Common-Support Restriction Rules

Why Make a Common-Support Restriction?

A propensity score can help describe how treatment assignment varies across measured baseline covariates. When the scores of treated and control units occupy substantially different regions, some comparisons rely on units with little or no counterpart in the other treatment group. A common-support restriction makes a specific design decision: it keeps units whose scores fall in a region represented in both groups, and removes units outside that region.

This tutorial focuses on two operational rules. The minimum-to-maximum intersection rule uses the observed score ranges in the treated and control groups. A prespecified score-range rule keeps units inside bounds chosen in advance. Neither rule, by itself, proves that the groups are comparable on all covariates or that a causal effect is identified.

Definition: Let \(A_i\in\{0,1\}\) indicate whether unit \(i\) is treated, let \(X_i\) be its measured baseline covariates, and let \(e(X_i)=P(A_i=1\mid X_i)\) be its propensity score. In practice, the score used for a restriction is often an estimated score, written \(\hat e_i\). A common-support restriction retains units in a specified score region shared, or intended to be shared, by treated and control units; it excludes units outside that region.

Calculating a restriction requires scores and treatment indicators. A causal interpretation requires more. In particular, the usual observational-study assumptions include consistency, conditional exchangeability given the measured covariates, and positivity for the population being analyzed. Conditional exchangeability states that, given \(X\), treatment assignment is independent of the potential outcomes. Positivity requires a nonzero probability of each treatment level for covariate values in the target population. A score restriction can narrow the population under study, but cannot repair unmeasured confounding or establish these assumptions from the observed scores.

The rules below are written for a finite data set with one score per unit. They can be applied to estimated scores, but the resulting boundaries are then boundaries of the observed estimated scores—not necessarily the exact population support of the true propensity score.

Minimum-to-Maximum Intersection Rule

Find the smallest and largest scores separately in the treated and control groups. The shared interval is bounded below by the larger of the two group minima and above by the smaller of the two group maxima. Keep a unit if its score lies in that interval, including the endpoints.

\begin{aligned} \hat e_{1,\min}&=\min_{i:A_i=1}\hat e_i, &\hat e_{1,\max}&=\max_{i:A_i=1}\hat e_i,\\ \hat e_{0,\min}&=\min_{i:A_i=0}\hat e_i, &\hat e_{0,\max}&=\max_{i:A_i=0}\hat e_i,\\ L&=\max(\hat e_{1,\min},\hat e_{0,\min}), &U&=\min(\hat e_{1,\max},\hat e_{0,\max}). \end{aligned}

When \(L\le U\), the retained set is \(L\le\hat e_i\le U\), for either treatment group. A score exactly at \(L\) or \(U\) is retained. When \(L>U\), the observed group ranges do not intersect, so this rule retains no units. In reporting, state the interval, the number retained in each group, and which units or scores were excluded.

Worked Example: Finding the Shared Observed Range

Suppose a hypothetical data set has these estimated scores. The unit labels are included to make the retention decision explicit.

Unit	Treatment group	Estimated score
T1	Treated	0.12
T2	Treated	0.28
T3	Treated	0.46
T4	Treated	0.72
C1	Control	0.08
C2	Control	0.20
C3	Control	0.54
C4	Control	0.81

The treated range is \([0.12,0.72]\), and the control range is \([0.08,0.81]\). Thus \(L=\max(0.12,0.08)=0.12\) and \(U=\min(0.72,0.81)=0.72\). The shared observed interval is \([0.12,0.72]\).

All four treated scores lie in that interval, so T1, T2, T3, and T4 are retained. Among controls, C2 at 0.20 and C3 at 0.54 are retained; C1 at 0.08 is below \(L\), and C4 at 0.81 is above \(U\). The restricted data set therefore contains 4 treated and 2 control units. This rule excludes the control observations beyond the treated group’s observed score range; it does not remove every observation that might be unusual on the covariates.

Prespecified Score-Range Rule

A prespecified rule sets lower and upper bounds, \(L_0\) and \(U_0\), before applying the restriction to the analyzed scores. It retains a unit when its score is within the chosen range, with endpoints included. The same bounds apply to treated and control units unless a different rule is explicitly defined.

R_i= \begin{cases} 1,&L_0\le \hat e_i\le U_0,\\ 0,&\text{otherwise}. \end{cases}

Here \(R_i=1\) indicates retention and \(R_i=0\) indicates exclusion. The bounds must be stated as part of the analysis rule, not inferred after seeing which units produce a preferred outcome estimate. A rule such as “retain scores from 0.10 through 0.80” is operationally clear. A description such as “remove extreme scores” is not, unless “extreme” has been translated into explicit bounds.

Worked Example: Applying Prespecified Bounds

Consider a second hypothetical data set with prespecified bounds \(L_0=0.15\) and \(U_0=0.70\). The scores are: treated units T1 = 0.09, T2 = 0.15, T3 = 0.43, T4 = 0.69, and T5 = 0.76; control units C1 = 0.11, C2 = 0.27, C3 = 0.70, and C4 = 0.74.

Apply \(0.15\le\hat e_i\le0.70\) to each unit. T1 is excluded because \(0.09<0.15\). T2, T3, and T4 are retained because \(0.15\le0.15\le0.70\), \(0.15\le0.43\le0.70\), and \(0.15\le0.69\le0.70\), respectively. T5 is excluded because \(0.76>0.70\).

For controls, C1 is excluded because \(0.11<0.15\); C2 and C3 are retained because their scores are between the bounds, including C3 at the upper endpoint; C4 is excluded because \(0.74>0.70\). The restricted data set has 3 treated and 2 control units. These bounds produce a well-defined retained set, but the counts alone do not show whether the covariates are balanced within it.

Why the Minimum-to-Maximum Rule Works—and What It Establishes

The rule is the direct interval intersection of two observed ranges. The treated scores occupy \([\hat e_{1,\min},\hat e_{1,\max}]\), while the control scores occupy \([\hat e_{0,\min},\hat e_{0,\max}]\). A score is in their intersection exactly when it is at least as large as both lower endpoints and no larger than either upper endpoint. Those two requirements reduce to \(\hat e_i\ge\max(\hat e_{1,\min},\hat e_{0,\min})\) and \(\hat e_i\le\min(\hat e_{1,\max},\hat e_{0,\max})\), which gives the stated \(L\) and \(U\).

[\hat e_{1,\min},\hat e_{1,\max}] \cap [\hat e_{0,\min},\hat e_{0,\max}] = [\max(\hat e_{1,\min},\hat e_{0,\min}), \min(\hat e_{1,\max},\hat e_{0,\max})]

If the lower endpoint exceeds the upper endpoint, no number can satisfy both inequalities. If it does not, every retained score falls within the observed range of both groups. This is an interval-level statement about the scores in this data set. It does not imply that every score value inside the interval occurs in both groups, that the covariate distributions overlap adequately, or that all confounders have been measured.

Worked Example: Detecting an Empty Intersection

In a hypothetical data set, treated scores are 0.62, 0.71, and 0.83; control scores are 0.10, 0.35, and 0.58. The treated range is \([0.62,0.83]\) and the control range is \([0.10,0.58]\). Therefore \(L=\max(0.62,0.10)=0.62\), while \(U=\min(0.83,0.58)=0.58\).

Since \(L=0.62>0.58=U\), the intersection is empty. There is no observed score satisfying \(0.62\le\hat e_i\le0.58\), so the rule retains zero treated and zero control units. It would be incorrect to report the interval as \([0.62,0.58]\) and proceed as though it were an ordinary range. The appropriate conclusion is that this rule finds no common interval between the two observed score ranges. Any alternative restriction or analysis would be a separate design decision.

Implementation Checks and Interpretation

A restriction is simple to calculate, but easy to implement incorrectly. For either rule, calculate the bounds first, apply the same stated inclusion criterion to every unit, and tabulate retained and excluded counts separately by treatment group. Keep track of excluded units or their score ranges so that the resulting population can be described.

Check endpoint handling. If the rule is inclusive, a score equal to a bound is retained. Use the same convention in code and in the written description.
Check treatment labels. Calculate minima and maxima within the correct treated and control groups; reversing the group labels changes the calculation.
Check the retained counts. Confirm that the retained units satisfy the rule and that all excluded units fail at least one bound.
Check the score scale. Apply bounds to the same score definition and scale used to state them. A probability score on the 0-to-1 scale is not interchangeable with its logit transform.
Describe the resulting population. A restriction changes which units contribute to the analysis. The estimated effect is therefore about the restricted population, not automatically the original study population.

The observed-range intersection is data-dependent: its bounds are determined by the most extreme observed scores in each group. A prespecified interval instead uses bounds set independently of the observed group extrema. Both can be stated as transparent rules, but they answer different operational questions. One asks where the two observed ranges intersect; the other asks which units fall within chosen bounds. Visual overlap plots can help reveal gaps and concentrations that range endpoints conceal, while covariate-support diagnostics examine compatibility in covariate space rather than just on the one-dimensional score scale.

Neither rule is a balance guarantee. Even when the score intervals intersect, the treated and control units retained within that interval may differ in important covariates. Conversely, the observed minimum and maximum are sensitive to sample size and individual extreme scores: a single observation can extend a range without creating substantial overlap in its vicinity. Treat the retained interval as a restriction rule, then evaluate the design and state its limitations rather than interpreting interval overlap as proof of comparability.

Common Mistake

Common mistake: “The treated and control score ranges overlap, so all relevant covariates are balanced and a causal effect is identified.” The first part only describes the endpoints of observed score ranges. It does not establish dense overlap at every score, covariate balance, conditional exchangeability, or correct score estimation. The correction is to report the rule and retained units precisely, then assess balance and the assumptions needed for a causal interpretation separately.

A related mistake is to use group-specific bounds while describing a common prespecified range. If the intended rule is \(L_0\le\hat e_i\le U_0\), apply that same interval to both groups. Any different procedure needs its own explicit definition; it is not the rule described here.

Check Your Understanding

Answer each question, then compare your reasoning with the short solutions.

Treated scores are 0.18, 0.40, and 0.77; control scores are 0.09, 0.31, and 0.68. Find the minimum-to-maximum intersection and identify retained units.
With prespecified inclusive bounds \([0.20,0.60]\), what happens to units scored exactly 0.20 or 0.60?
If the treated range is \([0.55,0.88]\) and the control range is \([0.12,0.49]\), what does the intersection rule retain?
Give one conclusion supported by a nonempty observed-range intersection and one conclusion it does not support.

Solutions. (1) The bounds are \(L=\max(0.18,0.09)=0.18\) and \(U=\min(0.77,0.68)=0.68\). All three treated units are retained; the first two controls, at 0.09 and 0.31, are not both retained: 0.09 is below the interval and 0.31 is inside it, while 0.68 is at the upper endpoint and is retained. Thus the retained units are all three treated units and controls at 0.31 and 0.68. (2) Both are retained because the bounds are inclusive. (3) \(L=0.55>U=0.49\), so the intersection is empty and no units are retained by this rule. (4) A nonempty intersection supports the statement that the observed group score ranges share an interval. It does not establish covariate balance or causal identification.