Rescue Medication and the Outcome Question
In a rheumatoid arthritis trial, participants whose symptoms remain severe may receive rescue medication, such as an additional anti-inflammatory treatment. Rescue use is clinically important, but it does not by itself make a participant’s later outcome missing. If the week-12 disease activity score is measured after rescue, that score is part of the participant’s outcome under the trial’s treatment strategy. The difficult case is when that post-rescue score is not observed.
This distinction determines the question being answered. A treatment-policy comparison asks about outcomes under assignment to each randomized treatment, with rescue use occurring as it did under the trial’s care rules. It does not ask what the outcome would have been if rescue had been prohibited. The latter is a different question and cannot be answered merely by omitting rescued participants.
The estimand includes outcomes after rescue whether or not rescue was used. The analysis should therefore retain participants in their randomized groups and use their observed post-rescue scores. Comparing only participants who did not use rescue, or treating rescue as if it were ordinary dropout, generally changes the target question and can introduce selection bias: rescue use is often related to worsening symptoms and prognosis.
Where the MNAR Assumption Enters
For a missing week-12 score, define \(M=1\) if the score is missing and \(M=0\) if it is observed. Let \(S=1\) indicate rescue medication before week 12. A missing score among \(S=1\) participants is not known simply because rescue occurred; its distribution must be modelled or varied in a sensitivity analysis.
A baseline imputation model might use randomized group, baseline DAS28, earlier DAS28 measurements, rescue status and timing, and other observed predictors of the week-12 score or its absence. Under a missing-at-random (MAR) assumption, after conditioning on the information included in that model, the distribution of missing scores is the same as the distribution of scores that would be observed for participants with those same recorded characteristics. MAR is an assumption, not something established by observing the data.
A pattern-mixture sensitivity analysis departs from that baseline assumption by assigning a shift to the imputed scores in a defined missingness pattern. Here, the clinically focused pattern is a missing week-12 score after rescue. Because higher DAS28 is worse, a positive shift represents worse outcomes than predicted under MAR.
The delta is not estimated from the observed scores in the missing pattern: those scores are unavailable. Its value encodes an assumption about how the unobserved outcomes differ from their MAR predictions. Other missingness patterns may be assigned separate shifts; setting their shifts to zero is also an assumption, not a data-driven finding.
Worked Examples
Worked Example: Classifying Observed Outcomes After Rescue
Suppose a hypothetical randomized trial measures DAS28 at baseline and week 12. A participant assigned to the experimental treatment has a high week-8 score, receives rescue medication at week 9, and has a recorded week-12 DAS28 of 4.1. Another participant receives rescue at week 9 but has no week-12 measurement.
For the first participant, 4.1 is an observed outcome after rescue. It remains in the experimental-treatment group’s week-12 outcome distribution for the treatment-policy estimand. The rescue event does not erase or replace the measurement.
For the second participant, the week-12 score is missing. Rescue status is relevant information for modelling or sensitivity analysis, but it does not reveal the score. A MAR imputation may condition on the recorded week-8 score and rescue information; a delta analysis can then ask how the conclusion changes if the missing post-rescue score is worse than that prediction.
Neither participant should be removed from the randomized comparison simply because rescue occurred. Removing both would discard the observed outcome in the first case and change the estimand in both cases.
Worked Example: Applying a Delta Shift to Missing Post-Rescue Scores
Consider hypothetical week-12 DAS28 data from 100 participants per randomized arm. In the experimental arm, 80 scores are observed, with mean 3.8; the other 20 are missing after rescue and have MAR-imputed mean 4.0. In control, 90 scores are observed, with mean 4.1; 10 scores are missing after rescue and have MAR-imputed mean 4.2. Use arm means to illustrate the point estimate, and apply a shift only to missing post-rescue scores in the experimental arm.
Under the MAR reference analysis, the experimental-arm mean is
The control-arm mean is
Thus the illustrative contrast is \(3.84-4.11=-0.27\) DAS28 units, favoring the experimental assignment. Now posit \(\delta_1=0.6\) DAS28 units for the 20 missing experimental-arm scores after rescue. Their adjusted mean becomes \(4.0+0.6=4.6\). The adjusted experimental mean is
Keeping the stated control assumption unchanged gives \(\bar{Y}_{0}=4.11\), so the adjusted contrast is \(3.96-4.11=-0.15\) DAS28 units. The assumed worse outcomes reduce the apparent benefit, but this point estimate still favors the experimental assignment. These calculations illustrate mean shifts; they do not supply a standard error or confidence interval.
Worked Example: Deriving a Tipping Value
Use the same hypothetical trial. Let \(\delta\) be the shift applied to the 20% of experimental-arm participants with missing post-rescue scores, with no shift applied to control. Each unit increase in \(\delta\) raises the experimental-arm mean by \(20/100=0.20\) units. Therefore, the contrast as a function of the assumed shift is
The point-estimate tipping value is the shift at which this contrast reaches zero. Solving \(-0.27+0.20\delta=0\) gives \(\delta=0.27/0.20=1.35\) DAS28 units. At that assumed shift, the two illustrative arm means are equal. Shifts larger than 1.35 make the experimental-arm mean higher in this simple calculation.
This is a tipping value for the point estimate, not necessarily for a confidence interval or a hypothesis test. A real sensitivity analysis would also compute uncertainty at each scenario and could define a separate tipping point for a specified inferential decision.
Justification, Assumptions, and Implementation
The weighting calculation follows directly from averaging observed and imputed values within each randomized arm. If a fraction \(q_a\) of arm \(a\) has missing post-rescue outcomes and each such outcome is shifted by \(\delta_a\), then, holding the other outcomes fixed, the arm mean changes by \(q_a\delta_a\). Consequently, when only the experimental arm receives a shift,
In the example, \(q_1=20/100=0.20\), so a shift of 0.6 changes the contrast by \(0.20(0.6)=0.12\), from \(-0.27\) to \(-0.15\). This identity explains why the same delta has a larger effect when a larger share of an arm’s outcomes is in the shifted pattern.
For an actual analysis, the MAR model must be appropriate for the outcome and repeated measurements, and it should include relevant observed predictors of week-12 DAS28 and missingness. The imputation model should preserve the randomized-arm structure and include rescue information when it helps describe outcome trajectories or missingness. There must also be enough observed data to support predictions for the missing pattern; severe lack of overlap makes those predictions extrapolations.
A typical analysis generates multiple MAR imputations, applies the chosen delta to the specified missing pattern in each imputed data set, and fits the prespecified treatment comparison in every completed data set. The estimates and their within-imputation variances are then combined using the usual multiple-imputation pooling rules. The examples above show only arm-mean arithmetic: their aggregate counts and means cannot determine the imputation-level variances needed for valid uncertainty estimates.
The delta specification should be stated before interpreting results: which arm and missingness pattern receive a shift, its sign and units, and the range of values examined. A useful display reports the contrast and uncertainty over a clinically defensible grid, including \(\delta=0\) as the MAR reference. The earlier material on pattern-mixture identifying restrictions and delta adjustments provides the general framework; here the key modelling choice is to tie the departure to missing post-rescue outcomes rather than to rescue use alone.
Diagnostics, Alternatives, and Limitations
Diagnostics cannot identify the true MNAR delta, because the relevant outcomes are unobserved. They can still reveal problems in the baseline model and in the sensitivity implementation. Check convergence and imputation distributions; compare observed and imputed values by randomized arm, rescue status, and prior DAS28; inspect whether imputed values are clinically plausible; and verify that the intended delta was applied only to the specified records. If shifted values fall outside the feasible range of the outcome or create implausible distributions, revise the model or use a method that respects the scale rather than silently accepting impossible values.
Interpret the sensitivity results alongside missingness counts and the timing of rescue. A large missing fraction after rescue, weak prediction from observed history, or a conclusion that changes under small clinically plausible shifts indicates substantial dependence on assumptions. A stable result over a broad, justified range is reassuring about that particular sensitivity dimension, but it does not prove that MAR holds or rule out other departures.
| Approach | Question or assumption | Relation to this analysis |
|---|---|---|
| Treatment-policy analysis using observed post-rescue outcomes | What are outcomes under randomized assignment with rescue available as in the trial? | Retain observed post-rescue scores; address missing scores separately. |
| MAR imputation | Are missing scores conditionally comparable to observed scores given recorded information? | Provides the reference imputation model, but the assumption is not testable from observed data alone. |
| Pattern-mixture delta sensitivity | How do conclusions change if a specified missing pattern differs from its MAR prediction? | Varies the unobserved post-rescue scores by an explicit, interpretable shift. |
| Analysis excluding rescued participants | What comparison is seen among participants who did not use rescue? | Does not generally estimate the treatment-policy effect because rescue groups are selected after randomization. |
A constant additive delta is deliberately simple. It assumes the same shift for all missing post-rescue outcomes in the specified pattern, even though departures may differ by prior severity, rescue timing, or randomized arm. A scalar shift also does not capture every possible change in variability or distribution shape. Multiple deltas or richer models may be appropriate, but additional parameters require defensible clinical interpretation and adequate data.
- Mistake: “Anyone who used rescue should be excluded because their outcome no longer reflects randomized treatment.” Correction: Under the treatment-policy question, the observed outcome after rescue is retained in the randomized arm; rescue is part of the care pathway.
- Mistake: “A rescued participant with a missing score should automatically receive a worse value.” Correction: Rescue does not determine the unobserved score. A worse-than-MAR shift is a sensitivity assumption that must be specified and varied.
- Mistake: “The delta is estimated from the observed rescued participants.” Correction: Observed post-rescue outcomes inform the outcome model, but they do not reveal the unobserved outcomes of participants whose scores are missing. Delta values represent unverifiable departures.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A participant receives rescue medication and has a recorded week-12 DAS28. Is that score included in a treatment-policy comparison?
- In the example, what change in the treatment contrast results from applying \(\delta=0.5\) to 20% of experimental-arm outcomes only?
- Why does excluding all participants who used rescue generally fail to answer the treatment-policy question?
- What does a point-estimate tipping value of 1.35 DAS28 units mean in the illustrative calculation?
Solutions. (1) Yes. It is the observed outcome under the assigned treatment strategy with rescue available. (2) The contrast increases by \(q_1\delta=0.20(0.5)=0.10\) DAS28 units, becoming \(-0.27+0.10=-0.17\). (3) Rescue use occurs after randomization and is related to clinical history; restricting to nonusers selects participants and changes the target comparison. (4) It is the assumed positive shift on missing experimental-arm post-rescue scores that makes the illustrative arm-mean contrast equal zero; it is not an estimated true shift or necessarily an inferential tipping point.