One-Period Derivation of Inverse Censoring Weights

Why a Single-Period Weight Is Needed

Suppose a study aims to describe an outcome \(Y\) for an eligible population, but \(Y\) is recorded only for people who remain uncensored through one follow-up period. If remaining observed is related to measured characteristics \(H\), the observed cases can have a different mix of those characteristics than the target population. When \(H\) also predicts \(Y\), the observed-case outcome distribution can therefore be misleading.

Inverse probability of censoring weighting corrects this selection by giving each observed person weight inversely proportional to their probability of being observed. The derivation below is for one follow-up period only. It requires that the measured history capture the relevant dependence between observation and outcome, and that every history value in the target population have a positive chance of being observed.

The outcome \(Y\) is understood to be defined for the target population, whether or not it was recorded. For example, it could be the outcome that would have been measured at the scheduled follow-up had the person not been censored. This is a missing-outcome setting: if the outcome is not meaningfully defined for some people, the target estimand must be specified before weighting can address anything.

Assumptions and the Weighting Identity

Two assumptions give the weight its correction property. First, conditional independent censoring requires \(R\) and \(Y\) to be conditionally independent given \(H\). In words, among people with the same measured history, observation does not select people according to their outcome. Second, positivity requires \(g(H)>0\) with probability one in the target population. To make the weighted expectations below well-defined, also assume \(Y\) is integrable, meaning \(E[|Y|]<\infty\).

Proof. Conditional independence gives $$ E[R f(H,Y)\mid H]=E[R\mid H]E[f(H,Y)\mid H] =g(H)E[f(H,Y)\mid H]. $$ Because \(g(H)>0\), division by \(g(H)\) yields $$ E\!\left[\frac{R f(H,Y)}{g(H)}\middle|H\right]=E[f(H,Y)\mid H]. $$ Taking expectations of both sides gives the claimed identity by the law of iterated expectations. The weighted quantity is integrable: applying the same argument to \(|f(H,Y)|\) gives \(E[R|f(H,Y)|/g(H)]=E[|f(H,Y)|]<\infty\). Thus the conditional expectation and subsequent expectation are well-defined. Setting \(f(H,Y)=Y\) gives the first identity. \(\square\)

This identity also establishes a distributional result, not just a mean correction. Define the weighted probability of an outcome event \(A\) by dividing its expected weighted observed count by the expected total weight. Since \(E[R/g(H)]=1\), applying the identity to \(f(H,Y)=\mathbf{1}(Y\in A)\) gives $$ \frac{E\!\left[R\mathbf{1}(Y\in A)/g(H)\right]} {E[R/g(H)]} =P(Y\in A). $$ Thus, at the population level, weighting the observed outcomes reconstructs the target outcome distribution. The same argument applies to functions of both \(H\) and \(Y\), so the weighted joint distribution is also the target joint distribution.

Why the Algebra Works Within and Across Strata

The weight does not alter the outcome distribution within a history stratum under conditional independent censoring. Its role is to restore each stratum’s representation in the target population. This can be seen directly in a discrete-history case. Write \(p_h=P(H=h)\) and \(g_h=P(R=1\mid H=h)\). Conditional independence implies that the observed outcome mean within stratum \(h\) is \(E[Y\mid H=h,R=1]=E[Y\mid H=h]\), whenever \(p_hg_h>0\). The expected weighted outcome contribution from that stratum is $$ E\!\left[\frac{RY\mathbf{1}(H=h)}{g(H)}\right] =\frac{p_hg_hE[Y\mid H=h,R=1]}{g_h} =p_hE[Y\mid H=h]. $$ Summing over strata recovers \(\sum_h p_hE[Y\mid H=h]=E[Y]\). In contrast, the complete-case mean combines stratum means using observed-case proportions \(p_hg_h/\sum_jp_jg_j\), which generally differ from the target proportions \(p_h\).

When the Weight Cannot Repair Selection

Positivity is essential, not a technical detail that can be ignored because a formula has been written down. If a history stratum has \(g(h)=0\), no one in that stratum contributes an observed outcome. The reciprocal \(1/g(h)\) is undefined, and the data contain no observed outcomes from which to recover that stratum’s outcome distribution without additional assumptions.

Near-violations also matter in practice. If \(g(H)\) is positive but very small for some people, their weights are very large. A few observations may then dominate a sample estimate, making it unstable even though the population identity holds. Inspect estimated observation probabilities and the distribution of the resulting weights; unusually large weights or sparse history strata warn that the result may be driven by limited support. Weighting also does not correct conditional dependence between \(R\) and \(Y\) left over after conditioning on \(H\), or errors in estimating \(g(H)\).

How This Approach Compares With Alternatives

A complete-case analysis uses only \(R=1\) observations without correcting their altered history distribution. It is valid for the target outcome mean under stronger conditions—for example, if observation is independent of \(Y\) without conditioning on \(H\)—but can be biased when observation and outcome are associated through \(H\). Inverse probability weighting instead keeps the target population as its reference and adjusts the contributions of observed people using \(H\).

Outcome-model approaches, including regression adjustment or multiple imputation, use a model for the outcome given measured variables to address missing outcomes. They differ from inverse probability weighting in the model they rely on: weighting models observation, whereas those approaches model outcomes or missing values. Neither strategy makes unmeasured selection disappear. In this one-period derivation, the essential comparison is that weighting reconstructs the target distribution by correcting the observed sample’s selection probabilities, under conditional independent censoring and positivity.

The identity is a population statement. In data, \(g(H)\) is usually unknown and must be estimated; substituting an estimated probability does not make the sample weighted mean equal the target mean exactly. Its performance depends on the adequacy of the observation-probability model and on having enough observations across the relevant histories. This tutorial’s result concerns one period; it does not derive the cumulative weights needed when observation must continue through multiple follow-up periods.