Fixed Truncation Thresholds and Asymptotic Bias

Why a Fixed Threshold Has an Asymptotic Consequence

Clipping large inverse-probability weights is often used to reduce the influence of observations with very small treatment probabilities. That can reduce variability, but it also changes the weights applied to the data. If the clipping threshold stays fixed as the sample grows, the change does not disappear: the estimator can converge to a different population quantity from the intended causal parameter.

The key distinction is between finite-sample error and <strong asymptotic bias. A noisy estimate may become more precise with a larger sample, but a fixed clipping rule can leave the estimator centered on the wrong limiting value. We derive that value for a simple binary-treatment mean estimator, then use it to identify when the discrepancy occurs.

Setup and Fixed-Cap Estimator

The target causal mean is \(\mu_a=E[Y^a]\); the average treatment effect is \(\tau=E[Y^1-Y^0]=\mu_1-\mu_0\). To identify these means from observed data, assume consistency (\(Y=Y^A\)), conditional exchangeability (\(Y^a\) is independent of \(A\) given \(X\), for each \(a\)), and positivity (\(p_a(X)>0\) almost surely for each arm). Assume observations are independent and identically distributed, \(E[|Y|]<\infty\), and the propensity probabilities in the estimator are the true probabilities. These conditions suffice for the probability limits below; estimated weights require additional analysis.

Write \(m_a(X)=E[Y^a\mid X]\). Under consistency and exchangeability, the conditional mean of the observed outcome among people with \(A=a\), given \(X\), is \(m_a(X)\). The clipped weight is bounded by \(c\), so the weighted outcome has finite absolute expectation under the stated integrability condition.

Probability Limit of the Clipped Mean

Proof. For one observation, the denominator contribution has expectation \(E[I(A=a)w_{a,c}(X)]=E[p_a(X)w_{a,c}(X)]=E[\min\{1,c p_a(X)\}]\). This is positive because positivity gives \(p_a(X)>0\) almost surely and \(c>0\). The contribution is bounded by \(c\), so the law of large numbers applies.

For the numerator, consistency and conditional exchangeability give \(E[Y\mid A=a,X]=m_a(X)\). Iterated expectation therefore yields \(E[I(A=a)w_{a,c}(X)Y]=E[p_a(X)w_{a,c}(X)m_a(X)]=E[q_{a,c}(X)m_a(X)]\). The absolute expectation is finite: \(w_{a,c}\leq c\) and \(E[|Y|]<\infty\). Thus the law of large numbers applies to the numerator as well. Since the denominator average converges in probability to a positive constant, the ratio converges to the ratio of these expectations by the continuous mapping theorem. This is the stated limit. \(\square\)

The factor \(q_{a,c}(X)\) shows what clipping does to the covariate distribution represented in the weighted arm. Without clipping, \(w=1/p_a(X)\), so \(p_aw=1\) and all covariate values receive equal weight in the target mean. With a fixed cap, values with \(p_a(X)<1/c\) instead receive factor \(c p_a(X)<1\). They remain represented under strict positivity, but are downweighted relative to the original target population.

When Does the Limiting Value Equal the Target?

Derivation. Since \(\mu_a=E[m_a(X)]\), subtract this quantity from the ratio defining \(\mu_{a,c}\): \[ \mu_{a,c}-\mu_a =\frac{E[q_{a,c}m_a]-E[q_{a,c}]E[m_a]}{E[q_{a,c}]} =\frac{\operatorname{Cov}(q_{a,c},m_a)}{E[q_{a,c}]}. \] The denominator is positive, so the difference is zero exactly when the covariance is zero. If \(m_a\) is constant, its covariance with any integrable \(q_{a,c}\) is zero. If no weights are changed, then \(q_{a,c}=1\) almost surely, which also makes the covariance zero. \(\square\)

For the average treatment effect estimated as \(\hat{\mu}_{1,c}-\hat{\mu}_{0,c}\), the limiting discrepancy is the difference between the two arm-specific discrepancies: \[ (\mu_{1,c}-\mu_{0,c})-\tau =\frac{\operatorname{Cov}(q_{1,c},m_1)}{E[q_{1,c}]} -\frac{\operatorname{Cov}(q_{0,c},m_0)}{E[q_{0,c}]}. \] Thus the direction of bias in a contrast cannot be inferred from clipping alone. Each arm can be shifted differently, and the shifts can reinforce or offset one another.

Interpretation, Diagnostics, and Alternatives

A fixed cap bounds each individual weight, but bounded weights are not by themselves a guarantee of consistency for the original causal target. They ensure that the specified estimator has a well-defined limiting weighted average under the theorem’s conditions. Whether that average is the desired causal mean depends on the relationship between the clipping factors and the conditional outcome means.

Useful diagnostics include reporting the cap and the proportion of weights affected, checking how estimates change across scientifically plausible fixed caps, and examining whether clipping materially changes weighted covariate distributions. These checks reveal sensitivity; they do not prove that a chosen cap removes bias. A small proportion of clipped observations can still matter if those observations represent covariate strata with substantially different conditional outcomes.

The main alternative is the untruncated IPW estimator: under the identification assumptions and suitable integrability, its population weight factor is \(p_a(X)/p_a(X)=1\), recovering \(E[Y^a]\). Its weights may be highly variable when treatment probabilities are small. Outcome standardization instead estimates conditional outcome means and averages them over the target covariate distribution; that approach has its own modeling requirements. Truncation trades some protection from extreme weights for a potentially altered target, rather than making the original estimator uniformly better.

The formula above is for an arm-specific Hájek mean with a clipped unstabilized inverse-probability weight and known propensity. A stabilized weight with a covariate-dependent numerator, a longitudinal product weight, or a different estimating equation can have a different limiting functional. The relevant calculation is always to take the expectation of the actual clipped weighted estimating equation; the simple \(q_{a,c}\) formula should not be transferred mechanically to those settings.

Common Pitfall