Why Two Weighting Problems Arise Together
A longitudinal marginal structural model (MSM) can target outcomes under specified treatment histories, even when treatment at each visit depends on prior measured history. In practice, however, the outcome may also be unobserved because a participant drops out. Treatment weighting addresses the measured time-varying confounding of treatment; censoring weighting addresses selection into the observed follow-up. If both processes are present, correcting only one does not generally correct the other.
The product weight follows from the fact that an observed record must both follow its treatment history and remain uncensored long enough for the outcome to be recorded. Its components have different denominators and different assumptions. The time horizon matters: for an outcome measured at visit \(k\), only censoring intervals that must be survived to reach visit \(k\) belong in the censoring weight.
Notation and the Combined Weight
Let visits be indexed by \(t=0,\ldots,K\). Let \(A_t\) denote treatment at visit \(t\), and let \(H_t\) denote the measured history available immediately before treatment at that visit, including baseline covariates and relevant prior treatments and outcomes. Write \(\bar A_k=(A_0,\ldots,A_k)\). Let \(R_t=1\) mean that the participant is still under observation and the visit-\(t\) outcome can be observed; \(R_0=1\). For \(t<k\), define the conditional probability of remaining observed through the next visit by \(q_t(H_t,A_t)=P(R_{t+1}=1\mid R_t=1,H_t,A_t)\). The treatment mechanism is \(g_t(a\mid H_t)=P(A_t=a\mid R_t=1,H_t)\).
The treatment factor reweights the observed treatment assignments so that, under the identifying assumptions below, measured past history no longer determines assignment in the weighted population. The censoring factor reweights retained participants to represent those who would have been lost under the measured censoring process. The product addresses both sources of selection in the same contribution; it does not make either assumption unnecessary.
Worked Example: Calculating Both Factors Through Visit 2
Suppose a participant follows the treatment path \((A_0,A_1,A_2)=(1,0,1)\), with corresponding treatment probabilities \(0.50, 0.40, 0.60\). The probabilities of remaining observed for the two intervals leading to visit 2 are \(q_0=0.80\) and \(q_1=0.75\). The participant is observed at visit 2.
The treatment weight is \(W^A_2=1/(0.50\times0.40\times0.60)=1/0.12=8.333\). The censoring weight is \(W^C_2=1/(0.80\times0.75)=1/0.60=1.667\). Therefore the combined weight is \(W_2=1/(0.12\times0.60)=1/0.072=13.889\).
The interval probabilities \(q_0\) and \(q_1\) enter because the participant must remain observed from visit 0 to 1 and from visit 1 to 2. There is no \(q_2\) in this visit-2 outcome weight. Including \(q_2\) would adjust for remaining observed after the outcome is already measured.
Identification of a Regime-Specific Mean
Let \(Y_k^{\bar a}\) be the potential outcome at visit \(k\) under the treatment regime \(\bar a=(a_0,\ldots,a_k)\). The following conditions are needed for the weighted identity. Consistency means that, when a participant receives \(\bar a\), their observed outcome equals \(Y_k^{\bar a}\). Sequential exchangeability for treatment means that at each visit, conditional on \(H_t\) among those still observed, treatment assignment is independent of the relevant future potential outcomes under the regimes being considered. Conditional independent censoring means that, conditional on \(H_t,A_t\) among those still observed, remaining observed to the next visit is independent of those potential outcomes. These are assumptions about the measured histories being sufficient to control the relevant selection.
There must also be positivity: for every history with positive probability under regime \(\bar a\), the regime’s treatment has positive probability, \(g_t(a_t\mid H_t)>0\), and, for each required censoring interval, \(q_t(H_t,a_t)>0\). The variables and probabilities must be well-defined and measurable, and the potential outcome must have a finite expectation. These conditions rule out histories where the needed treatment or continued follow-up is impossible.
Proof. Consider the distribution of the histories and outcome along a participant’s path through visit \(k\). In the observed data, a path following \(\bar a\) and observed at visit \(k\) is selected by the treatment probabilities \(g_t(a_t\mid H_t)\), for \(t=0,\ldots,k\), and the retention probabilities \(q_t(H_t,a_t)\), for \(t=0,\ldots,k-1\). By the chain rule, the conditional probability of those treatment and retention events along the path is the product of these factors. Sequential exchangeability and conditional independent censoring ensure that, conditional on the measured history at each step, these selection factors do not further select on the relevant potential outcome. Multiplying by their reciprocals therefore cancels the probabilities of treatment-path selection and continued observation.
More explicitly, under the intervention setting each \(A_t=a_t\), the conditional distributions of the measured covariate and outcome history are the same as their observed conditional distributions given the preceding history, by consistency and treatment exchangeability. Conditional independent censoring gives the same equality for the history among those retained, after conditioning on the measured predictors of censoring. The observed-data path probability contains the additional factors \(\prod_{t=0}^{k}g_t(a_t\mid H_t)\prod_{t=0}^{k-1}q_t(H_t,a_t)\). The indicator selects precisely the treatment path \(\bar a\), and \(R_k\) selects records with the outcome observed. The reciprocal product cancels those additional factors, leaving the intervention distribution of \(Y_k^{\bar a}\). Integrating \(Y_k\) against that distribution gives \(E[Y_k^{\bar a}]\). Positivity makes the reciprocals finite on the target histories, and integrability permits the expectations. \(\square\)
This result is a Horvitz–Thompson identity: it identifies a mean through inverse probabilities, not a guarantee that any one finite-sample weighted estimate will equal the target. In a sample of \(n\) independent participants, its direct estimator is $$ \widehat\mu_k(\bar a)=\frac{1}{n}\sum_{i=1}^n \frac{I(\bar A_{ik}=\bar a)R_{ik}Y_{ik}} {\left\{\prod_{t=0}^{k} \widehat g_t(a_t\mid H_{it})\right\} \left\{\prod_{t=0}^{k-1}\widehat q_t(H_{it},a_t)\right\}}. $$
Worked Example: A Weighted Regime Mean and Its Ratio Version
At visit 1, consider a target treatment path \((a_0,a_1)=(1,0)\). Two of four participants followed this path and were observed at visit 1. For the first, the treatment probabilities along the path are \(0.50,0.50\), the interval retention probability is \(0.80\), and \(Y_1=8\). For the second, these values are \(0.40,0.50,0.50\), respectively, and \(Y_1=4\). The other two participants do not match the regime and contribute zero to the indicator-based estimator.
For the first matching participant, the combined inverse probability is \(1/(0.50\times0.50\times0.80)=5\), so the weighted outcome contribution is \(5\times8=40\). For the second, it is \(1/(0.40\times0.50\times0.50)=10\), giving contribution \(10\times4=40\).
The Horvitz–Thompson estimate is \((40+40+0+0)/4=20\). With only four participants, this estimate is highly variable and can exceed the range of the observed outcomes; that is not an algebraic error. A commonly used normalized, or Hájek, version divides the sum of weighted outcomes by the sum of weights among matching observed participants: \((40+40)/(5+10)=80/15=5.333\). This ratio is often more stable in a small sample, but it is not the same finite-sample estimator as the Horvitz–Thompson estimate.
Using the Product in an MSM Estimating Equation
A marginal structural model specifies a mean outcome under treatment histories, such as \(m(k,\bar a;\beta)=E[Y_k^{\bar a}]\) under a model indexed by parameter \(\beta\). With observed treatment histories and censoring, a weighted estimating equation can take the form
where \(D_{ik}(\beta)\) is the chosen derivative or estimating-equation vector. For multiple visits, contributions can be summed over \(k\), using the corresponding visit-specific outcome and weight. The equation uses the realized treatment probabilities for each participant’s observed history; the indicator for a fixed regime appears when estimating a regime-specific mean directly. Under a correctly specified marginal mean model and the assumptions for identification, this reweighting targets the marginal treatment-history relationship rather than a conditional outcome regression adjusted for the full time-varying history.
Stabilized weights replace some denominator probabilities with numerator probabilities. For example, a stabilized treatment factor may use probabilities conditional on baseline variables, and a stabilized censoring factor may use retention probabilities conditional on a reduced history. The numerators help control weight variability, but they are not arbitrary constants: they define which associations are retained in the weighted population. For an MSM conditional on baseline variables, the numerator choices and the model must preserve that target. The unstabilized identity above is the clearest statement of the inverse-probability correction; a stabilized estimating equation requires its corresponding target and numerator specification.
Worked Example: A Combined-Weight Treatment Contrast
Suppose four participants have a single treatment \(A\), a visit-1 outcome \(Y\), treatment probability \(g(A\mid H)\), and retention probability \(q_0(H,A)\). All four are observed at visit 1. Their records are:
| Participant | \(A\) | \(Y\) | \(g(A\mid H)\) | \(q_0(H,A)\) |
|---|---|---|---|---|
| 1 | 1 | 8 | 0.50 | 0.50 |
| 2 | 1 | 6 | 0.40 | 0.80 |
| 3 | 0 | 3 | 0.50 | 0.80 |
| 4 | 0 | 5 | 0.60 | 0.50 |
The combined weights are \(1/(0.50\times0.50)=4\), \(1/(0.40\times0.80)=3.125\), \(1/(0.50\times0.80)=2.5\), and \(1/(0.60\times0.50)=3.333\). In a weighted saturated model, the treatment-group means are weighted outcome averages. For \(A=1\), the mean is \((4\times8+3.125\times6)/(4+3.125)=50.75/7.125=7.123\). For \(A=0\), it is \((2.5\times3+3.333\times5)/(2.5+3.333)\), approximately \(24.167/5.833=4.143\). The weighted contrast is approximately \(7.123-4.143=2.980\).
The treatment component changes representation across treatment assignment; the censoring component changes representation across retention. These four numbers illustrate the calculation, not evidence that the assumptions hold or that a particular causal contrast is identified in an actual study.
Comparisons, Diagnostics, and Limits
A treatment-only weight can address measured treatment confounding but leaves selection from informative dropout uncorrected. A censoring-only weight can address measured dropout but leaves treatment assignment confounding uncorrected. Complete-case analysis discards the censoring correction altogether; it is generally not justified when remaining observed depends on measured history related to the outcome. The combined product is appropriate when both processes require correction and the relevant assumptions hold. It does not replace a well-defined estimand or careful temporal ordering.
A practical check is to inspect each factor and the combined weight separately. Confirm that the treatment probabilities correspond to the observed treatments and that the censoring probabilities correspond only to intervals required to observe the outcome. Check for zero or very small fitted probabilities, unusually large products, and whether some histories have little or no support. A factor near zero can produce a large reciprocal; multiplying several moderate reciprocals can also yield a very large combined weight. Such patterns signal potential instability or positivity problems, although their absence cannot establish exchangeability, model correctness, or absence of unmeasured predictors.
The method is limited by the quality of the measured histories, correct specification (or otherwise adequate estimation) of treatment and censoring mechanisms, and positivity. If dropout depends on unmeasured information about future outcomes after conditioning on \(H_t,A_t\), the censoring factor does not remove that selection. If a treatment or retention probability is structurally zero in a target history, no finite inverse weight can recover the missing counterfactual information. Weight truncation may reduce variance, but it changes the estimator and can introduce bias; it does not repair a violated identification assumption.
Common Mistake: Extending the Censoring Product One Visit Too Far
- Incorrect reasoning: “The outcome is at visit \(k\), so include the retention probability \(q_k\) and use \(R_{k+1}\) in the weighted expression.” This adjusts for continued follow-up after the visit-\(k\) outcome has already been recorded.
- Correction: With \(R_k=1\) denoting observation of \(Y_k\), use \(R_k\) and \(\prod_{t=0}^{k-1}q_t(H_t,A_t)\). The last required interval takes a participant from visit \(k-1\) to visit \(k\). Keep the outcome-observation definition and the censoring product’s horizon aligned.
- Another error: Treating the product as one undifferentiated correction. Keep treatment probabilities and retention probabilities conceptually separate; each requires its own model, positivity, and conditional exchangeability assumptions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For an outcome \(Y_3\) observed at visit 3, which censoring probabilities belong in the censoring weight under the convention used here?
- At visit 1, suppose \(g_0=0.50\), \(g_1=0.25\), and \(q_0=0.80\) along a specified treatment path. What is the combined unstabilized weight?
- Which component of the combined weight addresses informative dropout, and which addresses treatment assignment?
- Why can an inverse-probability weighted finite-sample estimate fall outside the observed outcome range?
- Name one key condition that fails when a target treatment has probability zero for a history occurring under the target regime.
Solutions. (1) Use \(q_0,q_1,q_2\), the intervals needed to reach visit 3; do not include \(q_3\). (2) The weight is \(1/(0.50\times0.25\times0.80)=10\). (3) The censoring factor corrects retention selection, while the treatment factor corrects treatment assignment selection. (4) The Horvitz–Thompson estimator is an unnormalized weighted average divided by the full sample size, and a small sample can yield a highly variable value. (5) Positivity fails, so the reciprocal treatment probability is undefined and the target history lacks observed treatment support.