Why the dropout mechanism matters
Suppose participants who feel worse while taking an antidepressant are more likely to miss the final assessment. The participants whose symptoms are observed may then be a selectively less-worsened subset of everyone randomized. A treatment contrast calculated from completers alone can therefore differ from the contrast for the full randomized groups.
This case study uses a single binary endpoint to make the modeling choices visible. The endpoint is whether symptoms have worsened by week 8. A value of \(Y=1\) denotes worsening and \(Y=0\) denotes no worsening. Let \(R=1\) mean the week-8 outcome is observed and \(R=0\) mean it is missing. The target contrast is the antidepressant-arm risk of worsening minus the control-arm risk of worsening in all randomized participants. A negative contrast favors the antidepressant on this endpoint, but does not by itself establish a causal mechanism for any observed difference.
The models below are deliberately simple. Assume randomization was implemented as intended, the binary outcome is defined the same way in both arms, and each participant contributes one endpoint. Within each arm, take the observed response rate and the observed worsening rate among completers as known for this illustrative calculation. The selection model additionally assumes a common relative completion probability for worsened versus non-worsened participants in both arms. This is a sensitivity assumption, not a fact established by randomization.
Illustrative observed data
Imagine a hypothetical trial with 100 randomized participants per arm. At week 8, 80 participants in each arm have an observed outcome. Among those with observed outcomes, 12 antidepressant participants and 20 control participants have worsened. Thus the observed completion rate is \(0.80\) in each arm, while the observed worsening rates are \(12/80=0.15\) and \(20/80=0.25\).
| Arm | Randomized | Observed | Worsened among observed | Observed worsening rate |
|---|---|---|---|---|
| Antidepressant | 100 | 80 | 12 | 0.150 |
| Control | 100 | 80 | 20 | 0.250 |
The observed completer contrast is \(0.15-0.25=-0.10\), or 10 percentage points fewer observed worsenings in the antidepressant arm. That calculation describes participants with observed outcomes. To extend it to all randomized participants, we must say something about the 20 missing outcomes in each arm.
Worked Example: A Selection Model With Lower Completion When Symptoms Worsen
Assume, as a sensitivity scenario, that within each treatment arm the probability of a week-8 outcome being observed for a participant who worsened is \(\rho=0.60\) times the corresponding probability for a participant who did not worsen: \(P(R=1\mid Y=1)=\rho P(R=1\mid Y=0)\). This specifies a relative completion probability, not a dropout odds ratio. Let \(q=P(Y=1\mid R=1)\) be the observed worsening rate and \(p=P(Y=1)\) the full-arm worsening risk.
By conditioning on the outcome among observed participants, \(q=\rho p/\{\rho p+(1-p)\}\). Solving this equation for \(p\) gives the selection-model risk:
For the antidepressant arm, use the unrounded inputs \(q=12/80=0.15\) and \(\rho=0.60\):
For the control arm, use \(q=20/80=0.25\):
The resulting full-arm contrast is \(p_T-p_C=0.2272727\ldots-0.3571429\ldots=-0.1298702\ldots\), or about 13.0 percentage points fewer worsenings in the antidepressant arm under this scenario. The model attributes more worsening to the full groups than the completer rates suggest, because worsening is assumed to reduce the chance of an observed outcome.
Check that the scenario can reproduce the observed 80% completion rate. Write \(a=P(R=1\mid Y=0)\), so \(P(R=1\mid Y=1)=0.60a\). The overall response probability is \(a\{(1-p)+0.60p\}=a(1-0.40p)\). Thus \(a=0.80/(1-0.40p)\). This gives \(a=0.880\) in the antidepressant arm and \(a=0.9333333\ldots\) in control; the corresponding completion probabilities among worsened participants are \(0.528\) and \(0.560\). All are between zero and one, as required.
Deriving the corresponding pattern-mixture description
A pattern-mixture analysis works with the outcome rates in the observed and missing patterns. Write \(m=P(Y=1\mid R=0)\), the worsening rate among participants whose week-8 outcome is missing, and let \(r=P(R=1)\). Partitioning the arm by response pattern gives a simple mixture identity. This identity is also a useful check on the selection-model calculation.
For example, the selection model above implies values for \(m\); it does not remove the need for an assumption. Given \(p\), the probability of both worsening and being missing is \(p\{1-P(R=1\mid Y=1)\}\). Dividing by the missing fraction \(1-r=0.20\) gives the missing-pattern worsening rate. This provides a direct way to express the selection assumption in pattern-mixture terms.
Worked Example: Translating the Selection Scenario Into Missing-Pattern Rates
Use the response probabilities calculated above. In the antidepressant arm, the worsening probability is \(p_T=0.2272727\ldots\) and the completion probability among worsened participants is \(0.528\). Therefore:
In control, \(p_C=0.3571429\ldots\) and the completion probability among worsened participants is \(0.560\):
The pattern-mixture calculation reproduces the risks: \(0.80(0.15)+0.20(0.5363636\ldots)=0.2272727\ldots\) for the antidepressant arm, and \(0.80(0.25)+0.20(0.7857143\ldots)=0.3571429\ldots\) for control. The pattern-mixture contrast is therefore the same \(-0.1298702\ldots\). In expected counts per 100 randomized participants, the missing group contributes about 10.727 worsenings in the antidepressant arm and 15.714 in control. These are expected counts under the model, not observed fractional people.
The equality is not a general claim that the two modeling approaches always make the same assumptions. It shows that when a selection model and a pattern-mixture model encode the same joint distribution of \(Y\) and \(R\), they give the same marginal risk. Their practical difference is often how the unobserved-data assumptions are parameterized and communicated.
How sensitive is the treatment contrast?
A sensitivity parameter should be varied over values that are scientifically interpretable and mathematically compatible with the observed data. Here \(\rho=1\) represents equal completion probabilities for worsened and non-worsened participants; values below one represent lower completion among worsened participants. The observed rates and response fractions stay fixed while the assumed selection relationship changes.
Worked Example: Comparing Feasible Selection Scenarios
Using \(p=q/\{\rho+(1-\rho)q\}\), calculate the arm risks and contrasts at three candidate values. The calculations use the exact observed rates \(q_T=0.15\) and \(q_C=0.25\), with results rounded only after calculation.
| Relative completion probability \(\rho\) | Antidepressant risk \(p_T\) | Control risk \(p_C\) | Contrast \(p_T-p_C\) |
|---|---|---|---|
| 0.80 | 0.1807 | 0.2941 | -0.1134 |
| 0.60 | 0.2273 | 0.3571 | -0.1299 |
| 0.50 | 0.2609 | 0.4000 | -0.1391 |
For instance, at \(\rho=0.50\), \(p_T=0.15/\{0.50+0.50(0.15)\}=0.2608696\ldots\) and \(p_C=0.25/\{0.50+0.50(0.25)\}=0.40\). The contrast is \(0.2608696\ldots-0.40=-0.1391304\ldots\), or about 13.9 percentage points. This scenario is just feasible with the 80% response fraction: in control, the implied completion probability for non-worsened participants is 1.00. Values of \(\rho\) that require a conditional completion probability above one cannot describe the supplied response fraction and should not be reported as fitted scenarios.
This grid shows that the estimated contrast changes with the unobserved-data assumption. It does not select the “correct” value of \(\rho\). A different sensitivity analysis could instead choose plausible values of \(m_T\) and \(m_C\) directly, then calculate each risk using \(p_T=0.80(0.15)+0.20m_T\) and \(p_C=0.80(0.25)+0.20m_C\). The choice should be justified clinically, and arm-specific values may be appropriate if dropout mechanisms plausibly differ by treatment.
Diagnostics, alternatives, and limits
Several checks help reveal when this analysis is going wrong. First, verify denominators: \(q\) is the worsening rate among observed participants, whereas \(p\) is the full-arm rate. Second, check that all implied conditional probabilities lie in \([0,1]\) and that the model reproduces the observed completion fraction. Third, verify the mixture identity \(p=rq+(1-r)m\) for each arm. Finally, check direction: if worsening is less likely to be observed, the missing group should generally have a higher worsening rate than the observed group under the assumed mechanism.
These checks detect arithmetic errors and incompatible assumptions, but they cannot verify that dropout truly depends on unobserved worsening in the specified way. Observed data can describe dropout rates and the symptom history measured before dropout; they cannot reveal the week-8 outcomes that were never recorded. The sensitivity parameter therefore needs a rationale, and conclusions should be shown across a defensible range rather than presented as if the chosen value were identified by the trial.
A baseline MAR analysis is a useful comparison: it uses observed information under a missing-at-random assumption, but it does not represent the particular dependence of dropout on the unobserved endpoint posited here. Selection models parameterize that dependence through \(P(R\mid Y)\); pattern-mixture models parameterize it through outcome distributions by response pattern. Both can support transparent sensitivity analyses, and they can agree when calibrated to the same joint distribution. Neither makes missing outcomes observed or removes dependence on assumptions. In a richer longitudinal analysis, measured symptom history can be incorporated, but this one-endpoint illustration should not be interpreted as a complete model for repeated symptom trajectories.
A common mistake is to conclude, “Dropout is more likely when symptoms worsen, so the missing outcomes must all be worsenings.” That does not follow. The models assign probabilities, not certainty: even with lower completion among worsened participants, some missing participants may not have worsened and some worsened participants may complete. Another mistake is to call the observed completer contrast the treatment effect for everyone. It is a contrast among participants with observed outcomes; extending it to all randomized participants requires a missing-data assumption.
Check Your Understanding
Use the hypothetical trial data above. Answers follow the questions so that each calculation can be checked.
- What is the observed completer contrast in worsening risk, and what population does it describe?
- Under the selection model with \(\rho=0.60\), calculate the control-arm risk using \(q_C=0.25\).
- If a pattern-mixture scenario sets \(m_T=0.50\), what is the antidepressant full-arm worsening risk?
- Why is a sensitivity parameter that implies \(P(R=1\mid Y=0)>1\) unacceptable?
- Can the observed data alone establish that week-8 worsening caused dropout? Explain briefly.
Solutions. (1) The contrast is \(0.15-0.25=-0.10\), or 10 percentage points fewer observed worsenings in the antidepressant arm; it describes participants with observed week-8 outcomes. (2) \(p_C=0.25/\{0.60+0.40(0.25)\}=0.25/0.70=0.3571\), approximately. (3) \(p_T=0.80(0.15)+0.20(0.50)=0.12+0.10=0.22\), or a 22% full-arm risk. (4) A probability cannot exceed one, so the proposed conditional model cannot reproduce the observed data and is not a valid scenario. (5) No. The missing week-8 outcomes are unobserved; observed data can inform the assumptions but cannot establish dependence of dropout on those unobserved outcomes.