea is local: if units close to the cutoff are otherwise comparable, a discontinuity in the outcome can reveal the effect of treatment. That argument depends not only on the assignment rule, but also on which running-variable values are actually observed.

If the running variable takes only a few values, there may be no observations arbitrarily close to the cutoff. If values are heaped, many units may share rounded scores, and the observed score may not locate them precisely relative to the threshold. A donut design deliberately removes observations near the cutoff. It can address concerns about those observations, but it also removes the observations most directly relevant to the usual RD comparison.

Definition: Let $R$ be the running variable, $c$ the cutoff, and $D=1\{R\geq c\}$ the sharp assignment rule. A conventional sharp RD effect at the cutoff is the discontinuity $$ \tau_{\mathrm{RD}}= \lim_{r\downarrow c} E[Y\mid R=r] - \lim_{r\uparrow c} E[Y\mid R=r], $$ where the first limit is from above and the second is from below. Under the sharp assignment rule and continuity of the conditional mean potential outcomes at $c$, this discontinuity identifies the average treatment effect for units at the cutoff. The limits require support approaching $c$ from both sides, or additional assumptions that justify extrapolation.

Here $Y$ is the observed outcome. The continuity condition concerns the conditional means of the potential outcomes: absent treatment, the expected outcome should not jump at $c$, and the expected outcome under treatment should not jump there either. The treatment assignment changes at the cutoff; other determinants of outcomes should not change discontinuously at the same point. Covariate-balance and placebo diagnostics can help assess implications of this design argument, but do not create missing support.

Discrete Running Variables: A Jump Is Not Automatically a Limit

A running variable is discrete when it can take only separated values, such as integer scores or a small set of rating categories. A discrete score can still support an RD analysis if its support becomes sufficiently dense near the cutoff in an appropriate sequence of settings or sample sizes. But in a fixed sample with a gap around $c$, the conventional limits cannot be read directly from observations at the cutoff.

A difference between the mean outcome at the nearest observed value below $c$ and the mean at the nearest value above $c$ is an adjacent-support contrast. It is not, by definition, the limiting discontinuity at $c$. It may combine a treatment effect with ordinary changes in outcomes across score values. Local polynomial fits can produce an estimate at the cutoff, but with sparse support that estimate depends strongly on the chosen functional form. The resulting precision calculations do not remove that identification dependence.

Worked Example: A Gap in the Score Support

Suppose a hypothetical sharp RD uses cutoff $c=0$, but the only observed score values are $-2,-1,1,2$. The corresponding mean outcomes are 8, 9, 13, and 14 outcome units. There are no observations at scores between $-1$ and $1$.

Adjacent-support contrast. The nearest observed means differ by $$ 13-9=4 \text{ outcome units}. $$ This is a comparison of the mean at score 1 with the mean at score $-1$. Since score changes by two units between these points, the contrast can reflect both treatment and the outcome trend with score. It is not itself the RD limit at zero.

Linear extrapolation. The two left points lie on the line $m_L(r)=10+r$, because $10+(-2)=8$ and $10+(-1)=9$. The two right points lie on $m_R(r)=12+r$, because $12+1=13$ and $12+2=14$. Under the assumption that these lines describe the conditional means all the way to zero, the extrapolated jump is $$ m_R(0)-m_L(0)=12-10=2 \text{ outcome units}. $$

The estimates 4 and 2 answer different questions: one is observed, the other depends on linear extrapolation. With a gap in the support, the observed data alone do not establish that the fitted lines continue to the cutoff. A local-randomization interpretation is another possible framework, but it requires a defensible window in which assignment is as-if random; it is not implied by discrete scores.

Heaping and Assignment Near the Threshold

Heaping occurs when measured values accumulate at preferred numbers, often because of rounding, reporting habits, or limited measurement precision. For example, a continuous measurement might be recorded to the nearest five units. A large mass at a threshold can make it difficult to distinguish units truly just below from units truly just above that threshold.

The relevant concern depends on how treatment is assigned. If assignment uses the recorded score, the rule may be sharp in that recorded score, but the support may be too coarse to approximate a limit. If assignment uses an underlying unrounded value while the analyst observes only a rounded value, the recorded score may not determine treatment exactly. The analyst should establish which variable governs assignment and how it is measured before interpreting a jump.

Worked Example: A Rounded Eligibility Score

Imagine a hypothetical program that assigns treatment using a reported score, with scores of 50 or higher eligible. Scores are recorded to the nearest five points. In a local table, the mean outcome is 20 units among 40 people recorded at 45, and 29 units among 70 people recorded at 50.

The observed adjacent-score difference is $$ \bar{Y}_{50}-\bar{Y}_{45}=29-20=9 \text{ outcome units}. $$ This is a valid descriptive contrast between the two recorded-score groups. It is not, by itself, a causal effect at the cutoff. The mass at 50 is assigned as a group, and there are no observed values that reveal how outcomes vary within the interval hidden by rounding.

If the program’s rule truly uses the reported score, treatment is determined by that score, but identifying a cutoff limit still requires support or additional structure. If the rule instead uses an underlying unrounded score, some people recorded at 50 could fall on either side of the true threshold. In that case, the recorded score alone does not establish sharp assignment. The nine-unit contrast cannot resolve which situation holds.

Heaping diagnostics should therefore be interpreted in context. A concentration of scores at a round number is not automatically evidence of manipulation: it may reflect a known recording process. Conversely, a conventional density or sorting diagnostic can be hard to interpret when the running variable is inherently discrete. The assignment rule, measurement process, and available score support all matter.

Donut Designs: What Is Removed, and What Must Be Assumed?

A donut design omits observations within a chosen distance of the cutoff. With donut radius $\delta>0$, it excludes units satisfying $|R-c|<\delta$, then estimates the outcome relationship using observations farther away. The motivation may be concern about manipulation, measurement error, or another process concentrated at the threshold.

The cost is direct: the omitted units are those closest to the comparison that defines the usual RD effect. If the exclusion leaves a fixed gap around $c$, the conditional means at the cutoff are no longer directly approached by observed values. An estimate at $c$ must then rely on assumptions about how the outcome relationships extend across the gap, such as a specified polynomial form. A fixed-radius donut does not, on its own, identify the original cutoff effect.

A useful asymptotic idea is to let the donut radius shrink toward zero while retaining observations approaching the cutoff from each side. Under appropriate smoothness, support, and sampling conditions, such a procedure can recover the usual local comparison. That argument differs from excluding a fixed neighborhood indefinitely. A donut can be a sensitivity analysis—showing how conclusions change when close observations are removed—without automatically being a solution to a design problem.

Identification and precision: Identification asks whether the assumptions and observed support determine the causal effect. Precision asks how variable an estimate is under a specified model. Removing near-cutoff observations can increase uncertainty, but a small standard error does not validate extrapolation across a support gap. Conversely, wider intervals do not by themselves imply that an estimate is unidentified.

Worked Precision Calculation for a Donut Exclusion

The calculation below isolates the precision cost under a stated linear model. It does not show that the model’s extrapolation is causally valid. Let $x=R-c$, and fit separate linear regressions on the left and right. Assume independent Gaussian errors with known variance $\sigma^2=4$ outcome-units-squared on each side. The intercept difference estimates the jump at $x=0$. These assumptions make the standard-error calculation exact for this example.

Worked Example: Removing the Cutoff Score

In a hypothetical data set, the left side has 20 observations each at $x=-3,-2,-1$, and the right side has 20 each at $x=0,1,2,3$. Suppose the fitted left and right lines have intercepts 10 and 12 outcome units, so the estimated jump is $12-10=2$ outcome units. Assume the fitted slopes remain the same and that, after deleting the right-side observations at $x=0$, the fitted intercepts also remain 10 and 12.

For a regression with an intercept and slope, the variance of the estimated intercept is $$ \operatorname{Var}(\hat{\beta}_0)=\sigma^2\left(\frac{1}{n}+\frac{\bar{x}^{\,2}}{S_{xx}}\right), \qquad S_{xx}=\sum_{i=1}^{n}(x_i-\bar{x})^2. $$

Before exclusion, right side. There are $n=80$ observations and $\bar{x}=1.5$. With 20 observations at each score, $$ S_{xx}=20\left[(0-1.5)^2+(1-1.5)^2+(2-1.5)^2+(3-1.5)^2\right] =20(5)=100. $$ Thus $$ \operatorname{Var}(\hat{\beta}_{0,R}) =4\left(\frac{1}{80}+\frac{1.5^2}{100}\right) =4(0.0125+0.0225)=0.1400. $$

Left side. There are $n=60$ observations at $-3,-2,-1$, so $\bar{x}=-2$ and $$ S_{xx}=20\left[(-3+2)^2+(-2+2)^2+(-1+2)^2\right]=40. $$ The left-intercept variance is $$ 4\left(\frac{1}{60}+\frac{(-2)^2}{40}\right) =4(0.01667+0.1000)=0.4667. $$ The left and right estimates are independent, so the jump variance before exclusion is $0.4667+0.1400=0.6067$, and its standard error is $\sqrt{0.6067}=0.7789$ outcome units.

After excluding $x=0$. The right side now has $n=60$ observations at $1,2,3$, with $\bar{x}=2$ and $S_{xx}=40$. Its intercept variance is $$ 4\left(\frac{1}{60}+\frac{2^2}{40}\right) =4(0.01667+0.1000)=0.4667. $$ The left side is unchanged, so the jump variance is $0.4667+0.4667=0.9333$, and the standard error is $\sqrt{0.9333}=0.9661$ outcome units.

Under the assumed linear fits, the estimated jump remains 2 outcome units, but its standard error rises from 0.7789 to 0.9661 units. With Gaussian errors and known variance, the corresponding 95% intervals are $2\pm1.96(0.7789)=[0.473,3.527]$ before exclusion and $2\pm1.96(0.9661)=[0.106,3.894]$ after exclusion. These intervals quantify uncertainty conditional on the model; they do not account for possible misspecification of the extrapolation across the omitted score.

How to Evaluate a Discrete or Donut Specification

A defensible analysis should separate the design question from the regression calculation. First establish how treatment is assigned and whether the recorded running variable is the one used by the rule. Then inspect the support on each side: the number of distinct values, their spacing, and whether there are observations approaching the cutoff. For heaped data, examine how rounding creates the recorded values and whether units at a mass point can be assigned unambiguously.

For a donut specification, report the excluded interval and the support that remains. Explain whether the estimate is based on observations approaching the cutoff or on extrapolation across a gap. Compare results across plausible exclusion radii, while making clear that stability across radii is a sensitivity check rather than proof of identification. Report uncertainty using an inference procedure appropriate to the estimator, and distinguish that uncertainty from uncertainty about functional form and the causal assumptions.

Map assignment and measurement.
State the cutoff rule, whether the running variable is discrete or rounded, and whether the recorded value determines treatment.

Check support around the cutoff.
Determine whether distinct values approach the cutoff from each side or whether a gap prevents a direct local comparison.

State what the estimator needs.
For a fixed donut gap, identify the extrapolation or other assumptions that connect observed outcomes to the cutoff.

Present sensitivity and precision separately.
Show how estimates and uncertainty change as observations are excluded, without treating a narrow interval or stable estimate as proof of identification.

Common Pitfall

Wrong reasoning: “The nearest observed scores have different mean outcomes, so their difference is the causal effect at the cutoff. If the closest observations look problematic, I can omit them and estimate the same effect more cleanly.” This reasoning mistakes a finite-support contrast for a limiting discontinuity and assumes that removing observations preserves identification.

Correction: The adjacent-score difference is descriptive unless additional assumptions connect it to the cutoff effect. A donut exclusion may remove problematic observations, but if it creates a fixed support gap, estimating the cutoff jump requires credible extrapolation or another explicit identification strategy. State that assumption, show the resulting precision cost, and interpret the estimate accordingly.

Key takeaway: Discreteness, heaping, and donut exclusions are not merely technical details of regression fitting. They determine whether the data approach the cutoff, whether assignment is observed correctly, and how much the causal conclusion depends on extrapolation.

Check Your Understanding

Answer each question by distinguishing observed comparisons, identifying assumptions, and sampling uncertainty.

Why is the difference between means at the closest observed scores not necessarily the sharp RD effect at the cutoff?
A score is rounded to the nearest five points, and treatment uses the unrounded score. What concern does this raise about assignment based on the recorded score?
What extra assumption is needed to interpret a fixed-radius donut estimate at the original cutoff when no observations remain near it?
In the precision example, why does deleting $x=0$ increase the right-intercept variance even though the number of observations remains substantial?
What does a stable estimate across several donut radii establish, and what does it not establish?