Current-Outcome Versus Lagged-Outcome Dropout Dependence

Why the Timing of Dropout Dependence Matters

In a longitudinal study, dropout can be associated with a participant’s outcome at the visit when follow-up ends, or with the participant’s previously observed outcome. These may sound like small variations in a dropout model, but they imply different distributions for outcomes that were never recorded. If outcomes continue to evolve after dropout, the assumptions also imply different post-dropout trajectories.

The distinction depends on exactly when dropout is defined. Here, \(R_{it}=1\) means participant \(i\)'s outcome at visit \(t\) is observed, and \(R_{it}=0\) means it is missing. Dropout is monotone: once a participant stops, later outcomes are also missing. We define \(D_{it}=1\) when a participant who was observed at visit \(t-1\) drops out before their visit-\(t\) outcome is recorded. Thus \(Y_{it}\) is unobserved for someone with \(D_{it}=1\), while \(Y_{i,t-1}\) is observed. A different study may record the visit-\(t\) outcome before classifying a participant as withdrawn; the model labels must then be adjusted to match that timing.

Write \(H_{i,t-1}\) for the observed outcome and covariate history through visit \(t-1\), and \(X_i\) for baseline covariates. A simple pair of candidate hazards, for participants still under observation after visit \(t-1\), is

These equations are illustrative specifications, not universal models. In an applied comparison, candidate models should use compatible baseline covariates and time effects so that the main difference is which outcome drives dropout. The coefficients need not be numerically comparable: the two outcome variables can have different distributions, and the model parameters may be weakly identified.

What Each Assumption Implies for Missing Outcomes

For a participant with history \(H_{i,t-1}\), let \(f_t(y\mid H_{i,t-1})\) denote the modelled distribution of the current outcome before conditioning on dropout. Under a current-outcome model, define \(\pi_t(y,H)=P(D_{it}=1\mid Y_{it}=y,H_{i,t-1}=H)\). Conditioning on dropout changes the outcome distribution by weighting each possible outcome by its dropout probability:

This follows from Bayes’ rule: the joint density of \(Y_{it}=y\) and dropout, given history, is \(f_t(y\mid H)\pi_t(y,H)\); dividing by the probability of dropout given that history normalizes the density. Outcomes with higher dropout probabilities consequently receive more weight among the missing current outcomes. If \(\pi_t(y,H)\) increases with \(y\), the missing outcomes tend to be higher than the pre-dropout outcomes at that history, though the size of the difference depends on both the outcome distribution and the dropout model.

Under a lagged-outcome model, at a fixed observed history the dropout probability does not vary with \(Y_{it}\). If \(D_{it}\) is conditionally independent of \(Y_{it}\) given that history, then

This is an ignorable, or missing at random, dropout assumption for the current outcome, provided the history includes the variables needed to make the conditional independence statement true and the usual distinct-parameter and likelihood conditions hold. It does not mean dropout is unrelated to outcomes: dropout can vary substantially with the observed last outcome. It means that within a given recorded history, dropout supplies no additional selection on the unobserved current outcome.

Consequences for Post-Dropout Trajectories

A dropout model alone does not specify later outcomes. Trajectory predictions also require an outcome model that describes how outcomes evolve across visits. For example, suppose the conditional mean at the next visit is \(E(Y_{i,t+1}\mid Y_{it})=6+0.5Y_{it}\), with the same relationship for participants after dropout. This is an extrapolation assumption: no later outcomes from the dropout group are available to verify it directly.

The effect may grow, shrink, or reverse over later visits. That depends on the outcome transition, on whether the effect of earlier outcomes persists, and on how the dropout model is specified at each later risk set. A current-outcome dependence assumption does not by itself imply a steadily increasing or decreasing trajectory.

Comparing Models Without Confusing Fit and Extrapolation

Dropout indicators and recorded histories are observed, but the outcome that would have been measured after dropout is not. Consequently, agreement in observed dropout rates does not establish agreement about missing outcomes. The following example separates those two questions.

The current-outcome model requires an outcome distribution and a selection mechanism to jointly describe the observed data and the missing outcomes. Because the current outcomes for dropouts are absent, the strength of information about the dependence parameter can be limited; in some specifications, additional restrictions or sensitivity assumptions are needed. The lagged model is more directly connected to observed histories, but its apparent convenience does not prove that the missingness assumption is true. Both models can be compared as alternative extrapolations, as in the earlier material on selection and pattern-mixture extrapolations, while keeping their observed-data implications distinct from their unobserved-data implications.

Assumptions, Diagnostics, and Failure Modes

A useful comparison begins by stating the assumptions explicitly. In particular, check that the visit order really makes \(Y_{it}\) unobserved at dropout and \(Y_{i,t-1}\) observed. Then specify the outcome model, the dropout risk set, and the history and covariates each hazard conditions on. If the lagged model omits an observed predictor of both dropout and the current outcome, its conditional-independence claim may be inadequate. If the current model assigns high dropout probability to outcomes that are extremely rare under the outcome model, estimates and predictions may be sensitive to model form.

Diagnostics can assess observed-data compatibility, but cannot directly test the current-outcome dependence for dropouts. Useful checks include comparing predicted and observed dropout rates across visits and observed-history strata, examining whether the fitted outcome model describes observed outcomes in relevant histories, and checking that simulated probabilities and predictions are numerically sensible. Large disagreement among plausible current-outcome specifications is a warning that conclusions depend on extrapolation. Conversely, similar fitted dropout rates do not show that the missing-outcome distributions agree.

Keep the main alternatives distinct:

• Current-outcome dependence permits dropout to select directly on the outcome that is missing at that visit. Its missing-outcome distribution follows from the selection weighting and the outcome model.
• Lagged-outcome dependence uses a recorded outcome to explain dropout. Given a sufficiently rich observed history, it implies no additional selection on the current outcome.
• Pattern-mixture specifications describe outcome distributions conditional on dropout pattern directly. They offer a different parameterization of extrapolation; they do not make the unobserved outcomes empirically known.

Neither dependence structure is universally preferable. The current-outcome model is scientifically relevant when people may leave because of their current, unrecorded condition, but the defining dependence is difficult to verify from observed data. The lagged model is easier to relate to recorded histories, but may miss dependence on a worsening current condition that has not yet been measured. Reporting both can clarify how conclusions change; it does not eliminate the need to justify the outcome model, covariates, timing, and sensitivity assumptions.