Common Errors in Censoring-Weight Construction

Why Censoring-Weight Construction Is Easy to Get Wrong

An inverse-probability-of-censoring weight represents people who remain observed by people with similar histories who have dropped out. That interpretation depends on more than taking a reciprocal. At each time, the probability must describe the correct risk set, condition on the history relevant to dropout, and enter the weight in the correct direction. The weight must also stop where the outcome stops being observed.

These errors can produce plausible-looking numbers while targeting the wrong weighted population. The focus here is construction: how to define the probabilities, what assumptions make them usable, and how to recognize the consequences of four frequent mistakes.

Define the Risk Set and the Weight

Consider discrete follow-up times $t=1,\ldots,K$. Let $C_t=1$ indicate that a person remains uncensored during interval $t$, conditional on having been observed through the start of that interval. Let $R_0=1$, and define $R_t=R_{t-1}C_t$. Thus $R_t=1$ means the person is still observed through time $t$. Censoring is monotone in this setup: once $R_t=0$, all later $R_s$, for $s>t$, are zero.

Write $H_t$ for the measured history available just before the censoring decision in interval $t$. It may include baseline variables, prior treatment, and prior outcomes. The conditional retention probability is $g_t(H_t)=P(C_t=1\mid R_{t-1}=1,H_t)$. It is defined among people who are still observed and therefore at risk of censoring in that interval. The cumulative probability of remaining observed through time $t$, along an observed history, is $G_t=\prod_{k=1}^{t}g_k(H_k)$.

Definition: The unstabilized inverse probability of censoring weight through time $t$ is $W_t=1/G_t$, for a person with $R_t=1$. A stabilized weight replaces each factor $g_k(H_k)$ by a numerator probability $q_k$, such as a retention probability conditional on a specified subset of baseline variables: $SW_t=\prod_{k=1}^{t}q_k/g_k(H_k)$. The numerator and denominator must refer to the same retention event and time interval.

These definitions do not, by themselves, guarantee unbiased estimation. For an outcome $Y_t$ that would be observed under continued follow-up, the usual justification requires consistency, sequential conditional independent censoring given $H_t$, and positivity. In particular, each relevant history among those at risk must have a positive chance of remaining observed: $g_t(H_t)>0$. The censoring model must also correctly estimate the conditional probabilities used in the weights. Positivity is a property of the data-generating process, not something created by fitting a model.

Derivation: Why the Risk Set Matters

The denominator at time $t$ is not the fraction of the original cohort retained at time $t$. It is the probability of remaining observed in interval $t$, conditional on having reached that interval and on the history used by the model. The following derivation shows the role of that conditioning.

Result: If $g_t(H_t)$ is the true conditional retention probability among people with $R_{t-1}=1$, then, at a fixed history $H_t=h$, the expected weighted retention indicator in interval $t$ is one: $$ E\left[\left.\frac{C_t}{g_t(H_t)}\right|R_{t-1}=1,H_t=h\right] =\frac{P(C_t=1\mid R_{t-1}=1,H_t=h)}{g_t(h)} =1. $$ This identity requires $g_t(h)>0$. It explains why the probability and the indicator in the weight must refer to the same at-risk decision.

If instead the denominator is an overall retention proportion that includes people who were already censored before interval $t$, its numerator is no longer the conditional probability in this identity. The weight then cannot restore the representation of the interval-$t$ risk set by that calculation.

Worked Example: Using the Original Cohort as the Risk Set

Suppose 100 participants enter follow-up. By the start of interval 2, 20 have already dropped out, so 80 are at risk. Of those 80, 60 remain observed through interval 2.

The correct conditional retention probability for interval 2 is $60/80=0.75$. The corresponding unstabilized factor is $1/0.75=4/3$. Among the 60 retained participants, this factor gives a weighted count of $60(4/3)=80$, matching the interval-2 risk set.

A common error is to divide the 60 retained participants by the original 100, obtaining $0.60$, then use $1/0.60=5/3$. That factor gives a weighted count of $60(5/3)=100$, not 80. It has mixed dropout before interval 2 with the interval-2 retention decision. If the target calculation concerns participants eligible for that interval, the extra 20 people do not belong in its risk set.

Derivation: Why Relevant Prior History Belongs in the Model

Conditional independent censoring is a statement about histories, not just an overall dropout rate. If prior outcomes predict both dropout and the outcome of interest, a pooled retention probability can average together groups with quite different censoring probabilities. A single pooled factor then fails to restore each group’s representation.

Result: Suppose $H_t$ takes values $h$ and that, in the risk set, $g_t(h)=P(C_t=1\mid R_{t-1}=1,H_t=h)>0$. Then $$ E\left[\left.\frac{C_t}{g_t(H_t)}\,I(H_t=h)\right|R_{t-1}=1\right] =P(H_t=h\mid R_{t-1}=1). $$ Indeed, condition first on $H_t$. For $H_t=h$, the conditional expectation of $C_t/g_t(H_t)$ is $g_t(h)/g_t(h)=1$; for other histories, the indicator is zero. Averaging over $H_t$ gives the stated risk-set proportion. If the weight uses a pooled probability instead, this cancellation generally fails within histories. $\square$

Worked Example: Omitting a Prior Outcome From the Censoring Model

Among 100 people at risk, let $H=1$ indicate a high prior outcome and $H=0$ a low prior outcome. There are 40 people with $H=1$, of whom 20 remain observed, and 60 with $H=0$, of whom 54 remain observed.

The history-specific retention probabilities are $20/40=0.50$ and $54/60=0.90$. Correct inverse factors are $2$ and $1/0.90=10/9$. Weighting the retained people separately gives $20(2)=40$ in the high-history group and $54(10/9)=60$ in the low-history group. The weighted group counts recover the original risk-set counts.

If the model omits $H$, the pooled retention probability is $(20+54)/100=0.74$, and every retained person receives the factor $1/0.74=50/37$, approximately $1.351$. The resulting weighted counts are $20(50/37)\approx27.03$ and $54(50/37)\approx72.97$, not 40 and 60. A pooled factor recovers the overall total $74(50/37)=100$, but does not recover the history composition. If $H$ predicts the outcome, that mismatch can bias the outcome analysis. A prior outcome that predicts dropout should therefore not be omitted merely because the pooled retention rate appears reasonable.

Check the Direction of a Stabilized Ratio

In a stabilized factor, the numerator describes the retention probability under a simpler conditioning set, while the denominator describes retention conditional on the richer history used to control informative dropout. The ratio is numerator divided by denominator, $q_t/g_t(H_t)$, not its reciprocal. The unstabilized factor is the special case with numerator one.

Worked Example: Reversing the Stabilized Ratio

Suppose the numerator retention probability for a particular interval is $q_t=0.90$, and the denominator conditional on a person’s history is $g_t(H_t)=0.60$. The correct stabilized factor is $0.90/0.60=1.50$. For a group with this history, 60% remain observed, so the expected weighted retention mass is $0.60(1.50)=0.90$, matching the numerator probability.

Reversing the ratio gives $0.60/0.90=2/3$. The resulting weighted retention mass is $0.60(2/3)=0.40$, which is not $0.90$. The direction error therefore changes the intended weighting rather than simply changing the scale. As a check, setting $q_t=1$ in the correct direction must give the unstabilized factor $1/g_t(H_t)=1.667$, not $g_t(H_t)$.

Stabilization is not a license to use an arbitrary numerator. Its conditioning set determines which measured features the numerator retains in the weighted data. In particular, a stabilized weight should be paired with an estimating procedure and target that are appropriate for that numerator; it should not automatically be treated as interchangeable with an unstabilized weight in every raw weighted mean.

Stop at the Last Observed Outcome

The product defining $W_t$ or $SW_t$ is indexed through the time of the outcome being analyzed. It is meaningful for a person only through intervals in which that person remains observed. A record that is censored before time $t$ does not contain the observed outcome $Y_t$, so it cannot contribute $W_tY_t$ at that time. Multiplying additional factors after dropout does not reconstruct that missing outcome.

Worked Example: A Weight Cannot Supply an Unobserved Outcome

Suppose a participant has retention probabilities $g_1=0.80$ and $g_2=0.75$, remains observed through interval 2, and then drops out before interval 3. Through interval 2, the cumulative retention probability is $G_2=0.80(0.75)=0.60$, and the unstabilized weight is $1/0.60=5/3$. This is the appropriate weight for an outcome observed at time 2.

At time 3, the participant has $R_3=0$, and $Y_3$ is unobserved. The participant therefore contributes no observed time-3 outcome term. Creating a time-3 record with a missing outcome and multiplying the earlier weight by another inverse retention factor does not change that fact; it merely creates a number attached to an unavailable outcome. An analysis must use observed outcomes at each time and apply weights only to the corresponding observed records.

Construction Checks, Alternatives, and Limits

Before using censoring weights, verify the construction against the data’s time ordering and observation rules. These checks do not establish that the censoring model is correct, but they can uncover errors in how it was built.

Risk-set check: At each time, confirm that the denominator model is fit among people with $R_{t-1}=1$, and that its outcome is the retention decision $C_t$ for that interval.
History check: Confirm that variables used to account for informative dropout were measured before the censoring decision. Including future information reverses the intended time order; omitting a measured predictor of both dropout and the outcome can leave informative censoring unaddressed.
Ratio check: Confirm that stabilized factors use $q_t/g_t(H_t)$. Setting the numerator to one should recover the unstabilized factor $1/g_t(H_t)$.
Follow-up check: Confirm that each outcome term has $R_t=1$, and that weight products stop at the outcome’s observation time.

Complete-case analysis is simpler, but it discards observed outcomes from anyone who drops out and is generally valid for a full-cohort target only under additional assumptions. Imputation instead models missing outcomes and requires its own assumptions. IPCW models the observation process and reweights observed outcomes; it does not eliminate the need for assumptions. Its validity depends on measured history adequately controlling the censoring process, correct probability estimation, and positivity. If a history has near-zero retention probability, inverse weights can become very large; if retention is impossible for a relevant history, the full-data target is not identified from these observed data by weighting alone.

A common mistake is to treat a reasonable overall retention rate—or a correctly normalized total weight—as proof that the censoring weights are valid. Neither checks whether the risk set, conditional history, ratio direction, or observation window is correct. Construction checks catch coding and specification errors; they cannot establish conditional independent censoring or rule out unmeasured predictors of dropout. More detailed weight-distribution diagnostics can reveal instability, but cannot repair a misspecified censoring model or a positivity violation.

Check Your Understanding

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

At the start of an interval, 50 people remain under follow-up. Forty remain observed through the interval. What is the interval-specific retention probability and the unstabilized factor?
Why can an overall retention probability fail when prior outcome history predicts both dropout and the outcome?
If $q_t=0.80$ and $g_t(H_t)=0.50$, what is the stabilized factor, and what is the expected weighted retention mass for that history?
A person drops out before the outcome at time $t$. Should that person receive a time-$t$ outcome contribution after multiplying by additional inverse retention factors? Explain.

Solutions. (1) The interval probability is $40/50=0.80$, and the unstabilized factor is $1/0.80=1.25$. (2) The pooled probability averages across histories and need not restore the representation of each history; if history predicts the outcome, that residual mismatch can bias an outcome estimate. (3) The factor is $0.80/0.50=1.60$; the weighted retention mass is $0.50(1.60)=0.80$, the numerator probability. (4) No. The time-$t$ outcome is unobserved, so there is no observed outcome term to weight. Later factors cannot supply the missing value.