Why Weight Multi-State Histories?
A multi-state outcome records a subject’s state as follow-up unfolds. For example, a patient may move from an initial state to treatment, then to recovery or death. At a specified time, the state-occupation probability is the probability that a member of the target population occupies a particular state. Loss to follow-up creates a problem when the chance of remaining observed depends on a person’s recorded state history or other predictors: the people still observed may not represent everyone whose state is of interest.
Censoring weights address this selection by giving more weight to observed histories that had a lower estimated chance of remaining uncensored. Unlike a competing-risk calculation that treats other event types as competing outcomes, multi-state analysis retains observed transitions as part of each subject’s history. The target is the marginal distribution across states at a chosen time, not merely the distribution among those still followed.
Model, Assumptions, and Estimator
For clarity, consider follow-up recorded at ordered visits \(t_0,\ldots,t_J\). Just before interval \(m\), let \(H_{im}\) contain the observed transition history and censoring predictors available for subject \(i\). It can include the current state, earlier transitions, and measured time-varying covariates. Predictors must be recorded before the censoring decision for that interval; information that would only be known after censoring cannot be used as if observed.
Write \(q_{im}\) for the conditional probability of remaining uncensored through interval \(m\), given that the subject was uncensored at its start and given \(H_{im}\). The probability of remaining uncensored through \(t_j\), conditional on the recorded history, is the product of these interval probabilities. In practice, a censoring model estimates each \(q_{im}\), and the estimated product gives the estimated censor-free probability.
The assumptions needed for this correction are substantive, not just computational:
- Consistency and well-defined states: the recorded state history corresponds to the state process being studied, with state definitions and visit timing applied consistently.
- Sequential conditional independent censoring: conditional on the measured history available before each interval, censoring does not carry additional information about subsequent state transitions. Put practically, the history and predictors in the censoring model must capture the measured factors that jointly predict loss and future state.
- Positivity: every relevant history pattern has a nonzero probability of remaining uncensored through the time of interest. If some histories are always censored, their later state distribution cannot be recovered from the observed data by weighting.
- Adequate censoring-model specification: the estimated conditional retention probabilities must represent the actual probabilities sufficiently well, including important time and history dependence.
For an observed subject, let \(I\{X_i(t_j)=k\}\) indicate occupation of state \(k\) at \(t_j\). A normalized inverse-probability weighted estimator is
Only subjects observed through \(t_j\) contribute, but their contributions are weighted. The denominator normalizes the weighted state counts to sum to one across states. The unnormalized Horvitz–Thompson version divides the weighted state count by \(n\); with known censoring probabilities its expectation is the target occupation probability under the assumptions above. The normalized version is often convenient because the estimated probabilities across states sum to one in the observed sample.
Why the Weighting Works
The core argument is that an observed history represents itself and other histories like it that were lost, in proportion to how unlikely it was to remain observed. To see this, condition on the full state history through \(t_j\), including the state at \(t_j\). Under sequential conditional independent censoring, the probability of being observed through \(t_j\), for that history, is \(G_i(t_j)\). With the true probability in the weight,
Taking expectations over histories gives \(E[R_i(t_j)I\{X_i(t_j)=k\}/G_i(t_j)]=p_k(t_j)\). Thus, when the assumptions hold, weighting removes the selection caused by the modeled censoring process in the population expectation. In data, \(G_i\) is estimated, so the result is an estimator rather than an exact identity; its reliability depends on adequate model fitting, sample size, and weight stability.
Worked Example: Estimating Occupation Probabilities at One Visit
Consider a hypothetical cohort of five subjects and a target visit \(t_2\). Four remain observed through that visit. Their estimated censor-free probabilities and states are shown below. The fifth subject is censored before the visit, so their state at \(t_2\) is unobserved.
| Observed subject | Estimated \(\widehat G_i(t_2)\) | Weight \(1/\widehat G_i(t_2)\) | State at \(t_2\) |
|---|---|---|---|
| 1 | 0.80 | 1.25 | 0 |
| 2 | 0.50 | 2.00 | 1 |
| 3 | 0.80 | 1.25 | 1 |
| 4 | 0.50 | 2.00 | 2 |
The total observed weight is \(1.25+2.00+1.25+2.00=6.50\). State 0 has weighted count \(1.25\), state 1 has weighted count \(2.00+1.25=3.25\), and state 2 has weighted count \(2.00\). Therefore,
These estimates sum to one. The unweighted proportions among the four observed subjects would be \(0.25\), \(0.50\), and \(0.25\). Weighting changes the state distribution because the observed histories with estimated retention probability \(0.50\) count more than those with probability \(0.80\). Whether that change corrects bias depends on the censoring assumptions; weighting alone does not establish them.
Use the Transition History, Not Just Baseline Risk
A censoring probability can change after a subject moves between states or accumulates time-varying predictors. Sequential weights therefore multiply conditional retention probabilities at each interval, using the information available at that interval. A single baseline-only probability can be inadequate if, for example, a transition into a severe state predicts both later loss to follow-up and subsequent state occupancy.
Worked Example: Building a Weight From Sequential History
Suppose three hypothetical subjects are observed through \(t_2\). Their estimated conditional probabilities of remaining uncensored are recorded for each interval using the state and predictors observed before that interval.
- Subject A remains in state 0: \(\widehat q_{A1}=0.90\) and \(\widehat q_{A2}=0.80\).
- Subject B moves from state 0 to state 1 after the first interval: \(\widehat q_{B1}=0.90\) and \(\widehat q_{B2}=0.50\), where the second probability reflects the updated history.
- Subject C also moves to state 1, but has a different measured predictor history: \(\widehat q_{C1}=0.90\) and \(\widehat q_{C2}=0.80\).
Their estimated censor-free probabilities and weights are
At \(t_2\), A is in state 0 and B and C are in state 1. The total weight is \(1.3889+2.2222+1.3889=5.0000\). Thus \(\widehat p_0(t_2)=1.3889/5.0000=0.2778\) and \(\widehat p_1(t_2)=(2.2222+1.3889)/5.0000=0.7222\). The calculation uses the transition-related history to update B’s second-interval retention probability; it does not treat the transition itself as censoring.
Diagnostics, Alternatives, and Limitations
Diagnostics should ask whether the model and data support the correction, not merely whether software returned an estimate. Examine estimated retention probabilities over time and across histories, the distribution of weights, and which subjects or histories dominate the weighted state counts. Very small \(\widehat G_i(t)\) produces a very large weight. Check that relevant transitions and predictors are measured before censoring, and compare the fitted censoring model with observed censoring patterns across important histories. The earlier material on positivity diagnostics and weight balance provides complementary checks.
A small hypothetical illustration shows how extreme weights can change an estimate. Suppose two observed subjects in state A each have \(\widehat G=0.90\), and two in state B each have \(\widehat G=0.20\). The unweighted proportions are one half in each state. The weights are \(1.1111,1.1111,5,5\), so the weighted estimate for A is
This large shift is possible because the observed state-B histories represent more people who were likely to be censored. It is also a warning: the weighted estimate relies heavily on only two observations. If weights were truncated at 3, the state-A estimate would instead be \(2.2222/(2.2222+6)=0.2703\). Truncation can reduce instability, but changes the estimator and can reintroduce bias; it is not a repair for histories with no chance of remaining observed.
A complete-case state proportion is simpler, but is generally biased for the target population when censoring is related to future state after accounting for neither history nor predictors. Standard multi-state estimation, such as the Aalen–Johansen estimator, is appropriate under its censoring conditions; it does not automatically correct informative loss to follow-up. Censoring-weighted occupation estimates instead target marginal state probabilities under the stated conditional censoring assumptions. Their limitations include unmeasured predictors of censoring and future state, misspecified censoring models, poor positivity, and sampling variability from estimating weights. Uncertainty calculations should account for weight estimation; the earlier material on variance estimation after censoring weighting is relevant.
Name the state and follow-up time, and define the population whose occupation probability is being estimated.
At each interval, estimate remaining uncensored conditional on the measured history available before that interval.
Multiply conditional retention probabilities, invert the product, and calculate normalized weighted state proportions.
Inspect positivity, weight concentration, model fit, and the plausibility of measured-history conditional independent censoring.
Check Your Understanding
Answer each question, then compare with the solutions below.
- At a visit, two subjects occupy state A with weights 2 and 1; one subject occupies state B with weight 3. Find the normalized occupation estimates.
- A subject’s estimated conditional retention probabilities are 0.80, 0.75, and 0.50 over three intervals. Find the cumulative censor-free probability and weight.
- Why might a model using baseline predictors alone be inadequate when transitions and covariates change over follow-up?
- What does a very large weight indicate, and why does truncating it not automatically solve the underlying problem?
Solutions. 1. The total weight is \(2+1+3=6\). Thus \(\widehat p_A=3/6=0.50\) and \(\widehat p_B=3/6=0.50\). 2. The cumulative probability is \(0.80(0.75)(0.50)=0.30\), so the weight is \(1/0.30=3.333\). 3. Later censoring may depend on updated state or covariates that also predict future state; baseline-only predictors may fail to capture that dependence. 4. A very large weight corresponds to a very small estimated probability of remaining observed, so a few histories may dominate. Truncation reduces their influence but changes the estimator and may add bias; it does not establish positivity or remove unmeasured confounding of censoring and future state.