From a Threshold to an Assignment
In a threshold-based design, a measured quantity determines which side of a policy boundary a unit falls on. That quantity might be an examination score, an age, or a risk score. The boundary is the cutoff; the measured quantity is the running variable. The threshold rule determines an assignment, but assignment and treatment received are not necessarily the same thing.
The unit \(i\) may be a person, school, clinic, or other entity, depending on what the policy assigns. For example, if a rule assigns schools based on a school-level score, the running variable and assignment are defined for schools, even if treatment is delivered to students within them. The unit and the level at which \(R_i\), \(Z_i\), and \(D_i\) are recorded should be made explicit.
A common convention for a rule assigning units at or above the cutoff is
where \(\mathbb{1}(\cdot)\) equals 1 when the condition in parentheses is true and 0 otherwise. This convention assigns a unit exactly at \(c\) to \(Z_i=1\). A rule assigning units below the cutoff instead might use \(Z_i=\mathbb{1}(R_i<c)\). The direction of the rule and how equality is handled are part of the design definition, not details to infer from a generic label such as “threshold program.”
It is often convenient to describe distance from the cutoff as \(R_i-c\). A negative value means the unit is below the cutoff and a positive value means it is above it. This centered distance does not change the assignment rule; it simply makes the boundary correspond to zero.
Sharp Assignment and Imperfect Compliance
The key distinction is whether observed treatment follows the assignment rule exactly. In a sharp design, treatment receipt is completely determined by assignment. In an imperfect-compliance design, assignment and receipt can differ: some assigned units do not receive treatment, or some unassigned units do.
The term “sharp” describes the assignment-to-treatment relationship; it does not mean that the outcome changes abruptly or that a causal effect has been established. Similarly, “fuzzy” describes imperfect alignment between assignment and receipt. It does not, by itself, establish that the threshold is a valid instrument for a causal analysis.
A useful descriptive quantity in an imperfect-compliance setting is the difference in observed treatment rates by assignment status:
In sample data, replace each probability with the corresponding fraction treated. This contrast describes how treatment receipt differs between the two assignment groups. It is not automatically a causal effect: by itself, it does not establish that assignment is otherwise comparable across groups or justify a causal interpretation.
Worked Example: A Score Rule With Perfect Adherence
Suppose, hypothetically, that a program assigns units to receive a service when their score is at least 60. The score is \(R_i\), the cutoff is \(c=60\), and the rule is \(Z_i=\mathbb{1}(R_i\geq60)\). The observed records are:
| Unit | Score \(R_i\) | Assignment \(Z_i\) | Treatment received \(D_i\) |
|---|---|---|---|
| A | 58 | 0 | 0 |
| B | 60 | 1 | 1 |
| C | 63 | 1 | 1 |
| D | 55 | 0 | 0 |
For unit A, \(58<60\), so \(Z_A=0\); the record has \(D_A=0\). For unit B, equality is assigned to the upper group under the stated rule: \(60\geq60\), so \(Z_B=1\), and \(D_B=1\). Units C and D are classified in the same way, giving assignments 1 and 0, respectively.
For all four units, \(D_i=Z_i\). Thus the observed records follow a sharp assignment pattern. This check concerns only the relationship between assignment and treatment receipt. No outcome data are needed to classify the assignment pattern, and the pattern alone does not establish a treatment effect.
Worked Example: Eligibility With Imperfect Take-Up
Consider a hypothetical program in which a risk score of 70 or higher makes a person eligible for an offered intervention. Here \(R_i\) is the risk score, \(c=70\), and \(Z_i=\mathbb{1}(R_i\geq70)\). Among 100 people below the cutoff, 18 receive the intervention; among 100 people at or above it, 72 receive it.
The observed treatment rate in the assigned group is
The observed treatment rate in the other group is
The difference is \(0.72-0.18=0.54\), or 54 percentage points. This is a descriptive contrast in treatment receipt between the two assignment groups in these hypothetical data. The 72 eligible people who receive treatment do not include every eligible person, and the 18 treated people below the cutoff show that treatment is not confined to the eligible group. Therefore \(D_i\neq Z_i\) for some units, and assignment is imperfect rather than sharp.
Eligibility is the rule-based assignment in this example; it is not the treatment itself. If the data recorded only eligibility, the treatment-rate contrast could not be calculated. Conversely, if the data recorded treatment receipt but not the eligibility rule or score, the records alone would not reveal how the threshold assigned people.
State the Rule Before Classifying the Design
A threshold design should be described in terms of four separate elements: the unit, the running variable, the cutoff rule, and the treatment measure. This prevents an administrative label such as “eligible” or “high risk” from being mistaken for observed receipt. It also makes the equality convention visible, which matters whenever values can equal the cutoff.
Specify what one observation represents and how the running variable \(R_i\) is measured, including its units and measurement time.
Give the cutoff \(c\), state which side is assigned, and specify what happens when \(R_i=c\). Define \(Z_i\) from that rule.
Explain what counts as \(D_i=1\), including the relevant treatment, time window, and data source.
Check whether \(D_i=Z_i\) for every unit. If not, describe which mismatches occur and summarize treatment rates by assignment status.
Worked Example: A Lower-Score Rule and a Tie at the Cutoff
Suppose a hypothetical service is offered to people with scores of 40 or less. The rule is \(Z_i=\mathbb{1}(R_i\leq40)\), so a score of exactly 40 qualifies. The following counts show treatment receipt:
| Score | Number of people | Number treated |
|---|---|---|
| 38 | 10 | 8 |
| 40 | 10 | 7 |
| 41 | 10 | 2 |
The assignment rule puts the 10 people with score 38 and the 10 people with score 40 into \(Z_i=1\), for 20 assigned people total. Of these, \(8+7=15\) are treated, giving a treatment rate of \(15/20=0.75\). The 10 people with score 41 are in \(Z_i=0\); \(2/10=0.20\) are treated. The difference in these group rates is \(0.75-0.20=0.55\), or 55 percentage points.
The scores and counts show imperfect assignment-to-receipt alignment. In particular, not every person with a qualifying score is treated, and some people with a nonqualifying score are treated. The computation summarizes these groups as defined by the policy rule; it is not an estimate of the effect of receiving the service. Because the score takes discrete values, the cutoff tie is not hypothetical: the rule must classify the people at 40 explicitly.
Checks, Comparisons, and Limits
The first diagnostics concern whether the written rule matches the actual assignment process. Check the policy documentation and data definitions against the recorded assignment indicator. Then compare \(Z_i\) with the running-variable rule and compare \(D_i\) with \(Z_i\). In an imperfect-compliance design, tabulating treatment counts by assignment group reveals the direction and magnitude of the observed treatment-rate difference. These checks describe the design; they do not test whether a causal analysis is justified.
Several data features can make the classification misleading:
- Wrong boundary or direction. A rule may assign scores below a cutoff, not above it. Reversing the inequality misclassifies units.
- Unspecified equality rule. If the rule does not say whether equality qualifies, tied values cannot be assigned unambiguously from the description alone.
- Mixing eligibility and receipt. Being offered, being eligible, enrolling, and receiving a completed treatment can be distinct events. Choose the variable that matches the design question and define it precisely.
- Inconsistent measurement. A score measured after treatment could be a different variable from the pre-treatment score that actually governed assignment.
- Multiple policies or exceptions. Different sites, dates, or groups may use different cutoffs or override the rule. A single indicator may then fail to represent the actual assignment process.
- Incomplete records. Missing scores, assignment status, or treatment receipt can prevent a reliable comparison. Do not silently classify missing values as untreated or unassigned.
Threshold-based assignment differs from random assignment because the assignment is generated by a rule based on \(R_i\), rather than by a random allocation. It also differs from matching, which constructs comparisons using observed characteristics rather than defining assignment through a cutoff. In an imperfect-compliance threshold design, \(Z_i\) can serve as a candidate instrument for \(D_i\), but a treatment-rate difference alone is not enough to establish instrument validity or identify an effect. The assumptions and identification arguments required for causal interpretation are separate from the task of defining the threshold and checking how treatment was assigned.
In particular, knowing that a treatment-rate jump is observed does not settle whether units near the cutoff are comparable in the relevant sense, whether the running variable is recorded or governed as intended, or what effect a later analysis targets. Those are substantive design and identification questions. The narrow but essential task here is to establish what the rule assigns and how closely observed receipt follows it.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A rule assigns units with \(R_i\geq25\), and one unit has \(R_i=25\). What is its assignment if equality is included? What additional observation is needed to decide whether assignment is sharp?
- In a hypothetical study, 48 of 80 assigned units and 12 of 60 unassigned units receive treatment. Calculate the difference in treatment rates and state what it describes.
- A policy assigns people with scores at or below 10, but the analyst defines assignment as \(Z_i=\mathbb{1}(R_i\geq10)\). Identify the error and state the appropriate rule if equality qualifies.
- Why does a 40-percentage-point difference in treatment rates by assignment status not, by itself, establish a causal treatment effect?
Solutions. (1) The unit has \(Z_i=1\) under the stated rule. To classify the assignment as sharp, observe whether its treatment receipt is \(D_i=1\), and check that \(D_i=Z_i\) for all units governed by the rule. (2) The rates are \(48/80=0.60\) and \(12/60=0.20\); the difference is \(0.60-0.20=0.40\), or 40 percentage points. It describes the observed treatment-rate contrast between assignment groups. (3) The inequality is reversed. If equality qualifies, use \(Z_i=\mathbb{1}(R_i\leq10)\). (4) The contrast is descriptive. It does not alone establish that assignment supports a causal comparison or that the assumptions for a causal interpretation hold.