IPCW Estimation of Fixed-Horizon Marginal Means

Why Weight Observed Outcomes at a Fixed Horizon?

Suppose a study enrolls a population at baseline and plans to measure an outcome at time \(K\). Some participants drop out before that measurement. The mean among participants who remain observed is generally not the population mean: remaining observed may depend on characteristics associated with the outcome. Inverse-probability-of-censoring weighting (IPCW) addresses this selection by giving more weight to observed outcomes from participants whose histories made continued observation less likely.

The target here is the marginal mean of the outcome at the fixed horizon in the baseline population, \(E[Y(K)]\). This is a mean under the population’s actual course of follow-up and covariate histories; IPCW for censoring alone does not define a treatment contrast or a treatment intervention.

A participant censored before \(K\) contributes zero to the numerator, not an imputed outcome. An observed participant contributes their outcome multiplied by the inverse of their estimated probability of being observed through the horizon. The denominator of this Horvitz–Thompson estimator is the baseline sample size \(n\).

Identification: Why the Weight Recovers the Marginal Mean

The argument requires a censoring model that conditions on enough history to account for informative dropout. Write the full data as the trajectory that would be available without censoring, including \(Y(K)\) and any covariates needed to describe the censoring process. Sequential independent censoring means that, among participants uncensored at \(t-1\), the censoring decision at \(t\) is independent of the relevant full-data future conditional on the history \(H(t-1)\). In particular, given the full trajectory, the probability of remaining uncensored at \(t\) is \(q_t\{H(t-1)\}\). This condition allows the censoring probability to vary with observed history; it does not require censoring to be marginally independent of the outcome.

The other central condition is positivity: each history with positive probability in the target population must have a positive probability of remaining uncensored through the horizon. For stable estimation, a stronger practical condition is often imposed: \(G(K)\ge\epsilon\) for some \(\epsilon>0\). The identification argument itself can allow probabilities arbitrarily close to zero if the expectation exists, but such cases can make estimation highly variable.

Proof. Let \(\mathcal F\) denote a participant’s full-data trajectory, including \(Y(K)\) and the histories determining the censoring probabilities. By the chain rule for conditional probabilities and sequential independent censoring, $$ P\{R(K)=1\mid\mathcal F\} =\prod_{t=1}^{K}P\{R(t)=1\mid R(t-1)=1,\mathcal F\} =\prod_{t=1}^{K}q_t\{H(t-1)\} =G(K). $$ The histories in this expression are the full-data values along the trajectory; for a participant observed through \(K\), they are the histories used to compute the weight. Since \(Y(K)\) and \(G(K)\) are determined by \(\mathcal F\), conditioning on \(\mathcal F\) gives $$ E\left[\left.\frac{R(K)Y(K)}{G(K)}\right|\mathcal F\right] =\frac{Y(K)}{G(K)}P\{R(K)=1\mid\mathcal F\} =Y(K). $$ Taking expectations on both sides proves the identity. Integrability follows from \(E[|Y(K)|]<\infty\), because the absolute conditional expectation of the weighted quantity is \(|Y(K)|\). \(\square\)

Worked Calculations

Estimated Weights and Consistency

In applications, the \(q_t\) values are usually estimated from observed follow-up data, for example by modeling the discrete-time censoring hazards. The resulting \(\widehat G_i(K)\) is the product of the estimated conditional retention probabilities along participant \(i\)’s observed history. Using estimated rather than known probabilities adds an estimation step to the argument.

Proof. First consider the estimator using the true probabilities. The summands \(R_i(K)Y_i(K)/G_i(K)\) are independent and identically distributed, and their expectation is \(E[Y(K)]\) by the identification theorem. Since \(Y(K)\) is bounded and \(G(K)\ge\epsilon\), their absolute values are bounded by \(\sup|Y(K)|/\epsilon\). The law of large numbers therefore gives $$ \frac{1}{n}\sum_{i=1}^{n}\frac{R_i(K)Y_i(K)}{G_i(K)} \xrightarrow{p}E[Y(K)]. $$ It remains to show that estimating \(G\) does not change the limit. On the event that both \(G_i(K)\) and \(\widehat G_i(K)\) are at least \(\epsilon/2\), $$ \left|\frac{R_iY_i}{\widehat G_i}-\frac{R_iY_i}{G_i}\right| \le |Y_i|\frac{|\widehat G_i-G_i|}{\widehat G_iG_i} \le \sup|Y(K)|\frac{2}{\epsilon^2}|\widehat G_i-G_i|. $$ Averaging over \(i\), the absolute difference between the estimated-weight and true-weight averages is at most \(\sup|Y(K)|(2/\epsilon^2)\sup_i|\widehat G_i(K)-G_i(K)|\), which converges in probability to zero. The event used in the bound has probability tending to one by the stated lower-bound condition. Combining this result with the convergence of the true-weight average proves the claim. \(\square\)

This is a sufficient consistency result, not a guarantee that any fitted censoring model works. In practice, consistency requires the estimated retention probabilities to represent the relevant conditional probabilities well enough. The uniform convergence condition stated here is one transparent way to make the plug-in argument; more specialized results can use other regularity conditions.

Diagnostics, Alternatives, and Limits

Inspect the estimated retention probabilities and the resulting weights before interpreting the estimate. Useful checks include the distribution and maximum of \(1/\widehat G_i(K)\), the number of participants observed at each time, and whether fitted retention probabilities approach zero in particular histories. Compare weighted and unweighted summaries of measured baseline or follow-up history among observed participants. Large weights concentrated in a few people, very small fitted probabilities, or little overlap between observed histories can indicate practical positivity problems. These checks can reveal instability; they cannot establish that all predictors of dropout have been measured or that the censoring model is correct.

The main alternative is a complete-case mean, which averages only outcomes observed through \(K\). It is generally not a population mean when observation depends on outcome-related history. Outcome regression offers a different route: model the horizon outcome using observed history and use that model to predict outcomes for the target population. IPCW instead models observation probabilities and reweights the observed outcomes. Each approach relies on assumptions about its modeled process; neither is automatically protected from misspecification. Combining models may be useful, but is outside this fixed-horizon derivation.

IPCW also has clear limits. If some target histories have zero probability of remaining observed, the target mean is not identified by these weights. If probabilities are positive but tiny, the estimator can be highly variable. Weight truncation can reduce extreme contributions, but generally changes the estimator and may introduce bias; it does not restore the original identification argument. Finally, IPCW for censoring alone does not adjust for treatment-confounding, intermittent missingness, or a competing event that changes the estimand. Those situations require an analysis matched to their own structure.