IPCW for Repeated Longitudinal Outcome Means

Why Weight Repeated Outcomes?

A longitudinal study may measure an outcome repeatedly, yet some participants stop contributing measurements before the final visit. If the chance of remaining observed depends on prior health or other measured history, the mean among those still observed can differ from the mean in the original cohort. The target here is a separate marginal mean at each scheduled time: the mean outcome that would be observed in the original baseline population if outcome follow-up were complete.

Inverse-probability-of-censoring weighting (IPCW) reweights observed participants to represent those with similar observed histories who have become underrepresented through dropout. Each time point has its own weight, accumulating the inverse probabilities of remaining uncensored through that time. This is not a model for the outcome trajectory; it is a way to recover its time-specific marginal means under stated assumptions.

Identification and Estimation

The history in the censoring model must include measured predictors of both dropout and the outcome that are available before each censoring decision. The essential condition is sequential independent censoring: conditional on the measured history, retention at each time does not further select participants according to their future outcomes. One precise sufficient formulation is that, for every \(k\leq t\), \(P(R_k=1\mid R_{k-1}=1,\text{full data})=q_k(H_k)\). Here “full data” includes the outcome process, whether observed or not. The conditional probability on the right is the same as the observed-history retention probability defined above.

Identification also requires positivity: for histories occurring in the target population, \(q_k(H_k)>0\) at every \(k\leq t\). If a history makes continued observation impossible, outcomes for that history cannot be recovered by reweighting. For finite-variance inference, a stronger practical condition is often needed, such as bounded weights or suitable moment conditions. The model used to estimate each \(q_k\) must also be correctly specified, unless another valid estimation strategy supplies consistent probabilities.

Proof. Conditional on the complete data, the sequential censoring condition gives \(P(R_k=1\mid R_{k-1}=1,\text{full data})=q_k(H_k)\). Since retention is monotone, repeated conditioning yields \(P(R_t=1\mid\text{full data})=\prod_{k=1}^t q_k(H_k)=G_t\). The product is determined by the full-data history, and is positive by assumption. Therefore, $$ E\left[\left.\frac{R_tY_t}{G_t}\right|\text{full data}\right] =\frac{Y_t}{G_t}P(R_t=1\mid\text{full data})=Y_t. $$ Taking expectations of both sides proves the first identity. Replacing \(Y_t\) by 1 proves the second. \(\square\)

The theorem leads to two common sample estimators. The Horvitz–Thompson estimator averages weighted outcomes over the original cohort, treating unobserved outcomes as contributing zero after multiplication by \(R_{it}\). The Hájek estimator divides the weighted outcome total by the total weight among those observed at that time. The latter is a normalized ratio and is often convenient in practice, but is not generally exactly unbiased in a finite sample.

With known probabilities, independent and identically distributed participants, and finite expectations, the Horvitz–Thompson estimator has expectation \(\mu_t\) by the theorem. For consistency of the Hájek estimator, a simple sufficient set of conditions is that the probabilities are known, the weighted outcome and weight have finite first moments, and the denominator expectation is positive. The law of large numbers gives convergence of the numerator average to \(\mu_t\) and of the denominator average to 1; the ratio therefore converges in probability to \(\mu_t\). With estimated probabilities, consistency additionally requires that their estimation does not prevent convergence of these weighted averages—for example, under suitable regularity conditions, consistent censoring-model estimates and controlled weights suffice.

Accumulating Weights Across Visits

For repeated outcomes, a participant observed at time \(t\) must have remained uncensored at every earlier time. The relevant weight is therefore a product, not just the inverse probability of retention at the final visit. Each factor uses the history available at that step. If a participant had an estimated retention probability of 0.8 at the first visit and 0.5 at the second, the probability of remaining observed through the second visit is \(0.8\times0.5=0.4\), and the cumulative weight is \(1/0.4=2.5\).

The identification result applies separately at every time point. It does not require the weighted mean to follow a particular curve, nor does it imply that a single weighted regression is needed. A sequence of time-specific weighted means estimates the marginal mean trajectory \((\mu_1,\ldots,\mu_T)\). If treatment is also being handled with weights, the treatment and censoring components must correspond to the intended estimand and be combined appropriately; censoring weights alone do not adjust treatment assignment.

Interpretation, Checks, and Limitations

IPCW and complete-case analysis answer the same marginal-mean question only under different circumstances. A complete-case mean at time \(t\) uses the participants observed then and is directly representative of the original cohort if continued observation is independent of \(Y_t\) without conditioning on history (or under other conditions that make the observed mean valid). IPCW instead allows dropout to depend on measured history, provided that history is sufficiently informative to satisfy sequential independent censoring and the retention probabilities are positive and correctly estimated.

A model for the outcome, such as a likelihood-based longitudinal model, may use a different route to account for incomplete measurements and relies on its own assumptions. IPCW makes the selection mechanism explicit, but it is not assumption-free and does not automatically outperform a correctly specified outcome model. Unlike simple imputation, it does not fill in each missing outcome; it reweights observed outcome records to represent the target population at each time.

Basic checks should establish that retention probabilities are defined for the histories actually encountered, that the risk set at each visit matches the specified time ordering, and that cumulative probabilities are calculated across all required visits. Very small estimated probabilities imply large weights and unstable estimates; this is a warning about limited information and potential positivity problems, not proof that the model is wrong. Conversely, plausible-looking weights do not establish that all predictors of dropout and outcome were measured or that the censoring model is correctly specified. Weight truncation can reduce the influence of extreme weights but changes the estimator and may introduce bias; it does not repair a lack of positivity.

The target also matters. The mean \(E[Y_t]\) refers to outcomes in the original baseline population under the full-data outcome process. If death or another competing event makes the outcome undefined, treating that event as ordinary dropout silently changes the estimand unless a meaningful outcome under that event has been defined. Likewise, IPCW for monotone dropout does not by itself solve intermittent missingness, where a participant may miss one measurement and later return.