Stabilization Followed by Truncation

Why the Scale of a Weight Matters

A truncation rule such as “replace every weight above 3 by 3” sounds unambiguous, but it is incomplete unless the weight scale is specified. An unstabilized inverse-probability weight and its stabilized counterpart are generally not the same number: stabilization multiplies the former by a numerator probability. Applying the same numerical cutoff to both can therefore flag different observations.

This distinction is separate from whether truncation is useful. The issue here is how the numerator distribution changes the observations affected by a given rule. To make the comparison precise, begin with a single binary treatment \(A\), covariates \(L\), propensity \(e(L)=P(A=1\mid L)\), and marginal treatment probability \(q=P(A=1)\), where \(0<e(L)<1\) and \(0<q<1\).

Definition: The unstabilized inverse-probability weight is \(W^U=A/e(L)+(1-A)/(1-e(L))\). The stabilized weight is \(W^S=Aq/e(L)+(1-A)(1-q)/(1-e(L))\). Thus, for a treated observation, \(W^S=qW^U\); for an untreated observation, \(W^S=(1-q)W^U\). The factors \(q\) and \(1-q\) are the numerator probabilities for the observed treatment group.

These formulas presume consistency, conditional exchangeability given \(L\), and positivity for the causal interpretation of inverse-probability weighting; in particular, treatment probabilities must be bounded away from zero and one in the population of interest for stable finite weights. The algebraic comparison below only needs the displayed weight definitions and positive numerator factors. It does not by itself establish identification or justify a particular truncation choice.

The Effective Cutoff Depends on the Numerator

Write \(C_u(w)=\min(w,u)\) for upper clipping at \(u>0\). For a treated observation, \(C_u(W^S)\) clips exactly when \(qW^U>u\), or equivalently \(W^U>u/q\). For an untreated observation, the corresponding raw-scale threshold is \(u/(1-q)\). Thus, using the same numerical upper cutoff on both scales does not apply the same rule to the raw weights.

Theorem: For \(c>0\) and \(0\leq a\leq b\), define two-sided clipping by \(C_{[a,b]}(w)=\min\{b,\max(a,w)\}\). Then \(C_{[a,b]}(cw)=cC_{[a/c,b/c]}(w)\). Consequently, clipping a stabilized weight to \([a,b]\) is equivalent to clipping its corresponding unstabilized weight to \([a/c,b/c]\) and then multiplying by \(c\), where \(c=q\) for treated observations and \(c=1-q\) for untreated observations.

Proof. If \(w<a/c\), then \(cw<a\), so the left side is \(a\), while the right side is \(c(a/c)=a\). If \(a/c\leq w\leq b/c\), then \(a\leq cw\leq b\), and both sides equal \(cw\). If \(w>b/c\), then \(cw>b\), so both sides equal \(b\). These cases cover all \(w\geq0\), proving the identity. \(\square\)

For upper clipping alone, the same argument gives \(C_u(cw)=cC_{u/c}(w)\). The translated cutoff is group-specific: \(u/q\) for treated observations and \(u/(1-q)\) for untreated observations. If \(q<1/2\), treated stabilized weights have the smaller multiplier, so their equivalent raw-scale cutoff \(u/q\) is larger than \(u/(1-q)\). This does not mean that treated observations are always clipped less: their raw weights may be larger. The actual clipped set depends on both the numerator and the observed denominator.

Worked Comparisons

Worked Example: A Marginal Treatment Probability of 0.4

Suppose four equally prevalent covariate strata have treatment probabilities \(e(L)=0.1,0.3,0.6,0.6\). Their average is \((0.1+0.3+0.6+0.6)/4=0.4\), so the marginal probability is \(q=0.4\). Compare upper clipping at \(u=3\).

For treated observations, the raw and stabilized weights at those propensities are, respectively, \(1/e(L)\) and \(0.4/e(L)\). At \(e(L)=0.1\), these are \(10\) and \(4\); the raw weight clips to \(3\), and the stabilized weight also clips to \(3\). At \(e(L)=0.3\), the weights are \(3.333\ldots\) and \(1.333\ldots\); the raw weight clips to \(3\), while the stabilized weight remains \(1.333\ldots\). At either \(e(L)=0.6\), the weights are \(1.667\ldots\) and \(0.667\ldots\), neither of which clips.

For untreated observations, the weight is \(1/(1-e(L))\), or \(0.6/(1-e(L))\) after stabilization. At \(e(L)=0.1\), these are \(1.111\ldots\) and \(0.667\); at \(e(L)=0.3\), they are \(1.429\ldots\) and \(0.857\); and at \(e(L)=0.6\), they are \(2.5\) and \(1.5\). None exceeds 3. In particular, the untreated row with \(e(L)=0.6\) has raw weight \(2.5\) and stabilized weight \(1.5\).

The raw-scale rule clips treated observations at \(e(L)=0.1\) and \(e(L)=0.3\). The stabilized-scale rule clips only the treated observations at \(e(L)=0.1\). This agrees with the translated raw thresholds: \(3/0.4=7.5\) for treated observations and \(3/0.6=5\) for untreated observations. The untreated raw weights in this example do not exceed 5.

Worked Example: A Marginal Treatment Probability of 0.8

Now take two equally prevalent strata with \(e(L)=0.7\) and \(0.9\). Their average is \(0.8\), so \(q=0.8\). With upper clipping at \(u=2.5\), a treated observation at \(e(L)=0.7\) has raw weight \(1/0.7=1.429\) and stabilized weight \(0.8/0.7=1.143\); neither clips. At \(e(L)=0.9\), the treated weights are \(1.111\) and \(0.889\), also neither clipped.

An untreated observation at \(e(L)=0.7\) has raw weight \(1/0.3=3.333\) and stabilized weight \(0.2/0.3=0.667\). The raw weight clips to 2.5, but the stabilized weight does not. At \(e(L)=0.9\), its raw weight is \(1/0.1=10\), which clips to 2.5; its stabilized weight is \(0.2/0.1=2\), which remains below 2.5.

Here raw clipping affects both untreated rows, while stabilized clipping affects neither. The reason is not that the denominators changed: the numerator for untreated observations is \(1-q=0.2\). The stabilized upper cutoff corresponds to a raw cutoff of \(2.5/0.2=12.5\) in that group.

Worked Example: Translating Two-Sided Bounds

Return to \(q=0.4\), and consider a treated observation with \(e(L)=0.05\). Its raw weight is \(W^U=1/0.05=20\), and its stabilized weight is \(W^S=0.4(20)=8\). Apply the same two-sided bounds \([0.5,5]\) to each scale. The raw weight clips to 5, and the stabilized weight also clips to 5.

Although the displayed clipped values happen to match, the rules do not clip the same range of underlying raw weights. By the scale identity, stabilized clipping to \([0.5,5]\) corresponds to raw clipping to \([0.5/0.4,5/0.4]=[1.25,12.5]\), followed by multiplication by \(0.4\). For example, if a treated observation has \(e(L)=0.2\), then \(W^U=5\) and \(W^S=2\). Raw clipping leaves 5 unchanged, while stabilized clipping leaves 2 unchanged; these outcomes are consistent with the translated interval. But a treated raw weight of 1.1 would be raised to 1.25 under the translated lower bound, giving a stabilized clipped value of 0.5, whereas applying \([0.5,5]\) directly to the raw weight leaves it at 1.1.

From One Treatment to Longitudinal Weights

For a treatment history \(\bar A=(A_0,\ldots,A_K)\), an unstabilized cumulative weight is a product of inverse treatment probabilities conditional on measured covariate and treatment history. A stabilized cumulative weight additionally multiplies by the corresponding numerator probabilities conditional on treatment history. For a given individual and observed history, this can be written \(W^S=W^U N\), where \(N\) is the product of the numerator probabilities along that history.

Theorem: For any fixed individual with positive numerator product \(N\), upper clipping the stabilized cumulative weight at \(u\) affects that individual exactly when \(W^U>u/N\). Two-sided stabilized clipping at \([a,b]\) is equivalent to raw-scale clipping at \([a/N,b/N]\), followed by multiplication by \(N\).

Proof. Since \(W^S=NW^U\) and \(N>0\), the condition \(W^S>u\) is equivalent, after division by \(N\), to \(W^U>u/N\). The two-sided statement follows from the scale identity proved above with \(c=N\). \(\square\)

Unlike the single-time binary-treatment case, \(N\) can differ across treatment histories and individuals. Consequently, one common stabilized-scale cutoff corresponds to different raw-scale cutoffs across histories. The numerator probabilities are part of the definition of the stabilized weight, not a universal constant that can be removed from a pooled truncation rule.

Diagnostics, Alternatives, and Limitations

A practical comparison should record the scale on which each rule is applied and identify the observations changed by it. Useful checks include the number and proportion clipped overall and by treatment group or history, the range of the weights before and after clipping, and whether the clipping set agrees with the translated thresholds. For estimated weights, calculate these quantities using the actual fitted numerator and denominator probabilities. A discrepancy between a reported clipping set and the condition \(W^U>u/q\) (or its relevant group-specific version) signals a scale or implementation mismatch.

The main alternatives are to truncate the unstabilized weights, truncate the stabilized weights, or choose scale-specific bounds that translate into the same effective raw-scale rule. The theorem shows how to make the last comparison exactly. Simply using numerically identical bounds is not a scale-invariant alternative: it generally changes which observations are clipped. Quantile-based rules also need their scale stated. Within a treatment group, multiplying every weight by the same positive factor preserves that group’s ranking; in a pooled sample, different treatment-group factors can change rankings between groups and hence can change pooled quantiles.

Clipping changes weights, so it generally changes the weighted estimating equation and can introduce bias relative to the analysis using untruncated weights. Stabilization alone and clipping are distinct operations: without clipping, the numerator rescales weights by treatment group or history, while clipping can alter individual weights unequally within those groups. The weight-scale comparison does not determine the causal estimand after clipping, nor does it identify an optimal threshold. It clarifies which observations the chosen rule modifies; the consequences for balance, bias, and variance must still be assessed in the analysis at hand.

Common pitfall: “Both rules cap weights at 3, so they clip the same extreme observations.” This is false when one rule caps raw weights and the other caps stabilized weights. For treated observations, a stabilized cutoff of 3 corresponds to raw cutoff \(3/q\), not 3; for untreated observations it corresponds to \(3/(1-q)\). State the scale, translate the cutoff when comparing rules, and verify the clipped observations rather than assuming the numerical bounds are comparable.

Check Your Understanding

Answer each question, then check the solutions below.

If \(q=0.25\), what raw-scale upper cutoff corresponds to stabilized-scale clipping at 2 for treated observations?
For \(q=0.7\), an untreated observation has raw weight 4. What is its stabilized weight, and would it be clipped at 2 under stabilized-scale upper clipping?
For a treated observation with \(q=0.4\), which raw weights are affected by stabilized two-sided bounds \([1,3]\)?
In a longitudinal analysis, why might the same stabilized-scale cutoff correspond to different raw-scale cutoffs for different individuals?

Solutions. (1) Divide by \(q\): \(2/0.25=8\). The treated raw weight is clipped under the corresponding rule exactly when it exceeds 8. (2) The untreated multiplier is \(1-q=0.3\), so the stabilized weight is \(0.3(4)=1.2\), which is not above 2 and is not upper-clipped. (3) The equivalent raw interval is \([1/0.4,3/0.4]=[2.5,7.5]\); raw weights below 2.5 or above 7.5 are changed by the corresponding stabilized two-sided rule. (4) The numerator product \(N\) can vary with treatment history, so the equivalent raw cutoff \(u/N\) varies as well.