Symmetric Propensity Score Trimming

Why Use a Symmetric Score Rule?

A propensity score near zero indicates that the covariate profile is associated with a very low probability of treatment; a score near one indicates a very high probability. Comparisons between treated and untreated units can be weakly supported in either tail. Symmetric propensity-score trimming removes both tails using one threshold: a unit is retained only if its score is at least $c$ and at most the complement $1-c$.

The symmetry is about the score scale, not about the number of treated and untreated observations removed. The lower and upper cutoffs are equally far from 0 and 1, but the observed data can have very different numbers of units in the two tails. The rule is a design restriction: it defines which units contribute to the analysis and, consequently, which population an effect estimate can describe.

Definition: Let $A\in\{0,1\}$ denote treatment, $X$ the pretreatment covariates, and $e(X)=P(A=1\mid X)$ the propensity score. For a threshold $c$ in the usual domain $0\leq c<0.5$, define the symmetric trimming indicator $$ S_c(X)=I\{c\leq e(X)\leq 1-c\}. $$ A unit is retained when $S_c(X)=1$; otherwise it is trimmed. Both endpoints are included. In a data analysis using estimated scores, the corresponding rule is $S_c(X)=I\{c\leq \hat e(X)\leq 1-c\}$.

At $c=0$, the interval is $[0,1]$, so no score in the mathematical range of a propensity score is removed. Positive thresholds remove scores strictly below $c$ and strictly above $1-c$. The usual rule is defined for $c<0.5$. If the inclusive interval rule is extended to $c=0.5$, its interval is $[0.5,0.5]$: a score exactly equal to $0.5$ is retained, and every other score is excluded. Thus $c=0.5$ is outside the stated domain of the usual rule; it does not turn equality at the boundary into exclusion.

What Population Does the Rule Retain?

Suppose the target is the average treatment effect among units whose true scores satisfy the rule. Writing $Y(1)$ and $Y(0)$ for the potential outcomes, that target is

\tau_c=E\{Y(1)-Y(0)\mid S_c(X)=1\}.

This is generally not the average treatment effect for the original population. Trimming changes the covariate profiles included in the analysis, so conclusions should name the retained population rather than imply that the result applies to everyone. This is the population-target issue emphasized in the earlier material on trimming.

The score interval has a direct positivity interpretation. Since $A$ is binary, $P(A=0\mid X)=1-e(X)$. Therefore, for any retained covariate profile,

c\leq e(X)\leq 1-c \quad\Longleftrightarrow\quad e(X)\geq c\ \text{and}\ P(A=0\mid X)\geq c.

For $c>0$, both treatment probabilities are bounded away from zero by at least $c$ in the retained set. This gives a positivity margin there; it does not establish that all other causal assumptions hold or that an estimated score is accurate. For a causal interpretation of $\tau_c$, consistency and conditional exchangeability, $Y(a)\perp A\mid X$ for $a\in\{0,1\}$, are also required on the target set, together with positivity there. Under these conditions, treatment groups can be compared within covariate profiles represented in the retained population.

Worked Example: Applying a 0.10 Threshold

Suppose ten hypothetical units have estimated propensity scores $0.03, 0.08, 0.12, 0.25, 0.48, 0.51, 0.77, 0.89, 0.94,$ and $0.97$. Apply symmetric trimming with $c=0.10$.

The upper cutoff is $1-c=1-0.10=0.90$. The inclusive retention interval is therefore $[0.10,0.90]$. The scores $0.12, 0.25, 0.48, 0.51, 0.77,$ and $0.89$ lie in that interval, so six of the ten units are retained. The scores $0.03$ and $0.08$ are below $0.10$; $0.94$ and $0.97$ are above $0.90$. Those four units are trimmed.

The retained fraction in this hypothetical sample is $6/10=0.60$, or 60%. This describes sample retention, not the proportion of the target population retained in a broader population. The interpretation of a subsequent effect estimate should be restricted to units meeting this score rule.

Thresholds, Endpoints, and Target Interpretation

The threshold controls the width of the retained interval: increasing $c$ narrows it symmetrically around 0.5. For $0\leq c_1<c_2<0.5$, every score retained at $c_2$ is also retained at $c_1$, because $[c_2,1-c_2]$ is contained in $[c_1,1-c_1]$. A stricter threshold can improve the minimum treatment-probability margin, but it also excludes more observations and changes the target population.

Worked Example: Comparing Thresholds on the Same Scores

Consider ten hypothetical estimated scores: $0.02, 0.07, 0.16, 0.29, 0.44, 0.56, 0.71, 0.84, 0.93,$ and $0.98$. Compare $c=0.05$ with $c=0.15$.

For $c=0.05$, the interval is $[0.05,0.95]$. It retains $0.07, 0.16, 0.29, 0.44, 0.56, 0.71, 0.84,$ and $0.93$, giving $8/10=0.80$, or 80% retained. The values $0.02$ and $0.98$ are outside the interval.

For $c=0.15$, the interval is $[0.15,0.85]$. It retains $0.16, 0.29, 0.44, 0.56, 0.71,$ and $0.84$, giving $6/10=0.60$, or 60% retained. The more restrictive threshold removes two additional units, $0.07$ and $0.93$, and retains a narrower score range.

The calculation shows the mechanical trade-off, not which threshold is preferable. The effect under $c=0.15$ pertains to a more restricted population than the effect under $c=0.05$; the two estimates need not represent the same causal quantity.

Endpoint inclusion matters in real data. With $c=0.10$, a score of exactly $0.10$ or exactly $0.90$ is retained. Exclusion is based on being below the lower cutoff or above the upper cutoff, not on being equal to either cutoff. In software, the intended condition is typically written with both comparisons inclusive.

Worked Example: Checking Boundary Values

A hypothetical set of estimated scores is $0.099, 0.100, 0.500, 0.900,$ and $0.901$. Apply the rule with $c=0.10$.

The interval is $[0.10,0.90]$. The score $0.099$ is below the lower cutoff and is trimmed. The scores $0.100, 0.500,$ and $0.900$ are in the interval, including the two boundary values, and are retained. The score $0.901$ is above the upper cutoff and is trimmed. Thus three of the five units are retained.

As a boundary check on the rule’s domain, $c=0.50$ is not part of the usual $c<0.5$ definition. If one nevertheless applies the inclusive interval formula at that value, only scores equal to $0.50$ fall in $[0.50,0.50]$; those scores are retained, while all others are excluded.

Justification and Practical Diagnostics

The key property of symmetric trimming follows directly from the binary treatment definition. On the retained set, $e(X)\geq c$ by the lower inequality. Also, $e(X)\leq 1-c$, so subtracting from one gives $1-e(X)\geq c$. Since $1-e(X)=P(A=0\mid X)$, both treatment probabilities are at least $c$. Conversely, if both $P(A=1\mid X)\geq c$ and $P(A=0\mid X)\geq c$, then $e(X)\geq c$ and $1-e(X)\geq c$, which implies $e(X)\leq1-c$. Together these establish the equivalence, not just one direction.

In practice, the score is usually estimated rather than known. Before interpreting a trimmed analysis, inspect the score distributions by treatment group and verify the actual retained counts and boundary values. Check that the intended cutoff was applied to the intended score, that scores fall in $[0,1]$, and that the retained set contains both treated and untreated units. Compare the covariate profiles of retained and trimmed units so the population shift is visible. An interval rule can remove extreme estimated scores, but it cannot certify correct model specification, covariate balance, or adequate support in every important subgroup. The earlier material on propensity-score overlap plots and tail diagnostics provides the relevant diagnostic context.

Symmetric trimming is distinct from an empirical common-support restriction based on the observed score ranges of the treatment groups. The symmetric rule uses a specified pair of cutoffs around 0.5; an empirical range-based rule is determined by the observed groups and may not produce cutoffs symmetric around 0.5. Neither rule should be mistaken for a guarantee of covariate-level support just because the retained scores appear to overlap. In either case, the restriction changes who is represented in the analysis.

Common Mistakes and Limitations

Reversing the upper cutoff. For lower threshold $c$, the upper cutoff is $1-c$, not $1+c$. A rule such as $e(X)\leq1-c$ removes the high-score tail at the complementary distance from one.
Dropping boundary values by accident. The definition uses $c\leq e(X)\leq1-c$. A score equal to either cutoff is retained.
Claiming that the number removed is symmetric. The cutoffs are symmetric around 0.5; sample counts in the two tails need not be equal.
Calling the trimmed estimate an effect for everyone. It targets the population satisfying the rule, which may differ substantially from the original population.
Treating estimated scores as exact. A fitted score can be misspecified or uncertain. Trimming those estimates does not by itself resolve estimation error or prove positivity in the true data-generating process.

A further limitation is that changing $c$ changes both the amount of retained data and the estimand. Comparing estimates across thresholds is therefore not simply a sensitivity check on one unchanged target. The threshold should be reported with its score definition, retained counts by treatment group, and the resulting target-population description. These details make the restriction auditable without implying that any one cutoff is universally correct.

Key takeaway: Symmetric trimming with $0\leq c<0.5$ retains scores in the inclusive interval $[c,1-c]$. For positive $c$, both treatment probabilities are at least $c$ within that score-defined set, but the analysis targets the retained population and still requires the causal assumptions and diagnostics relevant to that target.

Check Your Understanding

Work through each question, then compare with the solutions below.

For $c=0.12$, state the retained interval and decide whether scores $0.12$, $0.88$, and $0.91$ are retained.
A sample has scores $0.04, 0.20, 0.30, 0.70, 0.80,$ and $0.96$. How many are retained at $c=0.20$, and what fraction of the sample is that?
Explain why both treatment probabilities are at least $c$ for a unit retained under the symmetric rule.
What does an effect estimate after trimming describe, and why is it not automatically the original-population average treatment effect?
Under the inclusive interval formula extended to $c=0.50$, what happens to scores equal to $0.50$?

Solutions. (1) The interval is $[0.12,0.88]$; $0.12$ and $0.88$ are retained at the inclusive boundaries, while $0.91$ is not. (2) At $c=0.20$, the interval is $[0.20,0.80]$; four scores, $0.20,0.30,0.70,$ and $0.80$, are retained, so the fraction is $4/6=0.667$, or about 66.7%. (3) Retention gives $e(X)\geq c$ and $e(X)\leq1-c$; the latter implies $1-e(X)\geq c$, and $1-e(X)=P(A=0\mid X)$. (4) It describes the effect among units whose scores satisfy the trimming rule. The restricted covariate distribution may differ from the original population, so the target may differ. (5) The extended interval is $[0.50,0.50]$, so scores exactly equal to $0.50$ are retained; all other scores are excluded. The usual rule’s stated domain remains $c<0.50$.