Why the Dropout Mechanism Matters
When a participant stops attending scheduled visits, later measurements are missing. The practical question is not only how many measurements are missing, but whether the chance of missingness is related to the measurements themselves. If participants with different health histories have different chances of remaining under observation, the observed records may not represent the full set of health trajectories. A precise description of the dropout mechanism helps clarify what an analysis assumes.
The names MCAR, MAR, and MNAR describe relationships between the observation pattern and the data that would have been available under complete follow-up. They are assumptions about how data become missing, not labels that can usually be assigned just by looking at a table of observed measurements.
Notation and Definitions
Suppose participant \(i\) has scheduled longitudinal measurements \(Y_{i1},\ldots,Y_{iJ}\), with larger \(j\) denoting later visits. Let \(Z_i\) denote fully observed baseline information, such as treatment group or age. For each measurement, let \(R_{ij}=1\) if \(Y_{ij}\) is observed and \(R_{ij}=0\) if it is missing. The vector \(R_i=(R_{i1},\ldots,R_{iJ})\) records the participant's observation pattern. Write \(Y_{i,\mathrm{obs}}\) for the measurements observed under that pattern and \(Y_{i,\mathrm{mis}}\) for the measurements that are missing.
In the definitions below, probabilities for a missingness pattern are conditional on the complete data values and the baseline information. The classification concerns whether that probability changes with the unobserved measurements after the observed information has been taken into account.
MCAR is the strongest of these three conditions: neither observed history nor unobserved outcomes alter the observation probability. If dropout probabilities differ across observed baseline groups, the mechanism is not strict MCAR as defined here, even if there is no further dependence on outcomes within those groups. It can be MAR because the group information is observed. In some settings, authors use the more specific phrase conditionally MCAR given baseline covariates; that qualification should be stated rather than silently treating the mechanism as strict MCAR.
For monotone dropout, a participant who misses a visit is assumed not to return for later scheduled visits. At visit \(j\), the observed history before that visit can be written \(H_{i,j-1}=(Z_i,Y_{i1},\ldots,Y_{i,j-1})\), for a participant observed through visit \(j-1\). Under MAR, the probability of dropping out at visit \(j\) may depend on \(H_{i,j-1}\), but it does not additionally depend on the missing value \(Y_{ij}\) or other missing outcomes. Under MNAR, it does. MCAR requires that the dropout probability not depend on the observed history either.
Three Worked Classifications
Worked Example: Dropout Independent of All Health Measurements
Consider a hypothetical cohort with a scheduled health score at each visit. Suppose each participant has a 0.10 probability of dropping out, regardless of baseline group, past scores, or the scores that would have been recorded later. Is this MCAR, MAR, or MNAR?
Let \(D_i=1\) indicate dropout by the next visit. The specified mechanism is \(P(D_i=1\mid Y_i,Z_i)=0.10\) for every complete outcome trajectory and baseline value. For example, for two different possible trajectories \(y_a\) and \(y_b\), \(P(D_i=1\mid Y_i=y_a,Z_i)=0.10=P(D_i=1\mid Y_i=y_b,Z_i)\).
The probability does not change with observed or unobserved measurements, so the mechanism satisfies MCAR. If 100 comparable participants were followed, the expected number dropping out would be \(100(0.10)=10\); this expected count is the same whether the cohort has mostly high or mostly low health scores. The label follows from the assumed mechanism, not from whether a particular realized sample happens to contain exactly 10 dropouts.
Worked Example: Dropout Depends on an Observed Past Measurement
In a hypothetical study, a prior visit's biomarker is observed for everyone still in follow-up. Define a high prior biomarker as \(H=1\). Suppose 40% of participants have a high prior biomarker, and the dropout probabilities before the next visit are 0.30 for \(H=1\) and 0.10 for \(H=0\). Assume that, within each prior-biomarker group, dropout does not depend on the current or later missing measurements.
The marginal dropout probability is \(P(D=1)=P(D=1\mid H=1)P(H=1)+P(D=1\mid H=0)P(H=0)\). Substituting the specified values gives \(P(D=1)=(0.30)(0.40)+(0.10)(0.60)=0.12+0.06=0.18\).
Dropout is associated with an observed past measurement: the probability is 0.30 rather than 0.10 for the high-biomarker group. It is therefore not MCAR. Under the stated assumption that the probability has no further dependence on missing measurements, it is MAR, not MNAR. To see how observed-history association can arise, the proportion with high prior biomarker among dropouts is \((0.30)(0.40)/0.18=0.12/0.18=0.667\), approximately 66.7%. This enrichment is compatible with MAR: the dropout differences are explained by information already observed.
Worked Example: Dropout Depends on Unobserved Current Health
Consider a hypothetical visit at which current health status is not recorded for participants who leave before the assessment. Let \(Y=1\) mean poor current health and \(Y=0\) mean good current health. Among participants with the same observed history, suppose 25% would have poor current health, and the dropout probabilities are 0.40 for \(Y=1\) and 0.10 for \(Y=0\).
The dropout probability averaged over current health is \(P(D=1)=P(D=1\mid Y=1)P(Y=1)+P(D=1\mid Y=0)P(Y=0)\). Thus, \(P(D=1)=(0.40)(0.25)+(0.10)(0.75)=0.10+0.075=0.175\). Among those who drop out, the proportion with poor current health is \(P(Y=1\mid D=1)=(0.40)(0.25)/0.175=0.10/0.175=0.571\), approximately 57.1%.
Because the current health value is unobserved for the people who drop out, yet the dropout probability differs according to that value even after fixing the observed history, this mechanism is MNAR. The calculation illustrates the definition; in an actual study, the current health values for dropouts are unavailable, so this difference cannot usually be calculated from the observed records alone.
A Short Justification: Why MAR Can Look Health-Related
The second example gives a useful general point. Let \(H\) denote an observed history group. Even if dropout depends only on \(H\), the overall dropout probability averages the group-specific probabilities:
If the terms \(P(D=1\mid H=h)\) differ across groups, dropouts will have a different distribution of \(H\) from participants who remain. That association is not evidence, by itself, that dropout depends on an unobserved current outcome. MAR allows dependence on observed history. The relevant question is whether dependence on missing outcomes remains after conditioning on all the observed information included in the definition and analysis.
How to Classify a Proposed Mechanism
A useful classification proceeds in a fixed order. First specify what is measured, when it is measured, and which values are missing at the dropout decision. Then write down the variables that determine the probability of dropout. Finally, separate those variables into observed information and values that would be missing for the participant who drops out.
Identify the visit or visits missed and distinguish the observed history from current and future unobserved measurements.
Include baseline information, prior measurements, and any current or future health value specified in the mechanism.
If dropout varies with observed values, strict MCAR does not hold. The mechanism may still be MAR.
If the mechanism depends only on observed information, it is MAR. If it still depends on missing outcomes, it is MNAR.
Diagnostics and Limits of the Labels
Observed data can provide useful checks, but they do not generally identify the missingness mechanism. For example, researchers can compare observed prior measurements and baseline characteristics between participants who later drop out and those who remain. A clear association between dropout and an observed history variable contradicts strict MCAR, subject to sampling variation and the quality of the measurements. It does not distinguish MAR from MNAR.
Likewise, finding no apparent association between dropout and the observed variables examined does not establish MCAR. The study may have limited sample size, the relevant observed predictor may not have been measured or modeled, or dropout may depend on an outcome that is missing precisely because of dropout. MAR is also relative to the information observed and included: if an observed predictor is omitted from the analysis, dropout may appear unexplained even though it depends on that predictor.
A common diagnostic is to model the probability of remaining under observation or dropping out using baseline variables and prior observed measurements. Such a model can reveal whether observed history predicts dropout and can help assess whether a proposed MAR assumption is plausible. It cannot test whether dropout also depends on unobserved current health. A model that fits the observed dropout patterns well is not proof of MAR.
Comparing the Mechanisms
| Mechanism | Can depend on observed history? | Can depend on missing outcomes after conditioning on observed data? | Illustrative dropout rule |
|---|---|---|---|
| MCAR | No | No | A fixed dropout probability for everyone |
| MAR | Yes | No | A higher dropout probability after a high observed prior biomarker |
| MNAR | Yes | Yes | A higher dropout probability with poor current health that is unobserved at dropout |
These categories describe assumptions, not a ranking of methods or a guarantee that one assumption is true. MCAR is a special, restrictive case of MAR. MNAR describes dependence left over on missing values after conditioning on observed information. The categories also depend on what the analyst defines as observed: a predictor measured in the study but omitted from a particular model may need to be added to the conditioning information before classifying that model's assumption.
Common Mistakes and Practical Significance
- Calling any health-related dropout MNAR. If dropout depends on a prior health measurement that was recorded, that dependence can be MAR. The key is whether dropout depends on an unobserved measurement after observed history is accounted for.
- Calling dropout MCAR because its cause seems unrelated to health. A scheduling issue may still be more common in a particular observed baseline group. If dropout probabilities differ by that group, strict MCAR does not hold.
- Treating an observed association as proof of MNAR. Different dropout rates across observed histories can occur under MAR, as the numerical example showed.
- Claiming that data prove MAR. The outcomes needed to assess dependence on missing values are unavailable. Observed data can challenge simple assumptions, but generally cannot verify MAR against MNAR.
The distinction matters because conclusions from longitudinal analyses can be sensitive to assumptions about why measurements are missing. MCAR, MAR, and MNAR make different claims about how observed participants relate to those with missing follow-up. In particular, classifying a mechanism as MAR does not mean dropout is harmless or unrelated to health; it means its relationship to dropout is represented by observed information rather than by the missing outcomes themselves. Any substantive classification should therefore be stated as an assumption and justified using the study's follow-up process and available measurements.
Check Your Understanding
For each question, classify the stated mechanism or explain what can be concluded. Then compare your reasoning with the solutions.
- Dropout occurs with probability 0.08 for every participant, regardless of baseline variables, observed scores, and missing scores. Classify the mechanism.
- Dropout is more likely after a participant has a high score at the previous visit, and the current unrecorded score has no additional effect. Classify the mechanism and explain why it is not MCAR.
- Researchers find that dropout is associated with a recorded baseline severity group. Does this establish MNAR? What is the strongest direct conclusion about strict MCAR?
- Among participants with the same observed history, dropout is more likely when their unobserved current health is poor. Classify the mechanism.
Solutions. (1) MCAR: the dropout probability does not depend on complete outcomes or baseline information. (2) MAR: dropout depends on an observed past score, so it is not MCAR, but it has no remaining dependence on missing values. (3) No. The association does not establish MNAR; it contradicts strict MCAR if the association reflects the population mechanism, and the mechanism could be MAR. (4) MNAR: dependence on the missing current health remains after conditioning on the observed history.