Dropout Driven By An Unobserved Future Trajectory

When the Reason for Leaving Lies Ahead

A participant may withdraw because they expect their condition to worsen, or because they expect to improve and see little reason to continue attending visits. In either case, withdrawal can depend on what the participant’s outcome trajectory would be after the last recorded visit. Those future outcomes are unavailable precisely for the participants who withdraw. This creates a form of selection that cannot generally be explained using only the recorded outcome history.

The central distinction is between a future trajectory that is unobserved and a future trajectory that is irrelevant. An unobserved outcome may still be part of the scientific target and may be associated with withdrawal. The fact that dropout is predictable in some broad sense, or that a participant reports a reason for leaving, does not by itself establish that the mechanism is ignorable.

The future outcomes here are part of a specified target outcome process, not a claim that outcomes after withdrawal are actually observed or necessarily unaffected by withdrawal. If the scientific target does not define outcomes after withdrawal, the future-trajectory statement needs a different target or an explicit model for what those outcomes mean. For the standard missing-data classification, treat the full target outcome vector as defined, with some of its values missing after dropout.

A Selection Model for Future-Trajectory Dependence

For monotone dropout, define \(D_{t+1}=1\) when a participant who is observed through visit \(t\) does not provide the outcome at visit \(t+1\), and \(D_{t+1}=0\) otherwise. The model is considered only for the risk set \(R_t=1\), meaning the participant remains under observation through visit \(t\). A compact selection model is

where \(Y_{t+1:T}=(Y_{t+1},\ldots,Y_T)\), \(S_t=g(Y_{t+1:T})\) is a prespecified summary of that future trajectory, and \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). For example, \(S_t\) might indicate whether a clinically important worsening would occur, or might measure the rate of change over later visits. The function \(\alpha_t(H_t)\) allows dropout risk to vary with visit and recorded history. If larger \(S_t\) means more future worsening, \(\beta>0\) represents higher dropout odds for worse future trajectories; if \(S_t\) instead measures improvement, the sign must be interpreted accordingly.

This model is a way to express a mechanism, not a procedure that can generally be estimated from the observed outcomes alone. Its assumptions include a defined target for the future outcomes, the stated risk set and dropout event, and a meaningful summary \(S_t\). It also assumes the model’s conditional probabilities are correctly specified if numerical probabilities are to be interpreted literally. A different function of the trajectory, or dependence on several features rather than one summary, may be more appropriate.

For comparison, MAR given the recorded history requires that, conditional on \(H_t\), the probability of dropout does not depend on the missing outcomes. In this setting the defining condition is \(P(D_{t+1}=1\mid R_t=1,H_t,Y_{t+1:T})=P(D_{t+1}=1\mid R_t=1,H_t)\). MCAR is more restrictive: dropout does not depend on the outcome history or future outcomes. A nonzero \(\beta\) in the model above generally violates MAR given \(H_t\), and hence is nonignorable for analyses that require this MAR condition. The classification always refers to the conditioning information and target outcomes being considered.

Why the Dependence Is Nonignorable

The key justification is a conditional-probability statement. If \(S_t\) is a function of future outcomes and the dropout probability varies with \(S_t\) after fixing \(H_t\), then the missingness mechanism depends on outcomes that are unobserved for dropouts. The relevant MAR equality therefore fails, except in special cases such as no actual variation in the dropout probability across possible values of \(S_t\).

A related view is to compare the distribution of future outcomes among dropouts with its distribution in the risk set. Write \(f(s\mid H_t=h,R_t=1)\) for the distribution of \(S_t\) before the dropout event, at a fixed recorded history \(h\). Bayes’ rule gives

When dropout probability varies with \(s\), the numerator weights some trajectories more heavily than others. The distribution among dropouts is therefore generally tilted relative to the distribution before dropout. This is the selection effect: the people whose future outcomes are missing need not represent the future-outcome distribution of the full risk set, even among participants with the same recorded history.

This result does not say that every sample will show an obvious difference, or that the association’s direction can be determined from observed outcomes. It says that under the stated model, the selection probabilities depend on missing outcomes. The observed data alone generally cannot identify that dependence without additional assumptions or information.

Separating Future-Trajectory Dependence From Current-Outcome Dependence

A last observed outcome can be poor and predict withdrawal, yet that alone does not establish dependence on an unobserved future trajectory. If dropout depends only on the recorded current outcome and other elements of \(H_t\), it is compatible with MAR given that history. Conversely, participants can have the same current outcome but different latent prognoses; if their dropout probabilities differ with those prognoses, the current outcome does not capture the relevant dependence.

The comparison with other mechanisms can be summarized as follows. The table refers to the stated target outcomes and conditions on the recorded history \(H_t\).

Diagnostics, Limits, and a Common Mistake

There is an important diagnostic limit: for a participant who withdraws, the future outcomes needed to confirm their future trajectory are not observed. Observed-data summaries can reveal patterns in dropout by recorded history, visit, treatment group, or recorded withdrawal reason, but they cannot directly compare dropout probabilities across the missing future trajectories. A plausible narrative about anticipation is not a measured trajectory, and a lack of an observed pattern does not establish that trajectory dependence is absent.

Useful checks include verifying the risk set and timing of dropout, examining whether dropout differs across the recorded histories available at that visit, and checking whether the proposed future summary has a defensible scientific interpretation. Such checks can expose errors in the setup or show that observed history predicts dropout. They cannot, by themselves, distinguish MAR from dependence on future outcomes that remain unobserved. Sensitivity analysis or additional information may be needed to examine how conclusions change under different assumptions about that dependence; no particular sensitivity procedure is required to define the mechanism.

Common mistake: “Participants with worse expected outcomes are more likely to leave, but because that expectation is predictable from their recorded history, dropout is MAR.” This is not necessarily correct. Predictability from recorded history is not the test. The test is whether, after conditioning on that history, dropout still depends on the actual target outcomes that are missing. If the recorded history fully captures the relevant dependence, the mechanism may be MAR given that history. If two participants with the same recorded history can have different future outcomes and different dropout probabilities, the dependence is not removed by conditioning on that history.

The mechanism is also not a causal claim that future outcomes make participants withdraw. A private expectation, an unmeasured disease feature, or another process could be associated with both withdrawal and the future trajectory. For the missing-data classification, the key issue is the conditional relationship between the dropout indicator and the unobserved target outcomes, not the causal story behind that relationship.