Why the Definition of a Pattern Matters
A dropout pattern is defined from a record of which scheduled outcomes are observed and which are not. That record can depend on practical choices: how late a visit may be and still count as attended, whether one missed visit marks dropout, and whether a participant who returns after a missed visit is classified as intermittent or as having dropped out at the first absence. These choices are not merely labels. They can change who belongs to each pattern, the proportion assigned to each pattern, and estimates calculated separately within those patterns.
Sensitivity analysis asks how much the analysis changes under defensible alternative definitions. It does not establish that one definition reveals a uniquely “true” pattern. Rather, it makes the operational choices visible and shows whether conclusions depend on them. The comparison is most informative when the visit schedule, target summary, outcome scale, and analysis population are held fixed except for the definition being varied.
A visit window is a rule about timing, not a claim that the outcome measured on a different date is interchangeable with the scheduled-visit outcome. For example, a protocol might count a measurement taken within seven days of a scheduled visit, whereas another analysis might use a 21-day window. A measurement outside the chosen window may still be recorded; whether it is eligible for a particular scheduled-visit summary is a separate decision that should be stated.
The comparisons below assume the same participants and scheduled visits are used under each definition, visit dates and outcomes are recorded consistently, and each rule is applied uniformly. Pattern-specific summaries must also be defined clearly—for example, the mean baseline outcome among members of each pattern. When an outcome is missing for some members of a pattern, its observed-data mean is calculated only among those with that outcome; it is not automatically the mean for the entire pattern. A definition change alone does not supply the missing outcomes or justify extrapolating to them.
Reclassification Changes Weights and Summaries
Suppose the analysis uses mutually exclusive patterns that cover all \(n\) participants. For a summary \(Y\) observed for every participant, let \(\bar{Y}_k\) be its mean in pattern \(k\). The overall sample mean is the pattern-weighted mean:
This identity follows by partitioning the sum over participants into pattern-specific sums. In each pattern, \(\sum_{i\in k}Y_i=n_k\bar{Y}_k\); therefore
If a new definition only reallocates participants, the overall mean of a universally observed outcome stays the same, even though weights and pattern-specific means may change. For a later outcome that is missing for some participants, the identity applies to a full-population mean only if pattern-specific means for the full pattern populations are available and justified. Weighting observed-data means does not, by itself, recover unobserved outcomes.
Worked Example: Narrow Versus Broad Visit Windows
Consider eight hypothetical participants with three scheduled follow-up visits, V1–V3. Every participant has a baseline outcome \(Y_0\). The visit-status table records whether each visit falls inside the stated window. For clarity, the observed sequences under a seven-day window are shown first; the broad-window sequence changes for G and H.
| Participant | \(Y_0\) | Days from target at V1, V2, V3 | Seven-day pattern | 21-day pattern |
|---|---|---|---|---|
| A | 4 | 1, 2, 1 | 111 | 111 |
| B | 3 | 2, 3, 2 | 111 | 111 |
| C | 5 | 1, 2, missed | 110 | 110 |
| D | 2 | 1, 1, missed | 110 | 110 |
| E | 4 | 1, missed, missed | 100 | 100 |
| F | 6 | 1, missed, 15 | 101 | 101 |
| G | 3 | 1, 20, missed | 100 | 110 |
| H | 5 | 1, 2, 18 | 110 | 111 |
Assume an outcome is counted at a visit only if it is within the specified window, and “missed” means no measurement. This rule makes G’s V2 outcome eligible under the 21-day window and H’s V3 outcome eligible under that window. Under seven days, their late measurements do not count as observations at those scheduled visits. The table therefore changes their pattern membership, not their baseline outcomes.
With the seven-day window, the pattern counts are \(n_{111}=2\), \(n_{110}=3\), \(n_{100}=2\), and \(n_{101}=1\). Dividing by eight gives weights \(0.250, 0.375, 0.250,\) and \(0.125\). The baseline means are:
The pattern-weighted baseline mean is \(0.250(3.500)+0.375(4.000)+0.250(3.500)+0.125(6.000)=4.000\). Under the 21-day window, the counts are \(3,3,1,1\) for patterns 111, 110, 100, and 101, respectively. The corresponding baseline means are \(4.000\), \(10/3=3.333\), \(4.000\), and \(6.000\). Thus the weighted result is \(0.375(4.000)+0.375(3.333)+0.125(4.000)+0.125(6.000)=4.000\). The baseline mean is unchanged, as it must be: all eight baseline values are unchanged and the patterns still partition the same participants. The pattern weights and within-pattern descriptions are not unchanged.
This example also shows why the visit-window convention must accompany a reported pattern count. Under the broad window, a late measurement counts as observed at its scheduled visit by definition; that does not show that the result would be identical under a model that accounts for actual measurement time.
Missed Visits and Dropout Coding
A missed visit need not mean permanent dropout. A participant may miss one scheduled assessment and return at a later visit. One defensible coding scheme defines the pattern by the full observed/missing sequence, preserving such a return as an intermittent pattern. Another summarizes participants by the first missed visit, even if they later return. These definitions answer different descriptive questions; neither should be substituted for the other without stating the change.
Worked Example: First Missed Visit Versus Intermittent Attendance
Suppose five hypothetical participants have three follow-up visits, all assessed using the same window. Their sequences and universally observed baseline outcomes are: A, 111 and 2; B, 111 and 4; C, 110 and 6; D, 100 and 8; E, 101 and 10. Compare two definitions. Definition 1 labels participants by the first missed visit: “no missed visit,” “first miss at V2,” or “first miss at V3.” Definition 2 retains the full sequence, including an “intermittent” pattern for a missed visit followed by a return.
Under Definition 1, A and B have no missed visit, D and E first miss at V2, and C first misses at V3. The respective counts are 2, 2, and 1, so the weights are \(2/5=0.400\), \(2/5=0.400\), and \(1/5=0.200\). Their baseline means are \((2+4)/2=3.000\), \((8+10)/2=9.000\), and \(6/1=6.000\).
Under Definition 2, A and B are pattern 111, C is 110, D is 100, and E is intermittent pattern 101. Each has weight \(1/5=0.200\), except 111, which has weight \(2/5=0.400\). The corresponding baseline means are 3.000, 6.000, 8.000, and 10.000. Thus, treating E’s first absence as the defining dropout event combines E with D and changes the “first miss at V2” baseline mean from 8.000 to 9.000.
Both weighted baseline calculations equal \(6.000\): under Definition 1, \(0.400(3.000)+0.400(9.000)+0.200(6.000)=6.000\); under Definition 2, \(0.400(3.000)+0.200(6.000)+0.200(8.000)+0.200(10.000)=6.000\). The equality is an arithmetic consequence of a complete partition and a baseline outcome observed for everyone. It does not imply that the pattern-specific summaries are interchangeable, or that later-outcome summaries would agree.
Compare the Consequences, Not Just the Labels
A useful sensitivity analysis records, for each candidate definition, both pattern sizes and the summaries that will inform interpretation. The next example illustrates how reclassification can affect a weighted summary when pattern-specific fitted means are supplied. These hypothetical fitted values are inputs to the calculation, not values identified from the pattern weights alone.
Worked Example: Combining Pattern Weights With Fitted Means
For a hypothetical late-visit outcome measured in points, an analyst reports three pattern-specific fitted means under each of two definitions. Under definition A, the weights are \(0.50, 0.30,\) and \(0.20\), and the corresponding fitted means are \(12, 9,\) and \(7\) points. The weighted estimate is
Under definition B, the weights are \(0.40, 0.35,\) and \(0.25\), and the fitted means are \(12, 8,\) and \(6\) points. The calculation is \(0.40(12)+0.35(8)+0.25(6)=4.80+2.80+1.50=9.10\) points. The difference is \(10.10-9.10=1.00\) point.
This difference reflects both changed weights and changed fitted means. To attribute it to either component, the analyst would need additional calculations holding one component fixed at a time; even then, the decomposition depends on the chosen reference. The example also requires a substantive qualification: if the late-visit outcome is missing for some participants, each fitted pattern mean relies on the specified model and its assumptions. The multiplication does not make an unobserved outcome empirically observed.
Diagnostics, Alternatives, and Limitations
For each candidate definition, construct a participant-level cross-tabulation of pattern assignments. The table shows which participants move, rather than only showing that a weight changed. Then report the new counts and weights, pattern-specific summaries with their units, and the resulting weighted summary when it is appropriate to calculate one. For a visit-window comparison, inspect the actual visit-time offsets and identify how many measurements lie near each window boundary. For a missed-visit comparison, list the number of intermittent sequences and the rule that assigns them.
Warning signs include a small pattern appearing or disappearing under a modest rule change; a large share of participants changing pattern; a pattern-specific estimate changing sharply because one or two participants move; and fitted summaries that depend heavily on sparse patterns. These are especially important when pattern-specific fits are unstable, as indicated by wide uncertainty or sensitivity to individual observations. Report uncertainty alongside estimates where available: stability of the point estimate does not guarantee stable precision.
The main alternatives are to define patterns using a strict monotone dropout rule, to retain the full visit-status sequence including intermittent patterns, or to use a prespecified visit window and vary it in a sensitivity analysis. These are alternative operational summaries, not interchangeable models. A continuous-time analysis of actual measurement dates may be more suitable when timing itself is scientifically important, but it answers a different modeling question and requires its own assumptions. The choice should be driven by the protocol and target question, not selected after seeing which definition produces a preferred result.
A pattern definition is also not a remedy for missing outcomes. Reclassification can alter the observed groups and their weights, but it cannot determine what missing outcomes would have been. Pattern-specific estimates for outcomes not observed throughout a pattern population need an explicit estimation strategy and assumptions. Sensitivity to pattern definitions should therefore be reported separately from sensitivity to assumptions used to estimate unobserved outcomes.
Common Pitfall
Correction: An unchanged mean for a fully observed outcome is guaranteed when the same individuals are partitioned in two different ways. It says nothing by itself about pattern-specific estimates, later outcomes with missing values, model-based extrapolations, or conclusions that depend on particular patterns. Compare membership, weights, within-pattern summaries, and the target estimate relevant to the scientific question.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In the eight-participant example, what are G’s V2 status and pattern under the seven-day and 21-day windows?
- Why does the pattern-weighted baseline mean remain 4.000 under both windows in that example?
- Under the five-participant first-miss definition, which participants belong to the “first miss at V2” group, and what is its baseline mean?
- Give two diagnostics that could reveal a consequential change in dropout-pattern definition.
- Under what condition can a pattern-weighted mean be interpreted as the overall mean of a fully observed outcome?
Solutions. (1) G’s V2 measurement is 20 days from target, so it is outside the seven-day window and inside the 21-day window. G is pattern 100 under seven days and 110 under 21 days. (2) All eight baseline outcomes are observed and unchanged, and both sets of patterns partition the same eight participants. The weighted pattern means therefore reproduce the same overall baseline mean, 4.000. (3) D and E first miss at V2. Their baseline mean is \((8+10)/2=9.000\). (4) A participant-level cross-tabulation reveals who changes pattern; pattern counts, weights, and within-pattern estimates reveal whether those moves change analysis summaries. Checking whether patterns become sparse is another useful diagnostic. (5) The patterns must be mutually exclusive and exhaustive for the same analysis population, and the outcome must be observed for every member; then weighting each pattern’s sample mean by its population share gives the overall sample mean.