Why the Order of Missing Visits Matters
In a longitudinal study, an outcome may be scheduled at several visits for each participant. Some participants provide outcomes at every visit; others stop contributing outcomes, while still others miss a visit and return later. These situations can look similar if we merely count how many outcomes are missing. Their order, however, distinguishes monotone dropout from intermittent missingness.
This distinction is made from the sequence of observation indicators across scheduled visits. It describes the pattern of observed and missing outcomes, not the reason for that pattern. A participant can have a monotone pattern whether the final missing visits arose from withdrawal, illness, or another circumstance; the indicator sequence by itself does not tell us which occurred.
The indicators are ordered by the scheduled visits \(1,\ldots,J\). A complete sequence of ones is monotone: no missing visit occurs, so the condition “once a missing visit occurs” is never violated. A sequence of zeros is also monotone, as are sequences that start with one or more observed visits and then consist entirely of zeros. Thus monotone dropout does not require that a participant first have an observed outcome; it is a property of the indicator sequence.
The word “dropout” in this classification refers to the shape of the observed-data pattern. It does not establish that a participant formally withdrew, permanently left the study, or was unable to attend a particular visit. Those are questions about the circumstances surrounding follow-up. Here, the operational question is narrower: after a missing scheduled outcome, is there any later observed outcome?
Checking the Sequence
A direct way to classify a pattern is to scan from the first scheduled visit to the last and look for a transition from \(0\) to \(1\). If one occurs, a missing visit was followed immediately by an observed visit, so the sequence is intermittent. If there is no such transition, the indicators never increase; once the sequence reaches zero, it stays at zero, and the pattern is monotone.
The same check can be written as an adjacent-visit condition. Since each indicator is either zero or one, \(R_{i,j+1}\leq R_{ij}\) allows the pairs \((1,1)\), \((1,0)\), and \((0,0)\), but not \((0,1)\). A transition from observed to missing is permitted; a transition from missing back to observed is exactly what violates monotonicity.
Why the Adjacent-Visit Rule Works
The equivalence between the verbal definition and the adjacent-visit inequality follows from the binary nature of the indicators. If all visits after a missing visit are also missing, then whenever \(R_{ij}=0\), the next indicator must be zero. Whenever \(R_{ij}=1\), the next indicator can be either zero or one. In both cases, \(R_{i,j+1}\leq R_{ij}\).
Conversely, suppose the inequality holds for every adjacent pair. If \(R_{ij}=0\), then \(R_{i,j+1}\leq 0\). Because an indicator cannot be negative, \(R_{i,j+1}=0\). Applying the same reasoning at visit \(j+1\) gives \(R_{i,j+2}=0\), and repetition shows that every later indicator is zero. Therefore the inequality for every adjacent pair ensures that a missing visit cannot be followed by a later observed visit. This proves the equivalence.
Worked Example: Classifying Several Indicator Sequences
In a hypothetical four-visit study, five participants have the following observation-indicator sequences:
A: \(1,1,1,1\). Every visit is observed. All adjacent pairs satisfy the inequality, so this complete sequence is monotone.
B: \(1,1,0,0\). The transition at visits 2 to 3 is \(1\) to \(0\), and the sequence remains zero afterward. Every adjacent pair satisfies \(R_{i,j+1}\leq R_{ij}\), so this is monotone dropout.
C: \(1,0,1,0\). At visits 2 to 3, the pair is \(0,1\), so \(1\not\leq 0\). The participant has a missing visit followed by an observed visit; this is intermittent missingness.
D: \(0,0,1,1\). The transition at visits 2 to 3 is \(0\) to \(1\), so the sequence is intermittent. The fact that the later visits are both observed does not undo the return after missingness.
E: \(0,0,0,0\). There is no observed visit, but there is also no missing-to-observed transition. The sequence is monotone under the indicator definition. This classification alone does not explain why all outcomes are missing.
The classification depends on the ordered pattern, not the number of ones. For example, B and C each have two observed outcomes, but B is monotone and C is intermittent.
Worked Records and Edge Cases
In practice, the indicator sequence is usually made from visit-level records. The first step is to order visits according to the study schedule and encode whether the outcome was observed at each visit. A visit should not be called observed merely because the participant was contacted or attended: the relevant indicator concerns whether the outcome under consideration was actually recorded.
Worked Example: Classifying Visit Records
Suppose a hypothetical study schedules a symptom score at visits 1 through 5. Participant L has recorded scores at visits 1, 2, and 3, but not at visits 4 and 5. Participant M has scores at visits 1, 3, and 5, but not at visits 2 and 4. Write and classify each indicator sequence.
For L, the indicators are \(1,1,1,0,0\). The adjacent pairs are \((1,1)\), \((1,1)\), \((1,0)\), and \((0,0)\). Each next indicator is less than or equal to the preceding one, so L has a monotone pattern.
For M, the indicators are \(1,0,1,0,1\). The pair at visits 1 to 2 is \((1,0)\), but the pair at visits 2 to 3 is \((0,1)\). Since a later score is recorded after a missing visit, M has intermittent missingness.
The visit numbering matters: the missing score at visit 2 cannot be erased by noting that M has three scores overall. Nor does a recorded score at visit 3 make the missing visit 4 a monotone pattern; the later score at visit 5 establishes a return after missingness.
Some sequences deserve particular care. A participant observed at visits 2 and 3 but not at visit 1 has sequence \(0,1,1\), which is intermittent according to this definition: a missing scheduled outcome is followed by an observed one. Whether visit 1 was actually part of that participant’s observation period is a design question. The indicator sequence must represent the visits that were scheduled and relevant for that participant; do not insert visits at which measurement was not intended.
Similarly, a complete sequence \(1,1,1\) is monotone even though there is no dropout. Monotonicity is not a synonym for “the participant dropped out.” It is a structural property that includes complete observations as a special case. If an analysis reports monotone dropout patterns, it may separately describe complete cases and patterns ending in missing outcomes, but that subdivision does not change the definition.
Worked Example: Distinguishing a Missed Visit From a Return
Consider two hypothetical participants scheduled for six visits. Participant P has indicators \(1,1,1,0,0,0\). Participant Q has indicators \(1,1,0,0,1,0\). Determine whether either sequence is monotone and identify the decisive transition.
For P, the adjacent pairs are \((1,1)\), \((1,1)\), \((1,0)\), \((0,0)\), and \((0,0)\). No pair is \((0,1)\), so \(R_{i,j+1}\leq R_{ij}\) at all five transitions. P has a monotone pattern, with the last observed visit at visit 3.
For Q, the indicators at visits 3 and 4 are both zero, but the pair from visit 4 to visit 5 is \((0,1)\). Therefore Q has intermittent missingness, even though visit 6 is missing again. A later return is sufficient to make the sequence intermittent; subsequent missingness does not restore monotonicity.
The “last observed visit” description is useful for P because all outcomes after visit 3 are missing. For Q, naming only the last observed visit (visit 5) would conceal the return after earlier missing visits. The full sequence, or at least the locations of all missing-to-observed transitions, is needed to represent the pattern accurately.
Diagnostics, Comparisons, and Limits
For data checking, the adjacent-pair rule provides a simple diagnostic. For each participant, inspect the indicators in scheduled order, or compute whether any adjacent pair equals \((0,1)\). A useful summary is the number of such transitions per participant: zero means monotone under this definition, while one or more means intermittent. If records are stored as outcome values rather than indicators, first verify that the coding distinguishes a truly missing outcome from a legitimate recorded value such as zero.
Common warning signs include visits sorted by calendar date rather than scheduled order, missing values mistakenly coded as zeros, and a visit marked observed even though the outcome was not collected. A sequence can also look monotone because only the final observed visit was retained in a summary, while the underlying records contain a later return. When possible, classification should be checked against the visit-level indicators rather than inferred from a single dropout label.
Monotone and intermittent patterns are not competing explanations for why data are missing. They classify when outcomes are observed. A monotone pattern can arise under different missingness mechanisms, and intermittent missingness does not by itself establish that missingness is random or nonrandom. The mechanisms and assumptions used for analysis are separate questions.
The distinction is useful because the pattern of available outcomes constrains which observations exist at each visit and how repeated records are organized. Monotone patterns have a simple “observed through some point, then missing” shape, whereas intermittent patterns include later observations after a gap. Still, classifying the pattern does not determine an appropriate statistical model, justify discarding participants, or prove permanent withdrawal. Those decisions require information beyond the sequence.
| Indicator pattern | Classification | Reason |
|---|---|---|
| \(1,1,1,1\) | Monotone | No missing visit occurs |
| \(1,1,0,0\) | Monotone | After the first zero, every later indicator is zero |
| \(0,0,0,0\) | Monotone | No zero is followed by a one |
| \(1,0,1,0\) | Intermittent | A zero is followed by a one |
| \(1,0,0,1\) | Intermittent | A later observed visit follows missing visits |
A frequent mistake is to define monotone dropout as “every observed visit is followed only by missing visits.” Taken literally, that would incorrectly rule out a sequence such as \(1,1,1,1\), because an observed visit is followed by another observed visit. The correct condition concerns what happens once a missing visit occurs: all later scheduled visits must also be missing. Equivalently, each next indicator must be less than or equal to the current one.
Another mistake is to call any participant with at least one missing outcome a dropout. A sequence such as \(1,0,1,1\) includes missingness but not monotone dropout; the later observed outcomes make it intermittent. The indicator pattern should be stated explicitly before using a label, especially when a single word such as “dropout” could be mistaken for a documented study event.
Check Your Understanding
Classify each pattern using the adjacent-visit rule, then compare your reasoning with the solutions.
- Is \(1,1,0,0,0\) monotone or intermittent? State the transition that determines the answer.
- Classify \(1,0,0,1,1\). Does the final run of observed visits change the classification?
- A participant has no recorded outcome at any of four scheduled visits. Is the indicator sequence monotone under the definition used here? Explain.
- Explain why \(1,0,1,0\) and \(1,1,0,0\) are not in the same category, even though each has two observed visits.
- What does an intermittent indicator sequence establish, and what does it not establish about the reason outcomes are missing?
Solutions. (1) It is monotone: the adjacent pairs are \((1,1)\), \((1,0)\), \((0,0)\), and \((0,0)\), with no \(0\)-to-\(1\) transition. (2) It is intermittent because visits 3 to 4 give the pair \((0,1)\); later observed outcomes do not erase that return. (3) Yes. The sequence \(0,0,0,0\) has no missing-to-observed transition, although this classification does not explain why the outcomes are all missing. (4) The first sequence contains a \(0\)-to-\(1\) transition from visit 2 to visit 3 and is intermittent; the second has no such transition and is monotone. The order, not just the count, determines the category. (5) It establishes that at least one missing visit is followed by an observed visit. It does not establish why the outcome was missing, whether the participant formally withdrew, or which missingness mechanism generated the pattern.