Treatment-Arm-Specific Dropout Mechanisms

Why Compare Dropout by Treatment Arm?

A randomized trial may have more withdrawals in one treatment arm than another. That difference matters for interpreting the trial, but by itself it does not reveal why participants withdrew or whether the missing outcomes are related to outcomes that were not observed. A larger dropout rate in one arm could arise because that arm has more adverse symptoms recorded before withdrawal, because its dropout process responds differently to the same observed history, or because withdrawal is related to an outcome that was never measured.

The distinction is important: a treatment-specific dropout rate is a descriptive feature of the observed follow-up, whereas a dropout mechanism describes how the chance of dropout relates to observed and unobserved information. Randomization determines how treatment is assigned; it does not ensure continued follow-up or make missing outcomes harmless.

Definition: Let \(A\) denote randomized treatment assignment, with \(A=1\) for the treatment arm and \(A=0\) for the control arm. Let \(D_{t+1}=1\) mean that a participant withdraws after visit \(t\), before the next planned visit; \(D_{t+1}=0\) means the participant remains under follow-up for that next visit. Let \(H_t\) denote information recorded by the time of the withdrawal decision, such as baseline covariates, prior observed outcomes, and symptoms recorded at visit \(t\). An arm-specific dropout probability is \(P(D_{t+1}=1\mid A=a,H_t)\), evaluated separately for \(a=0\) and \(a=1\).

This definition describes dropout given observed information. To classify the mechanism, also consider whether the dropout probability depends on an outcome that is not observed. In this tutorial, \(Y_{t+1}\) denotes the outcome intended for the next visit, which is unavailable if the participant withdraws before that visit.

Key distinction: Dropout is MAR conditional on treatment arm and observed history when, after conditioning on \(A\) and the relevant recorded history \(H_t\), its probability does not additionally depend on the missing outcome \(Y_{t+1}\). It is MNAR when that probability still depends on \(Y_{t+1}\), or other missing outcome information, after conditioning on what was observed. The mechanism may differ between arms while remaining MAR within each arm.

Rates Are Not Mechanisms

A simple comparison begins with the proportion who withdrew in each randomized arm. These proportions should use a clearly defined denominator and follow-up window: for example, all participants randomized to each arm and withdrawal before the same planned visit. The comparison can show that dropout was more frequent in one arm, but it cannot by itself establish whether the mechanism was MAR or MNAR.

Even if the conditional dropout rule is identical across arms, the overall rates can differ because the arms have different distributions of observed histories. For a binary symptom \(S_t\), the arm-specific rate can be written as a weighted average:

P(D_{t+1}=1\mid A=a) =\sum_{s\in\{0,1\}} P(D_{t+1}=1\mid A=a,S_t=s)\,P(S_t=s\mid A=a).

The weights \(P(S_t=s\mid A=a)\) describe how common each symptom status is in that arm. Thus, two arms can have different overall dropout rates even if their symptom-specific withdrawal probabilities are the same, simply because symptoms occur at different rates. Conversely, equal overall dropout rates do not show that the underlying conditional mechanisms are the same.

Worked Example: Comparing Overall Dropout Rates

Suppose a hypothetical trial randomizes 100 participants to each arm. Before the same planned follow-up visit, 30 treatment-arm participants and 15 control-arm participants withdraw. Assume every participant has the same opportunity for follow-up in this comparison, and withdrawal status is known for all 200 participants.

The observed dropout proportions are:

\hat p_1=\frac{30}{100}=0.30,\qquad \hat p_0=\frac{15}{100}=0.15.

The treatment-arm rate exceeds the control-arm rate by \(0.30-0.15=0.15\), or 15 percentage points. The rate ratio is \(0.30/0.15=2.00\): in this hypothetical data set, the treatment-arm dropout proportion is twice the control-arm proportion.

This supports a descriptive statement about these participants and this follow-up window. It does not show that treatment assignment itself caused withdrawal, nor does it tell us whether withdrawal depends on a missing outcome. To classify the mechanism, we would need information about the histories preceding withdrawal and a reasoned assessment of whether relevant outcomes were actually observed.

Classifying Symptom-Driven Withdrawal

Suppose an adverse symptom is measured and recorded at visit \(t\), before the decision about attending the next visit. If withdrawal is associated with that recorded symptom, the symptom belongs in \(H_t\). A mechanism that depends on treatment assignment and this recorded history can be MAR, provided it does not additionally depend on the missing next-visit outcome once the recorded history is taken into account.

The timing and completeness of symptom measurement are essential. If the symptom is only reported by participants who return, or if its severity between visits is not recorded, then the recorded symptom may not capture the information driving withdrawal. Calling the mechanism MAR merely because a symptom appears in a case note or withdrawal reason is not sufficient. The earlier material on Interpreting Recorded Reasons for Withdrawal distinguishes a reported reason from the full information that may have influenced a withdrawal decision.

Worked Example: Dropout Following a Recorded Adverse Symptom

Consider a hypothetical trial with 100 participants in each arm. At visit \(t\), symptoms are recorded for everyone. In the treatment arm, 30 participants report the adverse symptom and 70 do not; in the control arm, the corresponding counts are 24 and 76. The table gives withdrawals before the next planned visit.

Arm	Symptom recorded at visit \(t\)	Withdrawals	Participants in group	Withdrawal proportion
Treatment	Yes	18	30	18/30 = 0.600
Treatment	No	6	70	6/70 = 0.0857
Control	Yes	12	24	12/24 = 0.500
Control	No	4	76	4/76 = 0.0526

Within each arm, withdrawal is more common among participants with the recorded symptom: 60.0% versus 8.57% in the treatment arm, and 50.0% versus 5.26% in the control arm. The overall rates are \(24/100=0.24\) for treatment and \(16/100=0.16\) for control.

If the symptom was measured before withdrawal, was recorded for everyone at risk, and the dropout decision depends only on treatment arm and observed history rather than on the missing next-visit outcome, this is consistent with MAR conditional on the recorded history. The numbers alone cannot verify the final condition: observed data cannot show that an unmeasured future response has no additional relationship with withdrawal. The symptom association is evidence about observed predictors, not proof of MAR.

Assignment-Dependent and Outcome-Dependent Withdrawal

Treatment assignment is observed for every randomized participant. Therefore, dropout may depend on \(A\) and still be MAR, provided any dependence on outcomes is captured by the observed history included in the conditioning set. An arm-specific mechanism allows, for example, participants with the same recorded symptom history to have different dropout probabilities in treatment and control.

By contrast, suppose participants with especially poor next-visit responses are more likely to withdraw, even after accounting for treatment arm and all relevant recorded history. Because \(Y_{t+1}\) is missing for those who withdraw, this is MNAR with respect to that outcome. The fact that participants or investigators give an observable reason for some withdrawals does not rule out additional dependence on an unobserved response.

Worked Example: Assignment Alone Versus an Unobserved Response

In a hypothetical trial, assume that withdrawal probabilities are 0.20 in the treatment arm and 0.10 in the control arm for every participant, regardless of their outcomes or observed history. The probability differs by \(A\), but not by the missing outcome. Since assignment is recorded for everyone, this mechanism is MAR conditional on \(A\). It is not necessary for the two arms to have identical dropout probabilities for the mechanism to be MAR.

Now consider a different hypothetical rule. Within each arm and at the same recorded history, participants whose unobserved next-visit response would be poor have a 0.40 probability of withdrawal, while those whose response would be good have a 0.10 probability. The dropout probability depends on \(Y_{t+1}\) even after conditioning on observed information, so this mechanism is MNAR.

The distinction can be expressed directly. Under the first rule, \(P(D_{t+1}=1\mid A,H_t,Y_{t+1})=P(D_{t+1}=1\mid A,H_t)\). Under the second, those conditional probabilities differ according to \(Y_{t+1}\). In practice the second dependence cannot be diagnosed conclusively from the observed outcomes alone, because \(Y_{t+1}\) is not observed for the participants who withdrew.

Arm-Stratified Diagnostics and Model Comparisons

A useful diagnostic is to summarize dropout within each arm across observed histories, such as symptom status, baseline risk, or prior outcome categories. This can reveal whether an overall arm difference is concentrated among participants with particular recorded histories. If data are plentiful, a regression model can summarize these associations. For binary dropout, one possible working model is:

\operatorname{logit}\{P(D_{t+1}=1\mid A,S_t)\} =\beta_0+\beta_A A+\beta_S S_t+\beta_{AS}A S_t.

Here \(S_t\) is a recorded symptom indicator. The interaction coefficient \(\beta_{AS}\) allows the symptom association with dropout to differ by arm. This is a model for dropout given observed information; it does not test whether dropout depends on the missing outcome. A fitted model cannot prove MAR.

Arm-stratified tables and a pooled model with treatment-by-history interactions answer related but not identical descriptive questions. Separate tables make within-arm patterns easy to inspect, but small cells can make the estimated proportions unstable. A pooled model can share information across groups, but its conclusions depend on its functional form and on whether it includes important interactions. Neither approach can recover unobserved responses without additional assumptions.

When comparing arms, check that the dropout event, time window, and denominator are consistent. Also inspect whether symptoms and prior outcomes were recorded before withdrawal for all participants at risk. If follow-up opportunities differ, or planned administrative censoring is counted as withdrawal, the rate comparison may be misleading. Sparse data, incomplete symptom measurement, and few participants in particular history groups also limit what can be inferred.

Define the event and risk set.
Specify the visit interval and count participants eligible to withdraw before that visit in each randomized arm.

Compare arm-specific rates.
Report the numerators, denominators, and proportions, with consistent follow-up definitions.

Condition on recorded history.
Compare rates across relevant symptoms and prior outcomes, separately by arm where data permit.

Classify cautiously.
Dependence on recorded history or assignment can be MAR; residual dependence on missing outcomes is MNAR. Observed summaries cannot establish the absence of that residual dependence.

Common Mistake and Practical Limits

Wrong reasoning: “The treatment arm had more withdrawals, so dropout must be MNAR in that arm.” This confuses the frequency of dropout with the information that predicts it. A higher rate can arise under a MAR mechanism because the treatment arm has more participants with recorded symptoms that predict withdrawal, or because withdrawal depends on assignment itself.

Correction: Compare dropout across arms and observed histories, then ask whether the mechanism could still depend on outcomes that are missing after conditioning on that information. The answer is an assumption about unobserved data, not a conclusion established by a difference in observed rates. Similarly, equal rates do not establish equal mechanisms: opposing differences across symptom groups can cancel in the overall proportions.

The comparison is useful for understanding follow-up and describing how observed predictors relate to withdrawal. Its central limitation is that MAR and MNAR cannot generally be distinguished using observed data alone. The earlier material on Ignorability Conditions for Likelihood-Based Outcome Analysis explains why the missingness assumptions matter for likelihood-based analyses; comparing arms does not remove the need to state and assess those assumptions.

Key takeaway: Treat arm-specific dropout rates as descriptive summaries, not mechanism classifications. A mechanism can differ by randomized arm and remain MAR if it depends only on assignment and observed history; it is MNAR when dependence on missing outcomes remains after conditioning on that information.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A trial has more withdrawals in treatment than control. Does this establish MNAR? Explain.
Participants who report an adverse symptom at a completed visit are more likely to withdraw before the next visit. Under what conditions is this compatible with MAR?
Withdrawal depends on randomized arm but not on any outcome after conditioning on arm. Classify the mechanism.
Why can two arms have different overall dropout proportions even if symptom-specific dropout probabilities are the same?
Can a regression of dropout on arm and recorded symptoms prove that dropout is MAR? Why or why not?

Solutions. (1) No. A rate difference does not reveal whether dropout depends on missing outcomes; compare observed histories and state the assumptions. (2) It is compatible with MAR if the symptom was recorded before withdrawal for participants at risk, relevant observed history is included, and no additional dependence on the missing next-visit outcome remains. (3) MAR conditional on randomized arm, because assignment is observed. (4) The arms may have different proportions of symptomatic participants, changing the weights in the overall rates. (5) No. Such a regression describes dependence on observed predictors; it cannot rule out dependence on unobserved outcomes.