Why the Timing and Recording of Symptoms Matter
In a clinical follow-up study, a participant may stop attending after symptoms worsen. That description alone does not determine whether attrition is missing at random (MAR) or missing not at random (MNAR). The key questions are when the worsening occurred, whether it was recorded and available for analysis, and what other information the dropout decision depends on after conditioning on the observed history.
For example, a participant might complete a symptom questionnaire, see that the score has worsened, and then withdraw. Alternatively, symptoms might worsen after the last recorded visit, and the participant might withdraw without providing another score. These situations differ because the worsening is observed in the first case and unobserved in the second. A withdrawal reason such as “symptoms worsened” is not enough to settle the classification: the timing and data record determine whether the relevant symptom information belongs to the observed history.
A conditioning history must respect chronology. If a score was completed and entered in the study record before the participant withdrew, it can be included in the observed history, even if it was collected during an unscheduled contact. If symptoms worsened after the last recorded score and the next score was never obtained, that worsening is not part of the observed history. Information known to a clinician or participant is not automatically available for analysis: it must be recorded in the data being analyzed to count as observed here.
The Classification Rule
Write \(H_t^{obs}\) for the observed history available at the dropout decision. It may include \(Y_t\) if that score was recorded before withdrawal. It does not include missing current or future symptom values. In this setting, the MAR condition says that, among participants still under follow-up with the same observed history, dropout does not additionally depend on the unobserved symptom values. In conditional-probability form:
Here \(Y_t^{mis}\) denotes a current symptom value that was not recorded; the later terms represent unrecorded future symptom values. The equality states that, once the available history is held fixed, changing the unobserved symptom values does not change the dropout probability. If dropout still depends on those unobserved values, the mechanism is MNAR relative to this history.
The assumptions in that statement are substantive. The history must contain the information actually used to determine dropout that is available in the analysis, and the comparison concerns participants with the same such history who are still at risk. Adding recorded symptom measures or other recorded predictors can change the classification: a dependence that appeared to involve an unobserved value may be explained by information that is in fact recorded. Conversely, a plausible association with recorded worsening does not establish that there is no additional dependence on unobserved worsening.
The distinction can be summarized in three cases. If dropout responds to recorded worsening only, it can be MAR conditional on that worsening and the other relevant recorded history. If dropout responds to worsening that was not recorded, it is MNAR when that dependence remains after conditioning on the available history. If it responds to both, the recorded component is accounted for by conditioning, but the residual dependence on unrecorded worsening makes the mechanism MNAR.
Worked Classifications
Worked Example: Withdrawal After a Recorded Worsening
Suppose a hypothetical study records a symptom score at each visit, and larger scores indicate more severe symptoms. Of 100 participants still under follow-up, 60 have a recorded score showing worsening since the previous recorded visit; 30 of these participants withdraw before the next scheduled visit. Among the other 40, whose recorded scores do not show worsening, 6 withdraw. Assume for this example that the withdrawal process depends on the recorded worsening category, but not additionally on any symptom value that remains unrecorded.
The dropout proportions in the two recorded-history groups are:
Thus, in this hypothetical data, 50.0% of participants with recorded worsening withdraw, compared with 15.0% without recorded worsening. Under the stated assumption, the relevant conditioning history includes the last recorded symptom score, whether that score shows worsening, the other available recorded history, and being at risk immediately before the decision. The process is MAR conditional on that history: the withdrawal probability varies with an observed measure, but does not also depend on unrecorded symptoms after conditioning on what was recorded.
This is not MCAR, because the stated process varies with recorded worsening. The rates show an association between the recorded history and dropout; by themselves, they cannot verify the assumption that no unrecorded worsening also affects withdrawal. The classification follows from the scenario’s stated mechanism, not from the two observed rates alone.
Worked Example: Withdrawal After Unrecorded Worsening
In a second hypothetical study, all participants in the example have the same relevant recorded history: their last recorded symptom score is stable, and they have the same recorded baseline and treatment information. Between visits, some experience worsening that is not reported or measured before withdrawal. Suppose the dropout process is 50% for those with this unrecorded worsening and 8.33% for those without it. For illustration, imagine 40 participants in each unobserved group, with 20 and 3 withdrawals, respectively.
The two dropout proportions are:
The stated 8.33% rate would correspond to \(3.33\) withdrawals out of 40, which is not a possible count. Therefore use the internally consistent hypothetical rates given by the counts: 50.0% versus 7.50%. With identical observed histories, the difference is attributed in the scenario to whether symptoms worsened without being recorded. Conditioning on the last stable score and the other available information does not remove that dependence. The mechanism is therefore MNAR relative to the recorded history.
The unobserved groups in this illustration describe the assumed data-generating process; they cannot generally be formed from the study’s observed records when the worsening was never measured. That limitation matters: the example clarifies what the MNAR claim means, rather than offering a direct observed-data test for it.
Worked Example: Both Recorded and Unrecorded Worsening
A third hypothetical process depends on both a recorded worsening category from the last completed visit and worsening after that visit which is not recorded before dropout. The table gives the number of participants and withdrawals in each combination. The unrecorded-worsening category is part of the hypothetical mechanism, not necessarily an observable grouping in a real analysis.
| Recorded worsening | Unrecorded worsening | Participants | Withdrawals | Dropout proportion |
|---|---|---|---|---|
| No | No | 20 | 2 | 2/20 = 0.100 |
| No | Yes | 20 | 8 | 8/20 = 0.400 |
| Yes | No | 20 | 6 | 6/20 = 0.300 |
| Yes | Yes | 20 | 14 | 14/20 = 0.700 |
Within the no-recorded-worsening group, the dropout proportion is 40.0% with unrecorded worsening and 10.0% without it. Within the recorded-worsening group, it is 70.0% with unrecorded worsening and 30.0% without it. Thus the stipulated process varies with unrecorded worsening even after comparing participants with the same recorded-worsening category. It also varies with recorded worsening: averaging across the two unrecorded categories gives \(10/40=25.0\%\) without recorded worsening and \(20/40=50.0\%\) with it.
The correct classification is MNAR conditional on the available history. Conditioning on recorded worsening accounts for that part of the process but cannot remove the remaining dependence on worsening that was never measured. The process has both observed-history dependence and unobserved-outcome dependence; MAR does not follow merely because one of those components is observed.
Why the Conditioning Argument Works
The classification rule follows directly from what the conditional probabilities compare. Hold fixed the fact that the participant is still at risk and the complete observed history at the decision. Then compare dropout probabilities across different possible values of the unrecorded symptoms. If those probabilities are equal, the unobserved symptoms provide no additional information about dropout once the observed history is known. If they differ, dropout depends on information outside that history.
for any relevant unobserved values \(u\) and \(v\), with the analogous condition applying to other missing outcomes. The equality is a statement about the dropout mechanism, not a claim that recorded worsening and dropout must be unrelated. In the first worked example, the probabilities differ across recorded-history groups, yet within each fixed observed history the assumed dropout probability does not vary with unrecorded symptoms. In the second and third examples, it does.
This argument also explains why knowing that dropout is predictable does not settle the classification. If recorded worsening predicts withdrawal, the mechanism may still be MAR, provided the conditioning history captures the relevant observed information and there is no further dependence on unobserved outcomes. The important question is what dropout depends on after conditioning, not whether any variable can predict it.
Diagnostics, Alternatives, and Limits
Observed-data diagnostics can check whether dropout rates differ across recorded symptom histories. For example, a table of dropout proportions by last recorded score or recorded change can reveal patterns worth investigating. These comparisons help assess whether a proposed history is informative and whether the timing of recorded measures has been handled correctly. They cannot show that dropout is independent of symptoms that were never observed: those values are not available for the required within-history comparisons.
A careful mechanism assessment therefore begins with the record and timeline, not just the stated reason for withdrawal. Ask whether the symptom report was collected before the decision, whether it appears in the analysis data, and whether it measures the worsening that could have influenced withdrawal. Also specify the conditioning history. “MAR” without saying what information is conditioned on is incomplete, because adding recorded history can change the relevant conditional statement.
The main contrast is between a MAR account, in which the relevant dropout dependence is captured by observed history, and an MNAR account, in which dependence on unobserved outcomes remains. These are assumptions about how data became missing, not labels that can be assigned to a dataset solely by inspecting observed outcomes or the recorded reason for leaving. The introductory material on Overlaps Missing Data gives the broader missing-data context; here the emphasis is the symptom-specific conditioning decision in a follow-up study.
A common mistake is to reason, “Symptoms worsened, so dropout is MNAR.” This is wrong when the worsening was recorded and is included in the conditioning history: dependence on that observed worsening can be MAR. The opposite shortcut is also wrong: “The last recorded score predicts dropout, so dropout is MAR.” Prediction by observed history does not rule out additional dependence on unrecorded worsening. In both cases, correct the reasoning by asking whether dropout still depends on symptom values that are missing after conditioning on all relevant recorded information.
Finally, the classification depends on the study’s measurement and recording procedures. If a worsening score is collected at an exit contact before withdrawal, it may be observed for that decision even when the scheduled visit is missed. If worsening occurs only after the last recorded measurement, it is unobserved for the analysis. Ambiguous chronology, incomplete recording, and an overly narrow conditioning history can therefore produce an incorrect classification. No observed-data table can, on its own, eliminate these uncertainties.
Check Your Understanding
For each question, state the conditioning history and classify the mechanism under the assumptions given.
- A participant completes and records a worsened symptom score, then withdraws. The stipulated dropout process depends on that score only. Is the mechanism MAR or MNAR relative to the recorded history?
- Two participants have the same recorded history, but one withdraws after symptom worsening that was never measured. The scenario stipulates that this unrecorded worsening affects withdrawal. Classify the mechanism.
- Dropout depends on both the last recorded worsening and a new, unrecorded symptom worsening. Does conditioning on the recorded score make the mechanism MAR?
- A study records “symptom worsening” as a withdrawal reason, but does not say when the worsening occurred or whether a symptom score was recorded. Can the mechanism be classified from that reason alone?
- Dropout proportions are higher among participants with worse recorded scores. Does this observed association prove that the mechanism is MAR?
Solutions. (1) MAR, conditional on the recorded score and other relevant recorded history, because the scenario stipulates no remaining dependence on unobserved symptoms. (2) MNAR relative to the shared observed history, because dropout still depends on unrecorded worsening. (3) No. Conditioning accounts for dependence on the recorded score but not the additional dependence on the unrecorded worsening, so the mechanism is MNAR. (4) No. The chronology and whether the symptom information was recorded are needed to define the conditioning history. (5) No. The association shows that recorded history predicts dropout, but cannot rule out additional dependence on unobserved symptoms; the mechanism requires a conditional assumption beyond that observed association.