Truncation Choices in Dialysis Dose Comparisons

Why Compare Truncation Choices?

In a dialysis dose comparison, a few patients may receive very large inverse-probability weights because their observed dose was unlikely given their measured characteristics. Those patients can have substantial influence on an estimated marginal transfusion risk. Capping the largest weights limits that influence, but it also changes how patients contribute to the weighted population. A capped estimate is therefore a sensitivity analysis, not an automatic correction.

Let $A=1$ denote a higher dialysis dose and $A=0$ a standard dose, and let $Y$ indicate transfusion during a specified follow-up period. Let $L$ denote measured pretreatment covariates. Under consistency, conditional exchangeability given $L$, and positivity, inverse-probability weighting can identify marginal risks under each dose strategy. For example, conditional exchangeability means $Y^a$ is independent of $A$ given $L$, for $a\in\{0,1\}$; positivity requires $P(A=a\mid L=l)>0$ for covariate values in the target population.

The treatment-specific inverse-probability weight is $1/P(A=a\mid L)$ for a patient whose observed treatment is $a$. In a sample, the weighted risk in arm $a$ is the weighted average of that arm’s outcomes. Here, truncation means replacing weights above a chosen upper cap $c$ by $c$, while retaining every patient. This is also called upper winsorization. It is different from trimming, which excludes patients or covariate regions.

Definition: For observed treatment $A_i=a$, let $w_i=1/\widehat P(A_i=a\mid L_i)$. Under an upper cap $c$, use $w_i^{(c)}=\min(w_i,c)$. The capped weighted risk in arm $a$ is $$ \widehat R_a(c)= \frac{\sum_{i:A_i=a} w_i^{(c)}Y_i} {\sum_{i:A_i=a} w_i^{(c)}}. $$ The corresponding risk difference, higher dose minus standard dose, is $\widehat R_1(c)-\widehat R_0(c)$.

A cap must be described precisely: it may be a fixed numerical value or a prespecified sample percentile. The same rule should be stated for both treatment arms, including whether a percentile is calculated separately by arm or across all weights. Different rules can produce different estimates. The examples below use fixed caps applied to the weights in each arm.

Worked Dialysis Comparisons

Worked Example: An Influential Patient With Transfusion

Consider an illustrative sample of five higher-dose patients. Their weights are $2,2,2,2,8$, and their transfusion outcomes are $0,0,0,0,1$. In the standard-dose arm, suppose five patients each have weight $2$, with outcomes $1,0,0,0,0$. Compare no cap with caps $4$ and $2$.

With no cap, the higher-dose weighted risk is $8/(2+2+2+2+8)=8/16=1/2$. The standard-dose risk is $2/10=1/5$. Thus the risk difference is $1/2-1/5=3/10=0.3000$.

With cap $4$, the higher-dose patient with weight $8$ now has weight $4$. The higher-dose risk is $4/(2+2+2+2+4)=4/12=1/3$. The standard-dose risk remains $1/5$, so the difference is $1/3-1/5=2/15\approx0.1333$.

With cap $2$, all five higher-dose weights equal $2$, so that risk is $2/10=1/5$. The standard-dose risk is also $1/5$, giving a difference of zero. In this example, the cap reduces the estimated contrast because the most heavily weighted patient had a transfusion.

Worked Example: The Same Weight Tail, a Different Outcome

Keep the same weights in both arms, but let the higher-dose outcomes be $1,1,1,1,0$: the patient with weight $8$ did not have a transfusion, while each of the four patients with weight $2$ did. The standard-dose outcomes still give risk $1/5$.

Without a cap, the higher-dose risk is $(2+2+2+2)/16=8/16=1/2$, and the risk difference is $1/2-1/5=3/10=0.3000$.

At cap $4$, the numerator stays $8$, while the denominator becomes $12$. The risk is $8/12=2/3$, and the difference is $2/3-1/5=7/15\approx0.4667$.

At cap $2$, the numerator is still $8$ and the denominator is $10$. The risk is $4/5$, and the difference is $4/5-1/5=3/5=0.6000$. Capping increased the estimated contrast here. It does not necessarily move an estimate toward zero or toward any preferred clinical conclusion.

Why a Cap Changes the Estimate

The direction of a one-patient change can be derived exactly. Consider one arm with positive weights summing to $S$, weighted outcome total $T$, and weighted risk $R=T/S$. Suppose one patient has weight $W>c$ and binary outcome $Y_j$. Capping that weight removes $d=W-c$ from the denominator and $dY_j$ from the numerator. The new risk is

R_c=\frac{T-dY_j}{S-d}, \qquad R_c-R =\frac{d(R-Y_j)}{S-d}.

To verify the second equality, substitute $T=SR$ and subtract:

R_c-R =\frac{T-dY_j}{S-d}-\frac{T}{S} =\frac{S(T-dY_j)-T(S-d)}{S(S-d)} =\frac{d(T-SY_j)}{S(S-d)} =\frac{d(R-Y_j)}{S-d}.

Since $S-d>0$, capping lowers the risk if $Y_j=1$ and $R<1$, and raises it if $Y_j=0$ and $R>0$. This is a statement about one arm’s risk, not necessarily the direction of the risk-difference change: the cap may also affect the other arm.

Result: Capping a weight changes both the weighted outcome total and the weight total. Its effect on an arm’s risk depends on the patient’s outcome relative to that arm’s current weighted risk. Therefore, a large weight alone does not tell you whether a cap will raise or lower the estimated risk.

What Population Does the Capped Risk Represent?

The uncapped inverse-probability risk has a direct standardization interpretation under the assumptions above. Capping generally changes that interpretation. To see why, write $e_a(L)=P(A=a\mid L)$, and let $m_a(L)=E[Y\mid A=a,L]$. For a binary outcome, $m_a(L)$ is the conditional transfusion risk. In the population, the capped arm-specific numerator has expectation

E\!\left[I(A=a)\min\{1/e_a(L),c\}Y\right] =E\!\left[\min\{1,c e_a(L)\}m_a(L)\right].

This follows by conditioning on $L$: $E[I(A=a)Y\mid L]=e_a(L)m_a(L)$, so multiplying by $\min\{1/e_a(L),c\}$ gives $\min\{1,c e_a(L)\}m_a(L)$. Applying the same step without $Y$ gives denominator expectation $E[\min\{1,c e_a(L)\}]$. Thus the capped risk converges, under suitable laws of large numbers and fixed $c$, to

R_a(c)= \frac{E[\min\{1,c e_a(L)\}m_a(L)]} {E[\min\{1,c e_a(L)\}]}.

With no cap, the factor is $1$, so this expression is $E[m_a(L)]=E[Y^a]$, the marginal risk in the target population. With a finite cap, covariate profiles with smaller $e_a(L)$ receive a smaller factor when $c e_a(L)<1$. Unless that factor is constant across $L$, or the conditional risk happens to make the reweighting immaterial, the capped risk is not generally the same marginal causal risk. Capping can reduce instability while introducing bias relative to the original target.

Important qualification: This population expression assumes true treatment probabilities and an upper cap on the unstabilized inverse-probability weights. Estimated weights, data-dependent percentile caps, and stabilized weights require their own large-sample analysis. The formula explains the mechanism; it does not guarantee that any particular capped estimate is unbiased.

Diagnostics and Practical Comparisons

A useful comparison reports the uncapped estimate and several caps chosen before inspecting which one gives the most appealing effect. For every choice, show the two arm-specific risks and the risk difference, not just the contrast. Also report the weight distribution by arm, including upper quantiles and the maximum, and identify whether a small number of patients accounts for much of the total weight. These summaries help reveal whether the result depends on a handful of records.

The effective sample size (ESS) is another measure of weight concentration. For weights $w_1,\ldots,w_n$ in one arm, define

\operatorname{ESS}=\frac{(\sum_i w_i)^2}{\sum_i w_i^2}.

It equals $n$ when all weights are equal and is smaller when weights are concentrated. It is a diagnostic summary, not the actual number of independent patients and not proof of adequate confounding control.

Worked Example: Weight Concentration Before and After Capping

For the higher-dose weights $2,2,2,2,8$, the total is $16$, and the sum of squared weights is $4+4+4+4+64=80$. Therefore, the ESS is $16^2/80=256/80=3.2$.

At cap $4$, the weights are $2,2,2,2,4$. Their total is $12$, and their squared weights sum to $4+4+4+4+16=32$. The ESS is $12^2/32=144/32=4.5$.

At cap $2$, all weights are $2$, so the ESS is $10^2/(4+4+4+4+4)=100/20=5$. The ESS rises as the influential weight is reduced in this example. That indicates less concentration, not that the cap has restored information about unsupported treatment choices.

Compare upper capping with the main alternatives. Keeping all weights preserves the inverse-probability weighting target under the stated assumptions but may yield high variance in a finite sample. Trimming excludes patients or covariate regions and therefore changes the analyzed population more explicitly; it can also discard clinically important patients. Modeling the outcome or using a doubly robust estimator offers a different strategy for handling instability, but each approach has its own assumptions and diagnostics. Overlap weights target a different population rather than simply capping the original weights.

Most importantly, a cap cannot repair structural nonpositivity. If one dose is never observed for a clinically relevant covariate profile, changing large observed weights does not supply the missing outcome information. Check whether extreme weights correspond to sparse treatment support, and report the population and assumptions to which the comparison can reasonably apply. A dramatic estimate change across modest caps is a warning that the conclusion is sensitive to a few records or weak support, not a reason to select the most favorable threshold.

Common Pitfall

Wrong reasoning: “The uncapped estimate is unstable, so choosing a cap makes the estimate reliable and keeps the original causal interpretation.” Capping does limit the maximum contribution of a patient, but the derivation above shows that it can change the covariate distribution represented by the weighted risk. It may reduce variance while adding bias, and it cannot resolve absent treatment support. The correction is to prespecify a small set of defensible thresholds, compare arm-specific risks and contrasts, inspect weight concentration and support, and describe the results as sensitivity to the truncation rule.

Key takeaway: Weight truncation is a transparent sensitivity analysis of influence. Report what was capped, how the threshold was chosen, how each arm’s risk changed, and whether the treatment support can sustain the intended comparison.

Check Your Understanding

Answer each question, then check the brief solutions.

In an arm with weighted risk $R=0.30$, what happens to the risk when one patient with outcome $Y=1$ has their weight capped?
Why can a cap increase a weighted transfusion risk even though every capped weight is no larger than before?
What does a rising ESS after capping establish, and what does it not establish?
Does upper capping exclude patients from the analysis? How does that differ from trimming?
Why does capping not solve structural nonpositivity?

Solutions. (1) The risk decreases, because $R_c-R=d(R-1)/(S-d)<0$. (2) If the capped patient had outcome $0$ and the current risk is positive, the denominator falls without a corresponding reduction in the weighted outcome total, so the risk rises. (3) It indicates less weight concentration; it does not prove adequate confounding control, independent information, or treatment support. (4) Capping retains the patient with a smaller weight; trimming removes selected patients or regions. (5) A cap changes contributions of observed patients but cannot provide outcomes for treatment choices that were never observed in a relevant covariate profile.