Why Trimming Changes the Question
Inverse-probability weighting and matching rely on comparing treated and untreated units with relevant covariate profiles. When some profiles occur almost exclusively in one treatment group, those comparisons depend on very few observations and can be highly sensitive to modeling and sampling variation. Trimming removes units in regions of limited overlap so that the remaining comparison is more credible.
That decision has a substantive consequence: an effect estimated after trimming is generally not the average treatment effect for the original population. It describes a population selected by the trimming rule. To interpret the result correctly, we need to specify both which units are retained and which treatment-effect target is averaged over those units.
The notation \(S(X)\) emphasizes an important design choice: the rule is based on pretreatment covariates, often through the propensity score, rather than on observed outcomes. The resulting population may differ in its covariate composition from the original one. If treatment effects vary with \(X\), changing that composition can also change the average effect.
Common Trimming Rules
There is no universally correct cutoff. The rule should be chosen with the scientific target and the observed overlap in mind, and its implications should be reported. Three common approaches are:
- Symmetric propensity-score trimming: retain units whose propensity scores fall within a chosen interval, commonly \(a\leq e(X)\leq 1-a\). A cutoff such as \(a=0.10\) removes units with scores below 0.10 or above 0.90. This is a rule, not a universal standard.
- Common-support trimming: retain units whose scores lie in a region represented in both treatment groups. A practical rule might compare observed score ranges or define a shared region using score bins or prespecified limits. The exact operational definition matters, especially when estimated ranges are influenced by a few extreme observations.
- One-sided trimming for a treated-group target: when the target is an average effect for treated units, remove treated units whose covariate profiles have no credible untreated comparators. A corresponding rule may remove controls outside the retained treated group’s relevant support. The target then concerns the treated units who remain, not necessarily all treated units.
Trimming is not the same as weight truncation. Trimming excludes units from the analysis and changes the population being represented. Weight truncation caps unusually large weights while retaining the units, which changes how they contribute to a weighted estimate. These procedures address related numerical concerns but do not generally target the same population.
Worked Example: Counting the Units Removed by Symmetric Trimming
Consider a hypothetical sample of 200 people divided into three covariate-profile groups. Their estimated propensity scores and treatment counts are shown below. Apply the rule \(0.10\leq \hat e(X)\leq 0.90\), where \(\hat e(X)\) is the estimated propensity score.
| Profile group | Estimated score | Treated | Untreated | Total |
|---|---|---|---|---|
| Low-score profiles | 0.04 | 2 | 38 | 40 |
| Middle-score profiles | 0.50 | 60 | 60 | 120 |
| High-score profiles | 0.96 | 38 | 2 | 40 |
| Total | — | 100 | 100 | 200 |
The low-score and high-score groups fall outside the interval, so the rule removes \(40+40=80\) people. It retains all 120 people in the middle group, including 60 treated and 60 untreated people. Thus the retained share of the whole sample is \(120/200=0.60\), or 60%. The retained share is also \(60/100=0.60\) for treated people and \(60/100=0.60\) for untreated people in this example.
The retained sample has equal treatment-group counts, but that balance is not guaranteed by symmetric trimming. More importantly, the retained sample consists entirely of the middle-score profile group. An estimate based on it represents that group under this rule; it does not automatically represent the 200-person population.
From the Retention Rule to the Estimand
A treatment-effect estimand specifies the population average being sought. For the original population, the average treatment effect (ATE) is \(E[Y(1)-Y(0)]\). For the population retained by a rule, the corresponding estimand is
This conditional expectation averages individual treatment effects over the covariate distribution of retained units. If the retained population is defined by \(e(X)\) falling in a specified interval, then the target population is the population with covariates in that interval. It is not simply “the people for whom the estimate is more stable”; it is a defined subset of the original population.
In particular, conditional independence means \((Y(1),Y(0))\) is independent of \(D\) given \(X\), and positivity means \(0<P(D=1\mid X)<1\) for the profiles being used. Restricting the analysis can make the positivity condition more plausible within the retained region, but it cannot repair unmeasured confounding or make a poorly specified treatment-assignment model correct.
Why the Average Effect Can Change
The effect in a retained population equals the original ATE only under additional conditions—for example, if the average treatment effect is constant across covariate profiles. If effects differ across profiles, trimming changes the weights assigned to those effects.
A short derivation makes the role of the target population explicit. Let \(m_d(X)=E[Y\mid D=d,X]\) for \(d\in\{0,1\}\). Under consistency and conditional independence, \(m_1(X)=E[Y(1)\mid X]\) and \(m_0(X)=E[Y(0)\mid X]\). Thus the conditional outcome contrast is the conditional treatment effect. Averaging it among retained units gives
The second equality follows from iterated expectation, because \(S(X)\) is determined by \(X\). The derivation shows both what trimming changes—the distribution of \(X\) being averaged over—and what it does not do: it does not alter the meaning of a treatment effect for a given covariate profile.
Worked Example: A Different Retained Population, a Different Average Effect
Suppose, hypothetically, that a population has three profile groups with shares 20%, 60%, and 20%. Assume their average treatment effects are respectively 2, 4, and 10 outcome units. These values describe assumed causal effects for the example, not results from a real study. The original-population ATE is
Now suppose a trimming rule retains only the middle group. In the retained population, that group makes up 100% of the covariate distribution, so
The two values differ by \(4.80-4.00=0.80\) outcome units because the groups with effects of 2 and 10 units are no longer represented in the target. The trimmed estimate is not necessarily biased for its own retained-population estimand; it answers a different question from the original ATE.
Assessing Composition and Choosing a Rule
A cutoff is informative only when paired with a description of who remains. At a minimum, report the rule, the number and proportion retained overall, the numbers retained in each treatment group, and how important pretreatment characteristics change. The same cutoff can remove very different types of people in different applications.
A useful comparison is to calculate retention across several plausible rules. For instance, a narrower symmetric interval may improve empirical overlap but remove more of the population. If the estimated effect changes across these rules, that variation may reflect changing target populations as well as changes in the observations used to estimate each effect. It should not be interpreted as a pure measure of statistical instability.
Worked Example: A One-Sided Rule for a Treated-Group Target
Suppose a hypothetical study has 80 treated people and 120 untreated people. A common-support assessment identifies 12 treated people whose estimated propensity scores are above the range with credible untreated comparisons. The analysis removes those 12 treated people and retains the 68 treated people with support, together with the 120 untreated people in the comparison region. What is the treated-group target after this rule?
The retained treated share is \(68/80=0.85\), or 85%. The analysis no longer targets the average effect for all 80 treated people. Its treated-group estimand is \(E[Y(1)-Y(0)\mid D=1,S(X)=1]\): the average treatment effect for treated people whose profiles satisfy the retention rule. The 12 excluded treated people are outside that target.
The count of 120 retained controls alone does not establish adequate comparison. Their covariate profiles must still represent the retained treated people sufficiently, and the identification assumptions must be credible. If the rule instead also removed controls, the comparison sample would change, but the treated-group target would remain defined by the retained treated people.
Common Mistakes and Practical Interpretation
- Calling the result the original-population ATE. Once the rule excludes covariate profiles, the estimate generally targets an effect conditional on retention. Name that population and estimand explicitly.
- Assuming symmetric cutoffs guarantee balance. The interval restricts propensity scores; it does not ensure that all covariates are balanced. Balance diagnostics remain necessary.
- Treating a conventional cutoff as a law. A threshold such as 0.10 is a choice, not a guarantee of adequate overlap. Explain why the rule is appropriate and show how it affects sample composition.
- Confusing trimming with weight capping. Trimming changes which units define the target. Capping changes the contribution of retained units. State which procedure is used.
- Interpreting a score as an outcome probability. The propensity score \(e(X)\) is the probability of treatment given covariates, not the probability of a particular outcome.
Trimming can improve the credibility of comparisons by focusing attention on regions where both treatment states are represented. Its cost is that conclusions may apply to a narrower population. A transparent report therefore treats the retention rule as part of the estimand: it states whom the analysis describes, how many units remain, and which covariate profiles are no longer represented.
Check Your Understanding
Answer each question by distinguishing the retention rule from the treatment-effect target.
- Under symmetric trimming at \(0.10\) and \(0.90\), which propensity scores are retained?
- A rule retains 70 of 100 treated people. What proportion of the treated group remains, and what should be stated about the target if the aim is a treated-group effect?
- Explain why a trimmed estimate need not equal the original-population ATE even if both are correctly identified.
- How does trimming differ from capping inverse-probability weights?
- Why does retaining only propensity scores between two cutoffs not, by itself, prove that covariates are balanced?