Why the Missing-Outcome Restriction Matters
In a longitudinal trial, the mean among patients whose outcome is observed at a visit is not automatically the mean that would have been observed for everyone assigned to the trial. Patients may stop reporting pain for reasons associated with their pain, treatment experience, or health. A pattern-mixture analysis describes the data by their observed follow-up patterns and makes assumptions about the outcomes missing within each pattern. Those assumptions determine which observed groups are used to stand in for unobserved outcomes.
This tutorial compares three mean restrictions in a hypothetical chronic pain trial. They are presented as explicit operational rules: a restriction assigns a value to a pattern-specific mean that is not identified directly from the observed outcomes. The rules are not consequences of the data alone, and different rules can produce different trial-wide summaries.
Pain scores in the example range from 0 to 10, with higher scores indicating worse pain. There are measurements at baseline and weeks 4, 8, and 12. The patterns are: \(C\), observed at every visit; \(D_8\), observed through week 8 but missing week 12; and \(B\), observed at baseline only. All numbers below are hypothetical group sizes and means. The groups are mutually exclusive and together include 40 patients.
| Pattern | Number of patients | Baseline mean | Week 4 mean | Week 8 mean | Week 12 mean |
|---|---|---|---|---|---|
| \(C\): complete through week 12 | 28 | 7.0 | 6.5 | 5.5 | \(132/28\) |
| \(D_8\): observed through week 8 | 4 | 7.5 | 7.0 | 6.5 | Missing |
| \(B\): baseline only | 8 | 8.0 | Missing | Missing | Missing |
The week-12 total for the complete cases is 132 pain-score points, so their observed week-12 mean is \(132/28=4.7142857\). At week 4, the observed total is \(28(6.5)+4(7.0)=210\) among 32 patients. At week 8, it is \(28(5.5)+4(6.5)=180\) among the 32 patients observed then. Thus, the observed week-4 and week-8 means are \(210/32=6.5625\) and \(180/32=5.625\), respectively. These summaries describe observed groups; they do not, by themselves, determine the missing groups’ means.
Three Mean Restrictions
The restrictions below leave each pattern’s observed means unchanged. They assign values only to pattern-by-visit cells that are missing. This distinction matters: for example, the observed week-4 mean in \(D_8\) remains 7.0 under all three rules, even if a rule says that an unobserved week-12 mean in \(D_8\) should be borrowed from another pattern.
These are restrictions on means, not claims that the individual missing outcomes are known. In this example, the represented pattern order from least to most incomplete is \(C, D_8, B\). For the neighboring-case rule, a missing mean in \(B\) can first borrow from \(D_8\); if \(D_8\) also has a missing mean at that visit, the restriction is continued to \(C\). This makes the rule explicit even when more than one successive pattern lacks the outcome.
Worked Example: Complete-Case Restriction
For \(D_8\), the week-12 outcome is missing. Under the complete-case restriction, use the complete-case week-12 mean:
Here \(\mu_{D_8,12}\) denotes the restricted mean for pattern \(D_8\) at week 12, and \(\bar{x}_{C,12}\) is the observed complete-case mean. The eight baseline-only patients also receive the complete-case mean at each of their missing follow-up visits: 6.5 at week 4, 5.5 at week 8, and \(132/28\) at week 12. Their observed baseline mean remains 8.0.
The week-4 trial-wide mean under this restriction is the pattern-size-weighted mean \( [28(6.5)+4(7.0)+8(6.5)]/40=262/40=6.55 \). At week 12, every pattern has restricted mean \(132/28\), so the trial-wide mean is also \(132/28\). The restricted week-12-minus-week-4 change is therefore \(132/28-6.55=-1.835714\), or about \(-1.836\) pain-score points. The negative sign denotes a decrease in mean pain score.
Worked Example: Neighboring-Case Restriction
For \(D_8\), the nearest less-incomplete represented pattern is \(C\), which has an observed week-12 mean. Thus, the neighboring-case rule gives \(\mu_{D_8,12}=132/28\). For \(B\), the nearest less-incomplete represented pattern is \(D_8\): its observed week-4 and week-8 means supply the week-4 and week-8 restrictions for \(B\).
The final equality at week 12 follows from successive application: \(D_8\) lacks a week-12 observation, so its restricted mean is linked to \(C\); \(B\)'s week-12 mean is linked to \(D_8\)'s restricted mean. The week-4 trial-wide mean is \([28(6.5)+4(7.0)+8(7.0)]/40=266/40=6.65\). The week-12 mean remains \(132/28\), giving a restricted change of \(132/28-6.65=-1.935714\), or about \(-1.936\) pain-score points.
This calculation relies on the stated pattern ordering and on there being an observed week-4 and week-8 mean in the nearest represented pattern. It does not assert that the \(B\) and \(D_8\) patients actually have the same unobserved outcomes as patients in the neighboring pattern.
Worked Example: Available-Case Restriction
At week 4, the available patients are the 28 in \(C\) and the 4 in \(D_8\). Their observed total is \(210\), so the available-case mean is \(210/32=6.5625\). Assign this mean to the missing week-4 outcome mean for \(B\). At week 8, the 28 patients in \(C\) and 4 in \(D_8\) contribute a total of 180, so the available-case mean is \(180/32=5.625\); assign it to \(B\)'s missing week-8 mean.
Only \(C\) has week-12 outcomes observed, so the available-case week-12 mean is \(132/28\). Assign that mean to both \(D_8\) and \(B\) at week 12. The restricted week-4-to-week-12 change must use the unrounded means:
Thus, under this restriction, the mean pain score decreases by about 1.848 points from week 4 to week 12. The calculation does not imply that every patient’s pain decreased, or that the restriction is true; it is the change in means implied by this particular assignment of missing pattern means.
Compare the Pattern Means and Trial-Wide Summaries
The table displays the restricted means at follow-up. Each observed mean is identical across columns; differences arise only in cells that were missing. Values are rounded for display, while the calculations above use the underlying fractions where needed.
| Pattern and visit | Complete-case restriction | Neighboring-case restriction | Available-case restriction |
|---|---|---|---|
| \(C\), week 4 (observed) | 6.5 | 6.5 | 6.5 |
| \(D_8\), week 4 (observed) | 7.0 | 7.0 | 7.0 |
| \(B\), week 4 (missing) | 6.5 | 7.0 | 6.5625 |
| \(C\), week 8 (observed) | 5.5 | 5.5 | 5.5 |
| \(D_8\), week 8 (observed) | 6.5 | 6.5 | 6.5 |
| \(B\), week 8 (missing) | 5.5 | 6.5 | 5.625 |
| \(C\), week 12 (observed) | 4.7143 | 4.7143 | 4.7143 |
| \(D_8\), week 12 (missing) | 4.7143 | 4.7143 | 4.7143 |
| \(B\), week 12 (missing) | 4.7143 | 4.7143 | 4.7143 |
The trial-wide means and changes are obtained by weighting each pattern-specific mean by its pattern size, then dividing by 40. Under the available-case restriction, this reproduces the observed available-case mean at each visit: at week 4, for example, \( [32(210/32)+8(210/32)]/40=210/32 \). This identity follows from assigning the same available-case mean to the missing group. It is not independent evidence that the missing group has that mean.
The methods coincide at week 12 here because only the complete-case pattern supplies an observed week-12 mean, and all three restrictions ultimately use that value for the missing patterns. They differ at week 4 and week 8, where the observed patterns provide more than one possible reference. In another data set, or with different observed means and pattern sizes, the resulting endpoint or change summaries could differ more substantially.
Justification, Diagnostics, and Limitations
The arithmetic behind a restricted trial-wide mean is a weighted average over mutually exclusive patterns. If \(n_r\) is the number of patients in pattern \(r\), and \(\tilde{\mu}_{r,t}\) is its observed or restricted mean at visit \(t\), then:
This is a direct consequence of summing the pattern-specific totals \(n_r\tilde{\mu}_{r,t}\) and dividing by the total number of patients. It is a useful check: every pattern must be counted once, and the weights must sum to the full trial size. A restriction supplies the missing \(\tilde{\mu}_{r,t}\) values; it does not alter the weighting identity.
The restrictions can be compared with each other and with observed pattern means as a diagnostic for sensitivity to assumptions. Large differences between observed neighboring patterns, such as the week-4 means 6.5 and 7.0 here, are a warning that treating a missing pattern’s mean as identical to one reference group may be consequential. They do not prove that the restriction is false, because the outcomes of the missing patients are not observed. Likewise, close observed means do not verify equality among missing outcomes.
A neighboring-case restriction may be hard to use when a pattern has few patients, when no less-incomplete represented pattern has the needed observed visit, or when a large gap separates the patterns. Successive borrowing can also propagate an assumption through several groups. The analyst should state the pattern order and each link, rather than use “neighboring” as an unexplained label. Available-case means can be unstable when few patients remain observed at later visits, and an available-case restriction can conceal the fact that the observed group at one visit differs from the observed group at another.
Complete-case analysis, in the narrow sense of reporting only patients with complete follow-up, is not the same as using complete cases as a reference to assign missing pattern means. The first discards incomplete patterns from the summary; the restriction illustrated here retains all patterns and borrows their missing means from \(C\). Other model-based strategies, including selection-model approaches, express assumptions differently. None makes unobserved outcomes empirically known. The restrictions here are mean-level assumptions; they do not specify every feature of the outcome distribution or account automatically for uncertainty in estimated means.
- Do not confuse observed and restricted means. The week-12 mean in \(D_8\) is not observed; \(132/28\) is its value under the stated restrictions.
- Do not round before subtracting. Here \(132/28-210/32=-1.848214\ldots\), which rounds to \(-1.848\), not \(-1.849\).
- Do not treat an available-case mean as universal by definition. It describes those observed at a visit; assigning it to missing patterns is an additional assumption.
- Do not claim a restriction is validated by observed data. Observed patterns can motivate and diagnose an assumption, but cannot reveal the outcomes missing within another pattern.
Check Your Understanding
Use the hypothetical trial and the restrictions defined above. Give answers in pain-score points where relevant.
- Under the neighboring-case restriction, what week-8 mean is assigned to pattern \(B\), and which observed group supplies it?
- Calculate the available-case week-8 mean from the observed pattern means and group sizes.
- Under the complete-case restriction, calculate the week-4 trial-wide mean using all 40 patients.
- Why does agreement among the three week-12 restricted means fail to demonstrate that the missing week-12 outcomes follow the complete-case mean?
Solutions. (1) \(B\)'s week-8 mean is set to \(D_8\)'s observed week-8 mean, 6.5, because \(D_8\) is the nearest less-incomplete represented pattern with that visit observed. (2) The available-case mean is \([28(5.5)+4(6.5)]/32=(154+26)/32=180/32=5.625\). (3) The complete-case restriction assigns 6.5 to \(B\) at week 4, so the weighted mean is \([28(6.5)+4(7.0)+8(6.5)]/40=(182+28+52)/40=262/40=6.55\). (4) All three rules use the sole observed week-12 mean, \(132/28\), as the reference for missing week-12 means in this example. Their agreement is therefore a consequence of the chosen rules and observed pattern structure, not evidence about outcomes that were never recorded.