Why Discrete Running Variables Need Special Care
A running variable may be a test score, an age recorded in whole years, or a count of prior events. Such variables often take only a limited set of values near the assignment cutoff, with many individuals sharing each value. A large sample can therefore contain many observations but little distinct information about how outcomes vary with the score near the cutoff.
In a regression discontinuity design (RD), the key comparison concerns outcomes just below and just above the cutoff. With a finely supported running variable, observations can approach the cutoff from both sides. With a discrete variable, the closest observed scores may instead be separated by a gap. Repeating those same score values can improve the precision of score-specific averages, but it does not fill the gap or, by itself, justify extrapolating across it.
For a sharp design with treatment rule \(D=1(R\geq c)\), the usual RD estimand is the difference between the right- and left-hand limits of the conditional outcome mean at \(c\):
Under the relevant continuity conditions, this limit contrast identifies a causal effect at the cutoff. The limits refer to values approaching \(c\) from the two sides, not simply to any two observed score groups. A discrete support may not provide observations that approach \(c\) arbitrarily closely. That distinction is central: a precise contrast between observed groups is not automatically a precise estimate of the RD estimand.
What the Support Can Identify
A basic support audit lists, for each nearby score, its distance from the cutoff, whether it is below or above the cutoff, and its observation count. It should make clear whether the cutoff itself is a support point, whether there is a gap around it, and how many distinct values contribute to a chosen local fit. Counts should not be used as a substitute for the support-point list.
The identification issue can be seen directly. Suppose the observed support has no values in an interval around \(c\). The data then reveal conditional means at support points outside that interval, but not the conditional mean function throughout the gap. Even if that function is assumed continuous, multiple continuous curves can agree at every observed support point and have different limiting values at \(c\). Thus continuity alone does not nonparametrically determine the two limits from those observations. A local polynomial can produce an extrapolated estimate, but its value then depends on the chosen functional approximation, not just on observed outcomes arbitrarily close to the cutoff.
If the score equals \(c\) for some individuals, those observations are assigned to the treated side under \(D=1(R\geq c)\). They provide information about the conditional mean at \(c\) for that side. They do not, by themselves, reveal the left-hand limit. If the closest lower score is separated from \(c\), obtaining that limit still requires extrapolation or additional identifying assumptions.
Worked Example: Auditing Test-Score Support
Suppose a hypothetical scholarship offer is assigned when a test score \(R\) is at least \(70\). Within a proposed bandwidth of two score points, the data contain the following values.
| Score | Side under the rule | Number of students |
|---|---|---|
| 68 | Below | 18 |
| 69 | Below | 42 |
| 70 | At cutoff; assigned offer | 55 |
| 71 | Above | 24 |
There are \(18+42+55+24=139\) students in this window, but only four distinct score values. There are two distinct values below the cutoff, one at the cutoff, and one above it. In particular, there are no observations with scores strictly between 69 and 70, or between 70 and 71.
The count of 55 at score 70 may make the mean outcome at that score relatively precise. It does not create 55 distinct values approaching 70 from below. The nearest observed score below is 69, and the mean at 69 is not the left-hand limit at 70 without a further model or assumption. The support audit therefore flags an identification and extrapolation concern even though the local sample contains 139 students.
Local Fits with Few Score Values
Local polynomial RD methods approximate the conditional mean on each side of the cutoff using observations within a bandwidth. With a discrete score, the fitted line or curve is informed by the distinct score values as well as by the counts at those values. A large count at one value can stabilize its sample mean, but it cannot establish the shape of the outcome relationship between that value and the cutoff.
A local linear fit illustrates the distinction. If there is only one distinct score value on a side within the bandwidth, that side's slope cannot be learned from variation in the running variable within that local sample. With two or more values, a slope can be calculated, but with very few values its shape may be highly sensitive to the inclusion of a single score. Higher-order fits require still more support and can be especially fragile when fitted to only a few distinct points. The fitted value at the cutoff should therefore be described as a model-based extrapolation when it is not anchored by nearby support.
A useful practical summary is to report, for each bandwidth considered, the distinct support values and their counts on both sides, along with the distance from the closest observed value to the cutoff. Also report how many distinct values actually contribute to each side of the fit. This gives readers information that a total sample size alone cannot convey. If a local fit changes substantially when one support point is omitted or the bandwidth is changed, that sensitivity is evidence that the result relies heavily on a small number of score values.
Worked Example: A Local Contrast Is Not Automatically an RD Effect
Consider hypothetical mean outcomes at four test scores, with offer eligibility beginning at score 70. Suppose the score-specific sample means are 10 at 68, 12 at 69, 18 at 70, and 20 at 71. For illustration, fit a separate straight line on each side using the two available score values, then compare the fitted values at the cutoff.
On the left, the line through \((68,10)\) and \((69,12)\) has slope \((12-10)/(69-68)=2\) outcome units per score point. Extrapolating one point to \(R=70\) gives \(12+2(70-69)=14\).
On the right, the line through \((70,18)\) and \((71,20)\) also has slope \((20-18)/(71-70)=2\). Its fitted value at \(R=70\) is 18. The resulting local linear contrast is \(18-14=4\) outcome units.
The arithmetic is unambiguous, but the causal interpretation is not. The left fitted value of 14 is an extrapolation from scores 68 and 69; no observations lie between 69 and 70. Moreover, each side's slope is determined by exactly two score-specific means, leaving no additional local values to assess whether a straight-line approximation is reasonable. The value 4 is a model-based RD estimate under the linear extrapolations used here. The observed score-group means alone do not establish that the cutoff caused a four-unit change.
Inference: Precision Does Not Repair Sparse Support
Inference has two distinct concerns. Sampling uncertainty describes how much an estimate would vary across repeated samples under a specified sampling model. Identification and approximation concerns describe whether the observed support and assumptions justify interpreting the estimate as the desired RD limit. A small standard error addresses the first concern, not the second.
For example, an exact confidence interval for a difference in two score-group means is possible under a specified probability model. That interval concerns the difference between those group means. It does not automatically become an interval for the RD limit contrast, which may require extrapolation. Likewise, increasing the number of individuals at the same few score values can narrow uncertainty about the group means while leaving the support gap unchanged.
Worked Example: A Precise Score-Group Difference with a Support Gap
Suppose a hypothetical dataset has 100 independent students at score 69 and 100 independent students at score 70. The mean outcome is 12 at score 69 and 18 at score 70. Assume, specifically, that outcomes within each score group are normally distributed with known variance 25, and that outcomes are independent across students. These assumptions make the sampling distribution of the difference in group means exactly normal.
The observed difference, score 70 minus score 69, is \(18-12=6\) outcome units. Its standard error under the stated model is
An approximately 95% normal-reference interval for the difference in these two score-group means is
Under the specified normal model, this interval has approximately 95% coverage for the difference between the population mean outcomes at scores 70 and 69. It is not a 95% interval for the RD limit contrast: the group at 69 is not the left-hand limit at 70. The design has no observed score between those values, and this calculation does not supply the missing extrapolation assumptions.
Conventional RD inference also relies on an approximation to the local behavior of the conditional mean. When the running variable has many support points close to the cutoff and those points become increasingly dense as the sample grows, standard local asymptotic reasoning may be appropriate under its other conditions. When support remains fixed at a few values, simply increasing the number of observations per value does not provide that increasingly fine local variation. Bias correction or a smaller reported standard error cannot manufacture it.
Repeated observations at a score should also be interpreted in light of how the score is generated and how observations are sampled. The fact that several individuals share a score does not by itself mean they are statistically dependent. Conversely, if the relevant sampling or assignment process creates dependence among observations, the standard-error method must reflect that structure. The support audit and the dependence assumptions answer different questions.
A Practical Assessment
The right conclusion is not that every RD with a discrete running variable is unusable. Rather, the design must be assessed in light of its actual support and the assumptions used to bridge any gaps. If the score has enough distinct values close to the cutoff, the usual continuity-based argument may be more credible. If it has only a few values, the analysis should be explicit about what comes from the data and what comes from modeling or design assumptions.
List each score near the cutoff and its observation count, separately identifying values below, at, and above the cutoff.
Record the nearest score on each side and count the distinct values contributing to each local fit. Do not use the total number of observations as a proxy for either quantity.
Explain whether the cutoff limits are supported by nearby values or whether the estimate depends on a functional-form extrapolation or another assumption.
Describe sampling uncertainty for the estimator actually calculated, and do not present it as resolving uncertainty about unsupported extrapolation.
Common Mistake
Wrong reasoning: “There are hundreds of observations near the cutoff, so the RD estimate must be reliable.” This confuses repeated observations at the same scores with a dense set of distinct score values and treats a narrow confidence interval as proof that the cutoff limits are identified.
Correction: Report both observation counts and distinct support points. Check whether the data actually approach the cutoff on each side, and state any extrapolation assumptions needed to estimate the limits. A large sample can make observed score-specific means precise without making the RD causal contrast nonparametrically identified.
Check Your Understanding
Answer each question by distinguishing observed score groups from the RD limits.
- A dataset has 500 observations at score 49 and 500 at score 50, with cutoff 50. How many distinct values are observed in these two groups, and what does the large count at 49 tell you about the left-hand limit at 50?
- Why does continuity alone not determine cutoff limits from observations at scores 68 and 70 if no scores occur between them?
- Under the rule \(D=1(R\geq c)\), which side receives observations exactly at \(c\)? Does their presence identify the limit from the other side?
- What is the difference between a narrow confidence interval for a score-group contrast and evidence that the RD estimand is identified?
- Name two support details that should be reported in addition to the total number of observations in a local RD sample.