Dropout and the History Available at a Visit
In longitudinal follow-up, dropout is often a sequence of decisions: at each visit, a participant who has remained under observation may or may not provide the next outcome. A useful question is not simply whether dropout is related to outcomes. It is which outcomes were already recorded when the dropout decision occurred? If dropout depends on that recorded history, the dependence can be compatible with missing at random (MAR), even though dropout is not completely random.
Consider a study with visits \(1,\ldots,T\). Let \(Y_{it}\) be the outcome for participant \(i\) at visit \(t\), and let \(R_{it}=1\) if that outcome is observed and \(R_{it}=0\) otherwise. For monotone dropout, once \(R_{it}=0\), outcomes at later visits are also unobserved. A participant is at risk of dropout before visit \(t\) if \(R_{i,t-1}=1\); at the first visit, all enrolled participants are at risk. Let \(X_i\) denote covariates recorded before follow-up, and let \(C_{i,t-1}\) denote any additional covariates recorded before the visit-\(t\) dropout decision. These covariates are included in the recorded history only when their values are available at that time.
The definition concerns the underlying probability mechanism, not just the observed pattern of responses in one sample. It also depends on timing: a measurement can be included in the conditioning history only if it was recorded before the decision to drop out. For example, a symptom score collected at visit \(t-1\) can be part of the recorded history for a decision before visit \(t\). A score that would have been collected at visit \(t\), but was not collected because the participant dropped out first, is not part of that history.
What MAR Allows—and What It Rules Out
MAR allows dropout probability to vary across recorded histories. Participants with a low recorded score, for example, may be more likely to miss the next visit than participants with a high recorded score. That dependence is compatible with MAR if, among participants with the same recorded history, the dropout probability does not additionally depend on the unobserved current or future outcomes.
By contrast, missing completely at random (MCAR) requires the observation process to be independent of the complete data values, under the chosen definition of the complete-data vector. MAR is less restrictive: it permits dependence on information that is observed. If dropout continues to depend on an outcome that is unobserved at the time of the decision, even after conditioning on the recorded history, the MAR condition fails.
| Mechanism label | What dropout may depend on | Key condition |
|---|---|---|
| MCAR | No complete-data values | Observation is independent of the complete-data values. |
| MAR given recorded history | Outcomes and covariates recorded before the decision | Given that history, dropout does not additionally depend on unobserved outcomes. |
| Not MAR given recorded history | At least one unobserved outcome, even after conditioning on the history | The conditional independence in the definition fails. |
These labels describe assumptions about the mechanism. Observed associations between dropout and recorded history can be informative, but they do not reveal whether dropout also depends on outcomes that were never observed. That part of the mechanism cannot generally be checked using observed data alone.
A Sequential Factorization
The sequential form of MAR can be understood by factoring the probability of a response pattern into visit-specific probabilities. Among participants who remain under observation through visit \(t-1\), define the response probability \(p_{it}\) as the probability of \(R_{it}=1\) given the relevant information. By the chain rule, the probability of remaining observed through visit \(k\) is the product of the successive conditional response probabilities. If the participant drops out before visit \(k\), the final factor is the conditional probability of not responding at that visit.
Here \(Y_i=(Y_{i1},\ldots,Y_{iT})\), and at \(t=1\) the risk-set condition is understood to hold for everyone. The chain rule is an identity; it makes no missingness assumption. Under sequential MAR, each visit-specific factor can be expressed using only the history recorded before that visit’s decision. Thus, for a participant still observed at \(t-1\), the response factor becomes
This factorization explains why the risk set matters: the visit-specific probability is defined among participants who have reached that visit under observation. It also shows precisely what MAR removes from the conditioning set: dependence on the unobserved outcomes, not dependence on the recorded history.
Worked Example: Dropout Probabilities Depend on the Last Recorded Score
Suppose a hypothetical study has a visit-2 dropout decision among participants with a recorded visit-1 score. Assume the underlying response probabilities are \(0.70\) for participants whose visit-1 score is below 50 and \(0.90\) for those whose score is at least 50. Further assume that within each score group these probabilities do not vary with the unobserved visit-2 or later outcomes.
For the lower-score group, the response probability is \(0.70\), so the dropout probability is \(1-0.70=0.30\). For the higher-score group, the dropout probability is \(1-0.90=0.10\). The probabilities differ across recorded histories, but the visit-1 score is available before the visit-2 decision.
Classification. Under the stated assumption that unobserved outcomes have no additional effect within either recorded-score group, this mechanism is MAR given the recorded history. If these are the actual conditional probabilities in the population, their difference also means the mechanism is not MCAR with respect to the complete-data values: dropout depends on the recorded score. The numerical probabilities and the assumption about unobserved outcomes, not a comparison of sample rates, justify the classification.
Worked Example: Different Sample Rates Do Not Establish the Mechanism
In a hypothetical follow-up study, 12 of 30 participants in a low-score group drop out before the next visit, while 5 of 50 in a high-score group drop out. Calculate the observed proportions and state what they establish.
The observed dropout proportion in the low-score group is $$ \frac{12}{30}=0.400, $$ or 40.0%. In the high-score group it is $$ \frac{5}{50}=0.100, $$ or 10.0%. The difference between these observed proportions is \(0.400-0.100=0.300\), or 30.0 percentage points.
These figures describe the sample: its observed dropout proportions differ across the two recorded-history groups. They are descriptively inconsistent with equal rates, but they do not establish that the underlying conditional dropout probabilities differ. Such a sample difference can occur by chance even if the underlying probabilities are equal. Nor do the counts tell us whether dropout depends on unobserved outcomes after conditioning on the recorded history. Therefore, the counts alone do not classify the underlying mechanism as MCAR or MAR. A mechanism classification would require assumptions or information about the underlying probabilities, including whether unobserved outcomes have an additional effect.
Worked Example: Dependence on an Unobserved Visit Outcome
Suppose a hypothetical study has two participants with the same recorded baseline covariates and the same recorded visit-1 history. The visit-2 dropout probability is \(0.50\) for participants whose visit-2 outcome would be low and \(0.10\) for participants whose visit-2 outcome would be high. Assume these are the actual probabilities under the mechanism.
Because visit-2 dropout occurs before the visit-2 outcome is recorded, that outcome is unobserved for those who drop out. The two probabilities differ despite identical recorded histories: \(0.50\ne0.10\). Thus the conditional dropout probability still depends on an unobserved outcome after conditioning on the recorded history.
Classification. This mechanism is not MAR given the recorded history. The conclusion follows from the stipulated underlying mechanism, not from sample proportions. If the visit-2 outcome had instead been recorded before the dropout decision, it could belong to the information available for conditioning; timing is essential to the classification.
Diagnostics, Assumptions, and Common Pitfalls
A useful diagnostic is to tabulate observed dropout proportions by summaries of the recorded history, such as categories of the last recorded score or a covariate measured before the decision. A visit-specific regression of response on recorded history can also describe which observed variables are associated with response. These analyses help reveal whether a model that assumes no relationship with recorded history is implausible. They do not prove MAR: the missing outcomes are unavailable for checking whether they would further predict dropout.
A practical analysis also requires adequate support. If a recorded-history group has no participants who remain observed, outcomes after that history cannot be learned directly from observed follow-up in that group. This is a positivity or overlap problem, not a redefinition of MAR. Even when MAR is a reasonable working assumption, very sparse risk sets can make estimates unstable.
The most common mistake is to treat unequal observed dropout proportions as proof that the true mechanism is not MCAR, or as proof of MAR. Both conclusions go beyond what the counts establish. The correction is to distinguish the observed sample pattern from the underlying conditional probabilities. A mechanism can be classified from specified probability assumptions; sample rates provide evidence about those assumptions, not certainty.
A second pitfall is conditioning on a variable simply because it exists in the study database. For the definition at visit \(t\), a covariate is part of the recorded history only if it was available before the dropout decision. A measurement scheduled for the missed visit cannot be treated as observed history for a participant who dropped out before it was collected.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Before visit 3, dropout probability varies with the recorded visit-2 score but, within score levels, does not depend on visit-3 or later outcomes. Under the stated mechanism, is this MAR given the recorded history?
- In a sample, dropout proportions are 18% and 11% in two recorded-history groups. What can and cannot be concluded from this difference alone?
- A variable is measured at visit 3, after the decision to attend visit 3 has already been made. Can it be part of the recorded history for that decision?
- At a fixed recorded history, the stipulated dropout probability is \(0.35\) when the unobserved current outcome is low and \(0.35\) when it is high. Does this dependence violate MAR as stated? Explain.
Solutions. (1) Yes. The visit-2 score was recorded before the visit-3 decision, and the stipulated mechanism has no additional dependence on unobserved outcomes within score levels. (2) The observed proportions differ by 7 percentage points in this sample. They do not alone establish unequal underlying probabilities, nor do they show whether dropout depends on unobserved outcomes after conditioning on history. (3) No. It was not available before the decision, so it is not in the recorded history for that decision. (4) No. The probabilities are equal across the two specified current-outcome values, so this mechanism does not show additional dependence on that outcome at the fixed history. MAR still requires that no other unobserved outcome affects dropout after conditioning on the recorded history.