Classifying Missingness In Clinical Trial Outcomes

Why the Reason for a Missing Outcome Matters

In a clinical trial, “missed visit,” “withdrew,” and “outcome not recorded” describe events in the data collection process. None, by itself, identifies a missingness mechanism. A visit may be missed because of a documented scheduling conflict, because symptoms have worsened but were not recorded, or for a reason unrelated to any participant information. These possibilities have different implications for analysis.

The classification depends on what information could drive whether an outcome is observed, not simply on when it went missing or on the administrative label attached to the event. It also depends on which variables are included in the analysis. A process that is MAR given a recorded symptom history might not be MAR if that history is omitted from the analysis.

Definition: Let \(Y_i=(Y_{i1},\ldots,Y_{iT})\) denote participant \(i\)'s trial outcomes over \(T\) occasions, and let \(X_i\) denote fully observed variables included in the analysis, such as baseline characteristics and randomized treatment assignment. Let \(R_i=(R_{i1},\ldots,R_{iT})\), where \(R_{it}=1\) if the outcome at occasion \(t\) is recorded and \(R_{it}=0\) otherwise. For a particular response pattern \(r\), write \(Y_{i,\mathrm{obs}}(r)\) and \(Y_{i,\mathrm{mis}}(r)\) for the outcomes observed and missing under that pattern. MCAR means the response pattern is independent of the outcomes and included covariates: \(P(R_i=r\mid Y_i,X_i)=P(R_i=r)\). MAR means \(P(R_i=r\mid Y_i,X_i)=P(R_i=r\mid Y_{i,\mathrm{obs}}(r),X_i)\). MNAR means that response still depends on one or more missing outcomes after conditioning on the observed outcomes and included covariates.

These definitions are relative to the specified analysis variables. If a documented adverse event is included in \(X_i\) or among the observed outcomes, response that depends on it may be MAR. If it is not recorded, or if response also depends on a missing outcome after conditioning on recorded information, the process may instead be MNAR. A mechanism is not established just by learning that participants with missing outcomes differ on an observed characteristic: such an association is compatible with MAR and rules out MCAR only if it reflects a genuine association in the population.

Classify the Response Process, Not the Visit Label

For longitudinal outcomes, a response pattern includes which occasions have recorded outcomes. General MAR is a condition for the probability of that entire pattern: given the outcomes observed under the pattern and the included fully observed covariates, the probability of the pattern does not depend on the missing outcomes. The observed outcomes used in this condition are not restricted to those measured before the missing occasion. For example, a later outcome may be observed even when an earlier visit was missed.

A narrower, often useful model assumes that response at occasion \(t\) depends only on information available up to the preceding occasion. Under that particular history-based model, a condition such as \(P(R_{it}=1\mid H_{i,t-1},Y_{it},X_i)=P(R_{it}=1\mid H_{i,t-1},X_i)\), where \(H_{i,t-1}\) is the observed history through occasion \(t-1\), is a sequential MAR condition. It is not the general MAR condition. General MAR conditions on the full set of outcomes observed for the response pattern, which can include later outcomes.

Worked Example: A Missed Visit Related to Recorded Prior Symptoms

Suppose a hypothetical trial has 120 participants scheduled for an outcome assessment at week 8. A symptom score from week 4 is recorded for everyone. Of 40 participants with a high week-4 score, 16 miss the week-8 visit; of 80 participants with a lower week-4 score, 8 miss it. Assume, for this example, that the chance of missing week 8 depends on the recorded week-4 score and baseline variables, but not on the unrecorded week-8 outcome once those variables are taken into account.

Among participants with a high week-4 score, the missing proportion is \(16/40=0.400\), or 40.0%. Among those with a lower score, it is \(8/80=0.100\), or 10.0%. There are \(16+8=24\) missed week-8 outcomes among 120 participants, so the overall missing proportion is \(24/120=0.200\), or 20.0%.

The differing proportions show that the response process is associated with an observed predictor. They are compatible with MAR under the stated assumption: the week-4 score is recorded and, by assumption, no missing week-8 value further drives response. This is not MCAR, since the response probability varies with observed information. The percentages alone cannot verify the assumption about the unrecorded week-8 outcomes.

Worked Example: A Missed Visit Related to a Later Observed Outcome

Consider a different hypothetical trial with 60 participants. Every participant has a recorded outcome at week 12, but some missed the week-8 visit. Among 30 participants with a high week-12 outcome, 9 missed week 8; among 30 with a lower week-12 outcome, 3 missed week 8. Assume the week-8 response process depends on the observed week-12 outcome, but not on the missing week-8 outcome after conditioning on the week-12 outcome and baseline variables.

The week-8 missing proportions are \(9/30=0.300\), or 30.0%, in the high week-12 group and \(3/30=0.100\), or 10.0%, in the lower group. Because week 12 is observed for everyone in this example, dependence on its outcome is dependence on observed information. The process can therefore be MAR, even though a sequential model that conditions only on prior history would not capture this particular dependence. The correct classification uses the full observed information for the response pattern, not just information available before week 8.

Withdrawal and Outcome Recording Are Different Events

Withdrawal from assigned treatment is not automatically missing outcome data. A participant may stop taking the study treatment but continue to attend assessments, or may withdraw from visits while still providing outcomes by telephone or record review. Classify the observation indicator for the outcome of interest: if the outcome was obtained, it is observed, regardless of whether treatment continued. If a post-withdrawal outcome was not obtained, classify that missingness according to the information that could drive its absence.

Worked Example: Withdrawal After a Documented Adverse Event

In a hypothetical trial, 200 participants are followed for a post-treatment outcome. A documented adverse event before withdrawal is recorded for 50 participants. The outcome is missing for 15 of those 50 and for 15 of the other 150 participants. Assume for this example that the probability of obtaining the post-treatment outcome depends on the recorded adverse-event status and baseline variables, but not on the unobserved post-treatment outcome once these are taken into account.

The missing proportions are \(15/50=0.300\), or 30.0%, among participants with the documented event and \(15/150=0.100\), or 10.0%, among those without it. There are \(30/200=0.150\), or 15.0%, missing outcomes overall. Under the stated assumptions, this is compatible with MAR because the adverse-event status is recorded. It is not MCAR because missingness varies with observed information.

The conclusion changes if withdrawal is driven by a worsening symptom that was not recorded and that also predicts the missing post-treatment outcome. That process may be MNAR: after conditioning on the observed data, the chance of a missing outcome still depends on an unobserved value or its unrecorded proxy. The fact that the participant withdrew does not decide between these explanations.

Unrecorded Outcomes and Information About the Recording Process

An outcome can be absent even when a participant attended the visit. A form may be left blank, a measurement may fail, or a specimen may not be processed. For classification, ask what information was available to the person or system deciding whether the outcome would be recorded. A recorded site, visit date, or workload measure can explain a response process under MAR if it is included and the missing outcome has no additional influence. If the unrecorded outcome itself affects whether it is entered, the process may be MNAR.

Worked Example: A Measurement Left Blank at a Busy Clinic

Suppose a hypothetical trial has 90 scheduled assessments. A clinic workload indicator is recorded for every assessment: 30 occurred on busy days and 60 on less busy days. The outcome is unrecorded for 9 assessments on busy days and 3 on less busy days. Assume the recording probability depends on this recorded workload indicator and baseline variables, but not on the missing outcome value after conditioning on them.

The missing proportions are \(9/30=0.300\), or 30.0%, on busy days and \(3/60=0.050\), or 5.00%, on less busy days. There are \(12/90=0.1333\), or 13.3%, missing outcomes overall. The differences are compatible with MAR given the recorded workload and baseline variables. If staff are more likely to leave a form blank when the participant's unrecorded value is extreme, however, the workload explanation alone is insufficient; the process may be MNAR.

Why the General MAR Condition Uses the Whole Pattern

The distinction between a general pattern condition and a history-based condition follows directly from the definition. For a fixed response pattern \(r\), the observed-data probability is obtained by summing or integrating over the missing outcomes. If the pattern probability depends only on the observed outcomes and \(X_i\), that factor can be taken outside this integration:

\begin{aligned} p(Y_{i,\mathrm{obs}},R_i=r\mid X_i) &=\int p(R_i=r\mid Y_{i,\mathrm{obs}},Y_{i,\mathrm{mis}},X_i) p(Y_{i,\mathrm{obs}},Y_{i,\mathrm{mis}}\mid X_i) \,dY_{i,\mathrm{mis}}\\ &=p(R_i=r\mid Y_{i,\mathrm{obs}},X_i) \int p(Y_{i,\mathrm{obs}},Y_{i,\mathrm{mis}}\mid X_i) \,dY_{i,\mathrm{mis}}\\ &=p(R_i=r\mid Y_{i,\mathrm{obs}},X_i) p(Y_{i,\mathrm{obs}}\mid X_i). \end{aligned}

The second line uses MAR for the specific pattern \(r\). It conditions on all outcomes observed under that pattern, including a later outcome if one is recorded. This calculation explains why leaving a later observed outcome out of the conditioning information can make an otherwise valid MAR process appear to violate a narrower sequential condition. The factorization is a property of the assumed response process; it does not show that MAR is true. In likelihood-based analyses, further conditions, including distinct parameters for the outcome and response models, are relevant to ignoring the response model, but those conditions do not make the mechanism empirically verifiable.

Diagnostics, Comparisons, and Limits

A practical classification begins by defining exactly which outcome is missing and the response pattern that represents its absence. Then list the information recorded for participants with and without the outcome: treatment arm, baseline measures, prior outcomes, documented events, later outcomes, site, and administrative records. A response model using observed predictors can reveal associations worth accounting for and can help identify a pattern compatible with MAR. A response model cannot determine whether the missing outcome itself also influenced response.

Check for sparse groups and lack of overlap. If, for example, every participant with a particular recorded adverse event has a missing outcome, the data provide little or no information about outcomes in that subgroup. A fitted response model may then be unstable, and analyses relying on observed-data comparisons require assumptions beyond simply naming a mechanism. Also check that the variable described as a “reason” for missingness was actually recorded for participants with missing outcomes; a reason documented only for those who returned may not describe the entire response process.

Classification	What could drive the response pattern?	Clinical-trial illustration
MCAR	Neither observed nor missing outcomes or included covariates.	An administrative recording failure occurs independently of participant information and outcomes.
MAR	Observed outcomes and included fully observed covariates, but not missing outcomes after conditioning on them.	Post-withdrawal outcome availability depends on a recorded adverse event and baseline severity.
MNAR	A missing outcome still affects response after conditioning on observed information.	A participant is less likely to provide an outcome when an unrecorded worsening symptom, related to that outcome, is present.

These categories are alternatives for describing assumptions about a response process, not labels that can reliably be assigned from the response rate alone. MCAR is the most restrictive. MAR allows dependence on observed information, including observed later outcomes; MNAR allows dependence on missing information after observed information is accounted for. Observed-data patterns may be consistent with more than one mechanism, so a plausible clinical explanation is not proof of the mechanism.

Common Mistakes

Calling every missed visit MNAR. A missed visit can be MAR if its probability depends only on recorded information, such as a prior score or a fully observed later outcome. The correction is to specify the potential drivers and check whether they are observed.
Calling a process MCAR because its reason is “administrative.” An administrative process may still depend on site, calendar time, or treatment arm. If these variables predict missingness, the process is not MCAR relative to them.
Treating treatment withdrawal as the same as outcome missingness. A participant can discontinue treatment and still provide outcomes. Define response for the outcome being analyzed, not for treatment adherence.
Equating general MAR with dependence only on prior history. General MAR conditions on the full observed information for each response pattern. A sequential, history-based condition is a more restrictive model and can omit observed later outcomes.
Claiming that a response model proves MAR. Observed predictors can be examined, but dependence on missing outcomes cannot be ruled out using observed-data associations alone.

Key takeaway: For each missing trial outcome, state the response pattern, the observed information that could drive it, and whether missing outcomes could still affect response after conditioning on that information. The reason for missingness matters only insofar as it specifies those dependencies.

Check Your Understanding

For each question, give a classification under the stated assumptions. The solutions follow each question.

A week-6 outcome is missing more often among participants with a high, recorded week-2 score. Assume the week-6 value has no further effect on response after conditioning on the score and baseline variables. What mechanism is compatible with this process?
Solution: MAR, because response depends on recorded information and, by assumption, not on the missing week-6 outcome after conditioning on it. The association with the week-2 score means the process is not MCAR relative to that score.
A week-4 visit is missed according to a week-8 outcome that is recorded for every participant. Assume no dependence on the missing week-4 outcome after conditioning on week 8 and baseline variables. Is this necessarily MNAR because week 8 occurs later?
Solution: No. It is compatible with general MAR because week 8 is observed for the response pattern. It may violate a narrower sequential condition that conditions only on prior history, but that narrower condition is not the general MAR definition.
A participant stops assigned treatment after a documented adverse event but completes all later outcome assessments. Are those later outcomes missing because of withdrawal?
Solution: No. Treatment discontinuation and outcome observation are distinct. The later outcomes are recorded, so they are not missing for this reason.
A laboratory outcome is more often left blank when the true, unrecorded value is extreme, even after conditioning on recorded site and baseline variables. Which mechanism is indicated by this stipulated process?
Solution: MNAR, because response still depends on the missing outcome after conditioning on the observed information.
Researchers find no association between missingness and the measured baseline variables. Can they conclude that missingness is MCAR?
Solution: No. There may be unmeasured observed-history variables or dependence on missing outcomes. Failure to find an association with selected measured variables does not establish independence from all outcomes and included covariates.