Inverse-Probability-Weighted GEE Under MAR

Why Weighting Is Needed

A marginal GEE targets a population mean relationship, such as \(E[Y_{it}\mid X_i]=\mu_{it}(\beta)\), using repeated outcomes from each cluster. If outcomes are missing, applying the complete-data estimating equation only to observed values can change the target or introduce selection bias. Inverse-probability weighting corrects the contribution of observed outcomes by how unlikely they were to be observed, under a suitable missing-at-random assumption.

The key is to weight by the probability that an outcome is observed in the target population—not automatically by a conditional probability of remaining observed at one particular visit. For monotone dropout, the probability of being observed through visit \(t\) accumulates across visits. For intermittent missingness, the relevant probability is the current response probability conditional on the observed history, provided that conditional probability identifies observation in the target population.

A Marginal Estimating Equation and Its Weighted Form

Write \(\mu_{it}(\beta)=E[Y_{it}\mid X_i]\), where \(X_i\) denotes the covariates in the marginal mean model. Let \(v_{it}(\beta)>0\) be the specified marginal variance and let \(d_{it}(\beta)=\partial\mu_{it}(\beta)/\partial\beta\), a column vector. A convenient complete-data marginal estimating function is the sum of visit-level contributions

The corresponding complete-data equation is \(\sum_{i=1}^m\psi_i(\beta)=0\). It is a valid marginal estimating equation when its expectation vanishes at the target coefficient; in particular, this holds if the marginal mean model is correct. Under missingness, replace each unavailable contribution by zero and inflate each observed contribution by the inverse probability of observing it:

This componentwise equation uses only \(Y_{it}\), its marginal mean contribution, and information available for the response model. It is a marginal GEE with a working-independence form for the estimating contributions; dependence among visits within a cluster is retained for inference by treating the cluster as the sampling unit. A full-data GEE contribution that mixes multiple outcomes through a non-diagonal working covariance matrix requires more care: if some outcomes in that contribution are missing, simply multiplying by a visit-specific weight does not generally recover the full-data equation.

To prove the identity, condition on \(X_i,H_{it},Y_{it}\). Under the stated MAR condition, the conditional expectation of \(R_{it}\) is \(q_{it}\), while \(u_{it}(\beta)\) and \(q_{it}\) are fixed given the conditioning variables. Therefore

Taking expectations on both sides gives the result. Integrability of the weighted contribution is needed for these expectations to be defined; positivity alone does not guarantee finite variance if response probabilities can be arbitrarily close to zero. Summing over visits shows that the weighted population equation has the same expectation as the complete-data equation. Thus, if \(E[\psi_i(\beta_0)]=0\), the weighted equation is unbiased at \(\beta_0\).

Monotone Dropout and Intermittent Missingness

For monotone dropout, define \(\pi_{is}=P(R_{is}=1\mid R_{i,s-1}=1,H_{is},X_i)\), with \(R_{i0}=1\). Under sequential MAR, each conditional probability depends on observed history and covariates, not on future unobserved outcomes after conditioning on that history. The probability that a participant from the original target population is observed at visit \(t\) is then

The equality follows by repeatedly applying the multiplication rule for conditional probabilities to the event of remaining observed at every visit through \(t\). The factors are evaluated along the participant’s observed history; that history is available for a participant observed at visit \(t\). The weight for the visit-\(t\) contribution is \(1/q_{it}\), not merely \(1/\pi_{it}\).

With intermittent missingness, a participant may miss one visit and return later. There is no single sequence of “remaining observed” probabilities. Instead, for each outcome contribution use its current response probability \(q_{it}=P(R_{it}=1\mid X_i,H_{it})\), defined for the relevant target population and observed history. The MAR identity applies if response at \(t\) is conditionally independent of \(Y_{it}\) given that history and the response probability is positive. If histories differ by missingness pattern, the response model must use the history actually available under each pattern. A probability defined only among people observed at the preceding visit does not, by itself, serve as the full-population probability for a later intermittent response.

Conditions for Consistent Marginal Coefficients

An unbiased population equation is not alone enough to guarantee that a computed root converges to the target coefficient. A standard consistency argument also requires regularity and identification conditions. One sufficient set of conditions is the following.

• Clusters are independent and identically distributed, with a fixed finite number of visits, or satisfy suitable extensions for increasing cluster sizes.
• The marginal mean model is correctly specified at a unique coefficient \(\beta_0\), so \(E[\psi_i(\beta_0)]=0\), and the weighted equation has the same expectation by the MAR identity.
• Response probabilities are correctly specified or known, positive, and sufficiently bounded away from zero for the required moments of the weighted contributions to be finite.
• The parameter space \(B\) is compact; the population map \(\Psi(\beta)=E[\psi_i^{\,w}(\beta)]\) is continuous and has the unique zero \(\beta_0\); and a uniform law of large numbers holds for \(\Psi_m(\beta)=m^{-1}\sum_i\psi_i^{\,w}(\beta)\).
• A sample root \(\hat\beta\in B\) exists with probability tending to one. More generally, for a noncompact parameter space, assume the explicit separation condition \(\inf_{\beta\in B:\|\beta-\beta_0\|\geq\epsilon}\|\Psi(\beta)\|>0\) for every \(\epsilon>0\).

For a proof, fix \(\epsilon>0\). By the separation condition, there is a \(\delta>0\) such that \(\|\Psi(\beta)\|\geq\delta\) whenever \(\beta\in B\) and \(\|\beta-\beta_0\|\geq\epsilon\). On the event that a root exists and is at least \(\epsilon\) from \(\beta_0\), its defining equation gives \(\Psi_m(\hat\beta)=0\), and hence

The uniform law of large numbers makes the right-hand side converge in probability to zero, so the probability of that event tends to zero. The probability that a root does not exist also tends to zero by assumption. Therefore \(P(\|\hat\beta-\beta_0\|\geq\epsilon)\to0\) for each \(\epsilon>0\), which is consistency. If \(B\) is compact and \(\Psi\) is continuous with a unique zero, the separation condition follows by taking the minimum of \(\|\Psi(\beta)\|\) over the compact subset of \(B\) at least \(\epsilon\) from \(\beta_0\). \(\square\)

When response probabilities are estimated rather than known, additional conditions are needed to ensure that replacing \(q_{it}\) by an estimate does not change the limiting equation. The weighted contributions must remain well behaved, including in regions where estimated probabilities are small. Weight estimation and its implications for inference require separate attention.

Interpretation and Common Pitfalls

IPW GEE aims to restore the marginal target by making observed outcomes represent similar outcomes that were less likely to be observed. It does not make missingness ignorable without assumptions: MAR must be credible given the recorded history, the response probabilities must be valid, and positivity must hold. Very small probabilities create large weights and unstable estimates, even when the estimating equation remains unbiased.

• Wrong: “At visit \(t\), the probability of responding is \(\pi_{it}\), so weight by \(1/\pi_{it}\).” For monotone dropout, \(\pi_{it}\) is conditional on reaching the visit; the original-population observation probability is \(q_{it}=\prod_{s=1}^t\pi_{is}\). Use its inverse for a visit-\(t\) contribution.
• Wrong: “MAR means the response probability is unrelated to the outcome.” MAR permits response to depend on observed history, which can include earlier outcomes. The needed independence is conditional on the specified observed information.
• Wrong: “An unbiased weighted equation automatically yields a consistent estimate.” Consistency also needs identification, a suitable law of large numbers, and a condition preventing roots from drifting into regions where the population equation is arbitrarily close to zero.
• Wrong: “Every complete-data correlated GEE can be recovered by weighting each observed outcome separately.” A score that combines several outcomes may not be computable from the observed subset. The componentwise marginal equation above avoids that issue; more elaborate correlated equations require their own derivation.