Why Dropout Changes Population-Averaged Inference
In a longitudinal study, dropout means that a participant stops contributing outcome measurements before the planned end of follow-up. If the chance of remaining observed is related to earlier outcomes, the people still observed at later visits may no longer represent the target population. A population-averaged estimate based only on observed outcomes can then describe the retained participants rather than the original population.
The central distinction is between the assumptions that make a missingness process ignorable for a specified likelihood and the assumptions that make an ordinary GEE estimating equation unbiased. These are not interchangeable. In particular, dropout can be missing at random (MAR) and ignorable for a correctly specified likelihood, yet still bias an unweighted GEE.
For a marginal mean model, write \(\mu_{ij}=E(Y_{ij}\mid X_i)\), with regression parameter \(\beta\). A complete-data GEE has the form \(\sum_i D_i^{T}V_i^{-1}(Y_i-\mu_i)=0\), where \(D_i\) is the derivative of the mean vector with respect to \(\beta\), and \(V_i\) is a working covariance matrix. With missing outcomes, the usual observed-data version uses the available components. If dropout selects participants according to their outcomes, the observed residuals need not average to zero in the way that equation requires.
How Outcome-Dependent Dropout Produces Bias
Consider a simple mean at a final visit. Let \(R=1\) indicate that the final outcome \(Y\) is observed. The observed-case mean estimates \(E(Y\mid R=1)\), whereas the population target is \(E(Y)\). Their difference is
This difference is generally nonzero when the chance of observation depends on \(Y\), directly or through observed outcome history. A sandwich (robust) standard error estimates uncertainty around the solution to the estimating equation; it does not change that equation's target or remove selection bias.
Worked Example: A Lower Observed Mean After Selective Dropout
Suppose 200 hypothetical participants have a binary outcome \(Y_1\) at the first visit: 100 have \(Y_1=1\) and 100 have \(Y_1=0\). At the final visit, the outcome \(Y_2\) is positive for 80% of those with \(Y_1=1\), and for 20% of those with \(Y_1=0\). The target final-visit positive proportion is therefore
Now suppose only half of the \(Y_1=1\) group remains observed at the final visit, while all of the \(Y_1=0\) group remains observed. Among the observed participants, there are 50 from the first group, with 40 positive outcomes, and 100 from the second group, with 20 positive outcomes. The unweighted observed-case proportion is
Thus the observed-case estimate is 10 percentage points below the population target of 0.500. The difference occurs because the group with the higher final-visit positive rate is underrepresented among those observed. This is a selection effect, not evidence that the target population's outcome rate is 0.400.
Ignorability Depends on the Model and the Goal
For likelihood-based inference, MAR can permit the missingness mechanism to be omitted from the likelihood under additional conditions. The outcome model must be correctly specified, and its parameters must be distinct from the parameters governing missingness. To see the key factorization, let \(Y_{\mathrm{obs}}\) and \(Y_{\mathrm{mis}}\) denote observed and unobserved outcomes, and let \(R\) denote the missingness pattern. Under MAR, the missingness probability \(g_\psi(R\mid Y_{\mathrm{obs}},Y_{\mathrm{mis}},X)\) does not depend on \(Y_{\mathrm{mis}}\). The observed-data likelihood contribution is then
The missingness factor contains no \(\theta\) under parameter distinctness, so it does not alter likelihood inference for the outcome-model parameters. This is the likelihood meaning of ignorable MAR. It does not assert that the observed participants are a random sample of the population, and it does not establish that an unweighted GEE is unbiased.
Worked Example: MAR Can Be Ignorable for a Likelihood but Not for Naive GEE
Suppose dropout at visit 2 is more likely for participants whose visit-1 outcome was low. The visit-1 outcome is observed for everyone, and dropout does not additionally depend on the unobserved visit-2 outcome after conditioning on visit 1 and baseline covariates. This is MAR: the dropout probability uses observed history, not the missing outcome itself.
For a correctly specified longitudinal outcome likelihood with parameters \(\theta\), and a distinct dropout model with parameters \(\psi\), the factorization above allows likelihood inference about \(\theta\) without modeling the dropout process. But a standard unweighted GEE may still be biased: among those observed at visit 2, low visit-1 outcomes are underrepresented, and visit-1 outcome can predict visit-2 outcome. Consequently the observed visit-2 residuals may not represent the population residuals.
If dropout instead depends on the unobserved visit-2 outcome even after conditioning on baseline covariates and observed history, the process is MNAR. The factorization no longer follows, and the observed data alone generally cannot identify the missingness dependence without additional assumptions. An outcome likelihood that simply ignores dropout is not justified by MAR in that setting.
A special, less problematic case is dropout explained entirely by fully observed covariates included appropriately in the marginal mean model. Conditional on those covariates, observation then does not select on the outcome residual. By contrast, dependence on observed prior outcomes is outcome-dependent selection for the later marginal mean, even though it meets the MAR definition.
Inverse-Probability Weighting for GEE
One way to recover population representation is inverse-probability weighting (IPW). For monotone dropout, define the cumulative observation probability
where the history in this expression is observed for participants who remain under observation through visit \(j\). Each observed outcome is weighted by \(1/\pi_{ij}\): an outcome from a history with a low probability of remaining observed represents more people in the original cohort. A weighted GEE using a working-independence form is
where \(D_{ij}\) is the derivative of \(\mu_{ij}\) with respect to \(\beta\), and \(A_{ij}\) is the working variance for that outcome. This equation is a useful basic form; correlated weighted GEE methods require a compatible construction of the weighted estimating equation rather than an arbitrary insertion of weights into every covariance formula.
The justification is the inverse-probability identity. Under MAR for the current outcome conditional on observed history, and with the correct observation probability, weighting observed contributions by \(R_{ij}/\pi_{ij}\) restores their full-cohort expectation. For an outcome contribution \(h(Y_{ij},X_i)\), conditional on the history used to calculate \(\pi_{ij}\), the selection probability cancels the weight:
For an intercept-only binary mean with a working-independence equation, the derivative and Bernoulli variance factors cancel, and the estimator reduces to the weighted proportion \(\sum R_{ij}Y_{ij}/\pi_{ij}\) divided by \(\sum R_{ij}/\pi_{ij}\).
Worked Example: Recovering a Mean With Known Observation Probabilities
Return to the 200-person hypothetical cohort. At the final visit, participants with \(Y_1=1\) have observation probability \(0.50\), so each observed outcome in that group receives weight \(1/0.50=2\). Participants with \(Y_1=0\) have probability \(1.00\), so their weight is \(1/1.00=1\). The weighted number of positive outcomes is \(2(40)+1(20)=100\), and the sum of weights is \(2(50)+1(100)=200\). Therefore,
This matches the population target in this constructed example. The result relies on the specified observation probabilities being correct, on the observation process being MAR given the history used, and on positive observation probability in every relevant history group. The calculation does not show that weighting can correct MNAR dropout without further assumptions.
Assumptions, Diagnostics, and Alternatives
IPW is not automatic protection from dropout bias. Its validity requires a correctly specified marginal mean model, an adequately specified model for observation probabilities, and positivity: every covariate and history pattern relevant to the target population must have a nonzero chance of being observed. If an observation probability is near zero, its inverse weight is large and a few outcomes can dominate the estimate. Estimated probabilities also add uncertainty, which should be reflected in inference using an appropriate method.
Useful checks include comparing predicted and observed retention across visits and histories, inspecting the distribution of weights and the largest weights, and checking whether some histories have little or no follow-up. Extreme weights signal weak overlap and potentially unstable inference; trimming or truncating weights may reduce variance but changes the procedure and can reintroduce bias. A robust GEE standard error can address dependence-related variance estimation under suitable conditions, but it cannot repair a misspecified observation model or a biased mean equation.
| Approach | How it handles dropout | Main condition or limitation |
|---|---|---|
| Unweighted observed-data GEE | Uses available outcomes without correcting selection | Generally needs outcome-independent observation, or another structure that makes the equation unbiased |
| Likelihood-based outcome model | Integrates over missing outcomes | MAR, distinct parameters, and a correctly specified outcome likelihood support ignorability |
| IPW GEE | Weights observed contributions by inverse observation probabilities | Needs MAR given modeled history, correct probabilities, and positivity |
| MNAR sensitivity or joint models | Explicitly represents dependence on unobserved outcomes | Requires additional, generally untestable assumptions |
No diagnostic can prove that dropout is MAR: the missing outcomes that would distinguish MAR from MNAR are not observed. When MNAR is plausible, conclusions should be examined under scientifically defensible alternative assumptions, rather than treating the MAR analysis as verified by fit alone. The choice between likelihood and weighting also depends on the target and modeling assumptions; one is not universally superior.
Common Pitfall
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Dropout at visit 3 depends on the observed visit-2 outcome but not on the unobserved visit-3 outcome conditional on history. Is this MAR or MNAR? Does that alone establish that an unweighted GEE is unbiased?
- For an observation probability of \(0.25\), what inverse-probability weight is assigned to an observed outcome?
- Name two conditions needed for an ignorable likelihood analysis under MAR.
- Why does a sandwich standard error not correct bias from outcome-dependent dropout?
Solutions. (1) It is MAR because dropout depends on observed history, not the missing outcome after conditioning on that history. This alone does not establish unbiasedness of an unweighted GEE; selection on earlier outcomes can change the distribution of later outcomes among those observed. (2) The weight is \(1/0.25=4\). (3) The outcome likelihood must be correctly specified and the outcome-model and missingness parameters must be distinct; MAR is also required for the factorization. (4) The sandwich estimates uncertainty for the estimator's estimating equation. It does not change the equation's expectation or undo selection bias.