Stabilized and Truncated Censoring Weights

Why Stabilize or Truncate Censoring Weights?

Inverse probability of censoring weights compensate for the fact that people who remain observed may differ from those who are censored. The denominator of a weight is the estimated probability of remaining uncensored, given the observed history that predicts censoring. If that probability is small, its reciprocal is large. A few such observations can then dominate a weighted estimate, making it sensitive to small changes in the data or in the censoring model.

Stabilization and truncation address different aspects of this problem. Stabilization replaces the numerator 1 in an ordinary inverse probability weight with a probability, usually based on a smaller set of variables. Truncation modifies weights above a prespecified limit or percentile. Neither operation repairs a misspecified censoring model or creates information for people with histories that have virtually no chance of remaining observed. The choices need to be described along with the weighted analysis they support.

Definitions and Assumptions

Suppose follow-up is divided into intervals $k=1,\ldots,K$. Let $U_{ik}=1$ if individual $i$, having remained uncensored up to interval $k$, remains uncensored through that interval, and let $U_{ik}=0$ if censoring occurs during it. Let $H_{ik}$ be the observed history available just before the censoring decision in interval $k$, and let $B_i$ be baseline information included in that history. Weights are used only for individuals still observed at the time of the outcome analysis.

Definition: For an individual uncensored through interval $t$, the unstabilized inverse probability of censoring weight is the product of the reciprocals of the conditional probabilities of remaining uncensored: $$ W_i(t)=\prod_{k=1}^{t}\frac{1}{p_{ik}}, \qquad p_{ik}=P(U_{ik}=1\mid U_{i1}=\cdots=U_{i,k-1}=1,H_{ik}). $$ A stabilized weight replaces each numerator 1 with a specified probability $q_{ik}$: $$ W_i^{S}(t)=\prod_{k=1}^{t}\frac{q_{ik}}{p_{ik}}, \qquad q_{ik}=P(U_{ik}=1\mid U_{i1}=\cdots=U_{i,k-1}=1,B_i). $$

The precise numerator model is a design choice, not a cosmetic detail. In this definition, baseline information $B_i$ is included in the history $H_{ik}$, and the numerator conditions on remaining uncensored through the previous interval. Both numerator and denominator probabilities refer to the same interval and the same prior uncensored risk set.

The identifying assumptions are those required for censoring weighting: the censoring model conditions on the relevant observed history, and, given that history, censoring is independent of the future outcome information being recovered. The probabilities must also be correctly specified or consistently estimated, and the denominator probabilities must be positive for histories represented in the target analysis. A stabilized weight does not make these assumptions unnecessary. The earlier material on independent censoring and constructing inverse probability of censoring weights gives the broader interpretation of these conditions.

In practice, probabilities are estimated from a censoring model. The products use the fitted denominator probabilities for the observed history and, for stabilization, fitted numerator probabilities under the chosen numerator model. If an individual is censored before time $t$, that person does not contribute an observed outcome at $t$; the weight is not a way to fill in that missing outcome directly.

Why the Weighting Identity Works

The central step is a conditional expectation over whether an individual remains uncensored in an interval. Because $B_i$ is included in $H_{ik}$, the numerator probability is determined by information in that history. Under the stated conditional censoring model, for a quantity $Y_i$ defined for the individuals at risk at interval $k$,

E\left[\left.\frac{U_{ik}q_{ik}}{p_{ik}}Y_i\right|H_{ik},\text{uncensored through }k-1\right] = q_{ik}E\left[Y_i\mid H_{ik},\text{uncensored through }k-1\right].

To see the step involving censoring, condition on the history and prior uncensored status. The conditional probability of $U_{ik}=1$ is $p_{ik}$, and conditional independent censoring allows the outcome quantity to be separated from that censoring decision. Thus

E[U_{ik}Y_i\mid H_{ik},\text{uncensored through }k-1] = p_{ik}E[Y_i\mid H_{ik},\text{uncensored through }k-1].

Multiplying by $q_{ik}/p_{ik}$ gives the first identity. For an unstabilized weight, the corresponding conditional weighted contribution recovers the full-history conditional outcome contribution, because $q_{ik}=1$. For a stabilized weight, the numerator factor remains: the weighted contribution follows the numerator censoring mechanism rather than removing censoring conditional on all of $H_{ik}$.

Across multiple intervals, the same argument is applied sequentially to the product of interval-specific factors. It does not justify conditioning on variables absent from the history or assuming that a numerator probability applies within a finer conditioning set than the one used to define it. In particular, if $B_i$ is not included in $H_{ik}$, one cannot automatically use this conditional identity given both $H_{ik}$ and $B_i$.

Worked Example: Calculating a Two-Interval Stabilized Weight

Consider a hypothetical individual who remains uncensored in two intervals. In interval 1, the fitted denominator probability is $p_{i1}=0.80$ and the numerator probability is $q_{i1}=0.90$. In interval 2, the corresponding probabilities are $p_{i2}=0.50$ and $q_{i2}=0.85$. Assume that each numerator uses baseline information included in the denominator history, and that these are probabilities for the appropriate prior-uncensored risk sets.

The stabilized weight through interval 2 is

W_i^S(2)=\frac{0.90}{0.80}\times\frac{0.85}{0.50} =1.125\times1.700 =1.9125.

The unstabilized weight is

W_i(2)=\frac{1}{0.80}\times\frac{1}{0.50} =1.250\times2.000 =2.500.

The stabilized weight is smaller here because both numerator probabilities are below 1. This example illustrates the calculation, not a guarantee that stabilization always reduces every weight or its variability. The assumptions needed to interpret either weight include valid probability models, appropriate conditioning on history, and positive denominator probabilities for the individual’s histories.

Truncating Weights: Rules and Consequences

A truncation rule must be specified before examining how it changes the desired result. A fixed upper cap $c$ replaces each weight $W_i$ by $\min(W_i,c)$. A percentile cap replaces weights above a specified sample percentile with the percentile threshold. For example, an 80th-percentile cap retains weights at or below the sample’s 80th percentile and sets larger weights to that threshold. The quantile convention, whether the cap is applied to stabilized or unstabilized weights, and whether it applies to the full sample or analysis risk set should be stated.

Capping reduces the largest individual contributions relative to the uncapped weighted analysis. It often reduces weight dispersion and can make an estimate less sensitive to a few observations. But it also stops those observations from receiving the full inverse-probability correction. Consequently, truncation generally introduces bias relative to the uncapped weighted estimating equation, and the direction of that change depends on the outcomes and histories of the capped observations. Lower weight dispersion does not guarantee a smaller sampling variance for every estimator or data-generating process.

The descriptive effective sample size is sometimes used to summarize weight concentration:

n_{\mathrm{eff}}=\frac{\left(\sum_i W_i\right)^2}{\sum_i W_i^2}.

This is a summary of the weights, not a substitute for the actual sample size or a complete variance calculation. Its value often increases after upper capping because the most dominant weights are reduced, but it does not establish that the resulting estimate is unbiased or that its inferential uncertainty is correctly quantified. Estimated weights also require appropriate variance treatment; that issue is distinct from selecting the truncation rule.

Worked Example: Comparing Two Prespecified Caps

In a hypothetical fixed-time analysis, five people remain observed. Their estimated weights are $1, 1.5, 2.5, 5,$ and $8$, and their observed binary outcomes are $0,0,1,1,$ and $1$, respectively. For this illustration, the weighted estimate is the weighted mean of the observed outcomes, $\sum_i W_iY_i/\sum_i W_i$. Compare no truncation, a fixed cap of 3, and an 80th-percentile cap. Define the sample 80th percentile by the nearest-rank rule, so it is the fourth ordered weight.

No truncation. The total weight is $1+1.5+2.5+5+8=18$, and the weighted outcome total is $2.5+5+8=15.5$. Therefore,

\hat{\mu}_{W}=\frac{15.5}{18}=0.8611.

The descriptive effective sample size is

n_{\mathrm{eff}}=\frac{18^2}{1^2+1.5^2+2.5^2+5^2+8^2} =\frac{324}{98.5}=3.289.

Fixed cap of 3. The capped weights are $1,1.5,2.5,3,3$, with total $11$. The weighted outcome total is $2.5+3+3=8.5$, so

\hat{\mu}_{3}=\frac{8.5}{11}=0.7727, \qquad n_{\mathrm{eff}}=\frac{11^2}{1+2.25+6.25+9+9} =\frac{121}{27.5}=4.400.

80th-percentile cap. Under the stated nearest-rank convention, the fourth ordered weight is 5. Capping the weight 8 at 5 gives weights $1,1.5,2.5,5,5$, with total $15$ and weighted outcome total $12.5$. Thus

\hat{\mu}_{80}=\frac{12.5}{15}=0.8333, \qquad n_{\mathrm{eff}}=\frac{15^2}{1+2.25+6.25+25+25} =\frac{225}{59.5}=3.782.

Both rules lower the weighted estimate relative to the untruncated value in this data set, because the largest weights belong to people with outcome 1. The fixed cap changes it more than the percentile cap. The unweighted mean is $3/5=0.6000$, but that is not a valid substitute for the weighted analysis merely because it is smaller. These calculations describe how the prespecified rules alter this particular weighted estimate; they do not show which rule has the best bias-variance trade-off in another setting.

Numerator Choice and the Target Analysis

Stabilization affects more than the numerical size of individual weights. If the numerator probability is common across individuals at a given time, its factor cancels from a normalized weighted mean at that time. If the numerator depends on baseline covariates, however, the weighted distribution can retain a baseline-dependent factor. A pooled weighted estimate that ignores this distinction may therefore target a different baseline mix from the original population.

Worked Example: A Baseline-Dependent Numerator

Suppose a hypothetical population has equal numbers in baseline groups A and B. For a single interval, use a stabilized numerator probability of $0.90$ in group A and $0.60$ in group B. Assume the denominator censoring model is correctly specified and the outcome is constant within each group: 0.20 in group A and 0.80 in group B. In expectation, the stabilized weighted outcome contributions are proportional to the group’s population share multiplied by its numerator probability and its outcome mean.

For group A, the contribution is proportional to $0.50\times0.90\times0.20=0.090$. For group B, it is proportional to $0.50\times0.60\times0.80=0.240$. The pooled normalized stabilized estimate is therefore

\frac{0.090+0.240}{0.50(0.90)+0.50(0.60)} =\frac{0.330}{0.750} =0.440.

The original population mean is $0.50(0.20)+0.50(0.80)=0.500$, not 0.440. The difference arises because the baseline-dependent numerator leaves different expected weighted mass in the two groups. A common numerator factor would cancel from the pooled normalized mean in this one-interval illustration. With a baseline-dependent numerator, the analysis needs to account for the baseline variables in a way consistent with its target, such as estimating within baseline groups and standardizing to the desired baseline distribution. Truncation can further change the contribution pattern, so the target and analysis must be considered together.

Practical Checks and Limitations

Before interpreting a weighted analysis, verify that the numerator and denominator refer to the same intervals and risk sets, that the numerator variables are contained in the denominator history, and that the selected truncation rule is stated precisely. Recalculate a few weights by hand from their interval-specific probabilities. A mismatch between cumulative products and reported weights can be a coding error rather than a statistical feature.

Large weights, a marked change in an estimate after capping, or a substantial difference between fixed and percentile caps are reasons to investigate the model and the target analysis. They do not, by themselves, identify the cause. A low fitted denominator probability may reflect a sparse history, a model that extrapolates poorly, or genuinely limited observation probability. A cap can conceal the numerical symptom without resolving the underlying lack of support. Positivity diagnostics and detailed assessment of weight balance are separate checks.

There is no universally optimal cap. Report the unstabilized or stabilized specification, numerator conditioning, truncation rule and percentile convention, and the estimate under the prespecified primary rule. Sensitivity comparisons are useful when they are clearly labeled as comparisons, not as a search for whichever cap gives the preferred answer. All conclusions remain conditional on the censoring assumptions and on how the weighted analysis defines its target.

Common Mistakes

Using a numerator with variables absent from the denominator history. The conditional identity for stabilization requires compatible conditioning; include the numerator variables in the denominator history and state the censoring assumptions.
Assuming stabilization always reduces every weight. A numerator probability is not necessarily smaller in every interval or for every individual; calculate the product rather than infer its size from the label.
Treating truncation as a harmless numerical fix. Capping changes the estimating contributions and can change the target or introduce bias; report and interpret the rule.
Calling effective sample size the number of independent observations. It is a descriptive weight-concentration measure, not the actual sample size or a complete variance estimate.
Choosing a cap after seeing which estimate is preferred. Prespecify the rule, and present alternative caps as sensitivity comparisons rather than as a replacement for the primary analysis.

Key takeaway: Stabilized weights replace the numerator 1 with a specified censoring probability; truncation caps the resulting weights. Both can affect variability, but truncation sacrifices some inverse-probability correction, and a baseline-dependent numerator or cap can alter the weighted analysis target.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

For one interval, $p=0.75$ and $q=0.90$. Calculate the unstabilized and stabilized weights for an individual who remains uncensored.
Why must the baseline variables used in a stabilized numerator be included in the denominator history for the stated conditional weighting identity?
Weights are $1,2,4$, and outcomes are $0,1,1$. Calculate the weighted mean with no truncation and with a fixed cap of 2.
Does a larger descriptive effective sample size after capping prove that the capped analysis has lower variance and no bias? Explain.
If the stabilized numerator depends on baseline group, what should be considered before interpreting a pooled weighted estimate?

Solutions. (1) The unstabilized weight is $1/0.75=1.333$; the stabilized weight is $0.90/0.75=1.200$. (2) The numerator factor must be determined by the history on which the conditional expectation is taken; otherwise its value or the censoring probability may differ after conditioning on the numerator variables. (3) Without truncation, the mean is $(2\times1+4\times1)/(1+2+4)=6/7=0.8571$. With cap 2, the weights are $1,2,2$, giving $(2+2)/(1+2+2)=0.8000$. (4) No. Effective sample size summarizes weight concentration; it neither establishes lower estimator variance in every setting nor removes the bias that capping can introduce. (5) Check whether the numerator leaves a baseline-dependent weighted distribution and whether the outcome analysis or standardization matches the intended target population.