Dropout After A Poor Last-Observed Outcome

Why the Last Recorded Score Matters

A participant may be more likely to leave a follow-up study after reporting severe symptoms. That fact alone does not make dropout depend on a missing outcome: if the severe score was recorded before the participant left, it is observed information. The classification depends on what outcomes and other variables are included in the conditioning history, and on whether dropout also depends on outcomes that were not observed.

Consider visits \(1,\ldots,T\). Let \(Y_{it}\) be participant \(i\)'s symptom score at visit \(t\), with larger values indicating worse symptoms, and let \(R_{it}=1\) if that score is observed and \(R_{it}=0\) otherwise. For simplicity, suppose dropout is monotone: after the first missed visit, no later outcomes are recorded. Let \(D_i=t\) mean participant \(i\) first misses visit \(t\), so the last recorded score is \(Y_{i,t-1}\). The participant is at risk of dropout before visit \(t\) if they were observed through visit \(t-1\).

The word “poor” describes the clinical direction of the score, not a special statistical category. The threshold for calling a score poor may be useful for communication, but the MAR classification does not require a threshold: a dropout probability can vary smoothly with the recorded score. The key is temporal order. \(Y_{i,t-1}\) is observed before dropout at visit \(t\); \(Y_{it}\) is missing for a participant who drops out before that visit.

Visit-Specific Dropout and the MAR Condition

For a participant at risk before visit \(t\), define the dropout probability conditional on the recorded history as

Here \(D_i\ge t\) means the participant has not dropped out before visit \(t\), and so their outcomes through visit \(t-1\) are available under the monotone-dropout setup. Dropout can be more likely for participants whose last score is poor: \(q_{it}\) may increase as \(y_{i,t-1}\) increases. This dependence is on an observed value. If, given this history, dropout has no further dependence on \(Y_{it}\) or other missing outcomes, the visit-specific mechanism is MAR.

For this conclusion to describe the whole dropout process, the assumption must apply at each visit where dropout can occur. In particular, the process must not have an additional dependence on a missing current or future outcome that is left out of the stated conditional probability. The fact that a participant’s prior score predicts dropout is compatible with MAR; it does not by itself show that the MAR assumption is true.

Worked Examples

Why the Classification Follows

The justification is the distinction between a recorded history and the outcomes still missing after dropout. For a participant who drops out before visit \(t\), the observed outcome history is \(Y_{i1},\ldots,Y_{i,t-1}\), while \(Y_{it},\ldots,Y_{iT}\) are missing. If the conditional dropout probability is a function only of the recorded history, then changing the values assigned to those missing outcomes does not change that probability.

The same reasoning must hold for the dropout and continuation probabilities at each subsequent visit. Under monotone dropout, the probability of a complete observation pattern can be written as a product of these visit-specific probabilities. If every factor depends only on information observed before its corresponding dropout decision, the product depends on the observed history for that pattern, not on the missing outcomes. This is why a score-driven dropout mechanism can be MAR when the score is recorded before dropout.

Assumptions, Comparisons, and Failure Modes

This classification relies on a clear visit order and accurate information about which outcomes were actually recorded. It also assumes that the relevant last score is available in the history for the participants at risk. If dropout is decided before the score is measured, or the score is missing for another reason, it cannot be treated as an observed predictor for that decision without further assumptions. With several recorded predictors, the same principle applies: dependence on recorded history is compatible with MAR if the mechanism has no additional dependence on missing outcomes.

The contrast with MCAR is that MCAR requires the observation pattern to be independent of outcomes, whereas MAR permits dependence on observed outcomes. The contrast with MNAR is that MNAR involves dependence on outcomes that remain unobserved after conditioning on the observed data. These labels describe a mechanism relative to a stated conditioning set, as emphasized in the earlier material on why mechanism labels depend on the conditioning set.

A frequent error is to say, “Dropout is related to symptoms, so it must be MNAR.” That reasoning does not check whether the symptom score was observed before dropout. The correction is to identify the decision time and ask whether the relevant score belongs to the recorded history. A second error is to see a strong association between poor scores and dropout and declare MAR established. That is also too strong: the association is consistent with observed-history dependence, but the missing outcomes cannot be used to rule out additional dependence on them.

Useful checks are therefore conceptual as well as empirical. Confirm that the purported predictor was measured before the dropout decision; distinguish the last recorded score from the score at the missed visit; and state which history is being conditioned on. Observed dropout rates can describe how dropout varies across recorded histories, but they cannot, by themselves, determine whether missing outcomes also influenced leaving. The MAR assumption is not directly confirmed by observing the recorded data alone.