MCAR In Longitudinal Follow-Up

What MCAR Means in Follow-Up Studies

In a longitudinal study, each participant may contribute outcomes at several scheduled visits, then stop attending. The central question for MCAR is not simply whether some outcomes are missing. It is whether the pattern of being observed is independent of the outcomes the participant would have had across the study. If participants with different outcome trajectories have different chances of dropping out, the missingness is not MCAR—even when the reason for dropout is not recorded.

Use $Y=(Y_1,\ldots,Y_T)$ for a participant’s complete outcome trajectory across $T$ visits, including outcomes that are not observed. Let $R=(R_1,\ldots,R_T)$, where $R_t=1$ means the outcome at visit $t$ is observed and $R_t=0$ means it is not. For monotone dropout, once observation stops, later outcomes are also unobserved. The earlier material on representing longitudinal observation and dropout indicators establishes this indicator notation; here the focus is the relationship between the full pattern $R$ and the complete outcomes $Y$.

Definition: Dropout is missing completely at random (MCAR) with respect to the repeated outcomes when the full observation pattern $R$ is independent of the complete outcome trajectory $Y$. Equivalently, for every possible observation pattern $r$ with positive probability and every possible complete trajectory $y$, $$ P(R=r\mid Y=y)=P(R=r). $$ The probability of a dropout pattern therefore does not vary with any of the participant’s observed or unobserved outcome values.

Independence here concerns the joint pattern, not merely each visit indicator considered separately. Also, the indicators do not have to be independent of one another. Under monotone dropout they are necessarily related: for example, observing a late visit implies that earlier scheduled visits were observed. MCAR says that the distribution of this pattern does not depend on the outcome trajectory.

This outcome-focused definition asks whether dropout depends on repeated outcomes. In the strictest use of MCAR, the observation pattern is independent of the entire complete-data record, including measured covariates as well as outcomes. If the target claim includes covariates, independence must hold for those too. The examples below state clearly when the claim is limited to outcomes.

Consequences of the Independence Condition

A useful implication is that participants in different dropout-pattern groups have the same distribution of complete outcomes. This result is a way to understand the definition, not a way to establish MCAR from ordinary follow-up data: the outcomes missing after dropout are usually unavailable for comparison.

Result: If MCAR holds and $P(R=r)>0$, then the distribution of the complete outcome trajectory is the same in pattern group $r$ as in the full study population: $f(y\mid R=r)=f_Y(y)$.

To see this, apply the conditional-probability formula to the joint distribution. For any $y$ in the support of $Y$,

f(y\mid R=r) =\frac{P(R=r\mid Y=y)f_Y(y)}{P(R=r)} =\frac{P(R=r)f_Y(y)}{P(R=r)} =f_Y(y).

The second equality uses MCAR. The conclusion follows because the factors $P(R=r)$ cancel. Thus, under MCAR, a participant’s complete trajectory is not systematically shifted toward higher or lower values because that participant belongs to a particular dropout-pattern group.

Worked Example: Dropout Generated Independently of Outcomes

Consider three follow-up visits after baseline. Suppose each participant who is still attending has a 0.10 probability of stopping before the next visit, generated by a random process that does not use any outcome values. Calculate the probabilities of the four possible last-observed visits, and assess MCAR.

Let $D=0$ mean dropout before the first follow-up visit, $D=1$ mean the first visit is the last observed, $D=2$ mean the second is the last observed, and $D=3$ mean all three follow-up visits are observed. Dropout at each step is determined by the same outcome-independent random rule. Therefore,

\begin{aligned} P(D=0\mid Y=y)&=0.10,\\ P(D=1\mid Y=y)&=(0.90)(0.10)=0.0900,\\ P(D=2\mid Y=y)&=(0.90)^2(0.10)=0.0810,\\ P(D=3\mid Y=y)&=(0.90)^3=0.729. \end{aligned}

The probabilities sum to $0.10+0.0900+0.0810+0.729=1.000$. Crucially, these expressions are identical for every possible $y$. For instance, the probability of $D=2$ is 0.0810 whether the trajectory is $y=(4,5,6)$ or $y=(8,3,2)$. Under the stated randomization assumption, dropout is MCAR with respect to the outcomes. The calculation does not show that real-world dropout is MCAR; it shows why an outcome-independent design mechanism would satisfy the definition.

When an Outcome Association Rules Out MCAR

MCAR is a strong condition. Since the definition requires independence from every outcome value, demonstrating dependence on even one outcome is enough to refute it. In practice, the most accessible evidence is often an association between dropout at a later visit and outcomes already observed at an earlier visit. Such an association concerns a component of $Y$, so it is inconsistent with outcome-based MCAR.

Worked Example: Dropout Rates Differ by Last Observed Outcome

In a hypothetical follow-up study, 40 participants have an observed outcome at visit 2. Before visit 3, 8 of 20 participants with a low visit-2 outcome drop out; among 20 participants with a high visit-2 outcome, 2 drop out. Compare the dropout probabilities and assess MCAR.

The estimated visit-3 dropout proportions are

\hat p_{\text{low}}=\frac{8}{20}=0.400, \qquad \hat p_{\text{high}}=\frac{2}{20}=0.100.

The difference is $0.400-0.100=0.300$, or 30 percentage points, and the observed dropout proportion in the low-outcome group is four times that in the high-outcome group: $0.400/0.100=4.00$. These sample quantities alone do not prove that the population probabilities differ; sampling variability remains. But if the underlying dropout probability truly differs by the visit-2 outcome, then $P(R\mid Y)$ varies with an outcome value, and MCAR is false.

The comparison is interpretable because both groups were observed at visit 2 and were eligible to be followed at visit 3. It does not require knowing visit-3 outcomes for participants who dropped out. It does, however, address dependence on the observed visit-2 outcome—not dependence on all unobserved outcomes.

What Complete-Outcome Validation Could Show

The strongest direct assessment would include complete outcome trajectories for people who otherwise would have dropped out—for example, because a separate validation process obtained their later outcomes. Such data make it possible to compare outcomes across dropout patterns. Without them, the distribution of unobserved outcomes among those who dropped out cannot usually be inspected directly.

Worked Example: Comparing Full Outcomes Across Dropout Groups

Suppose a hypothetical validation study obtains visit-3 outcomes even for participants who stopped attending. Among 30 participants who dropped out before visit 3, 18 have a high visit-3 outcome and 12 have a low outcome. Among 70 who remained in follow-up, 21 have a high outcome and 49 have a low outcome. Compare the high-outcome proportions across patterns.

Among participants who dropped out, the high-outcome proportion is $18/30=0.600$. Among those retained, it is $21/70=0.300$. The difference is $0.600-0.300=0.300$, or 30 percentage points. Under MCAR, the distribution of complete outcomes should be the same across these groups; these observed proportions show a substantial discrepancy in this hypothetical validation data.

The same association can be expressed as dropout proportions by visit-3 outcome. There are $18+21=39$ high-outcome participants, of whom 18 dropped out, and $12+49=61$ low-outcome participants, of whom 12 dropped out:

\hat p_{\text{dropout}\mid\text{high}}=\frac{18}{39}=0.462, \qquad \hat p_{\text{dropout}\mid\text{low}}=\frac{12}{61}=0.197.

These calculations describe a hypothetical validation sample; they are not a test that definitively proves dependence in a population. If the full trajectories were not obtained, the visit-3 outcomes for dropouts would be unknown and this direct comparison could not be made from the usual follow-up records.

Diagnostics, Alternatives, and Limitations

A practical assessment uses the evidence in a deliberate order. First, examine whether the study design makes an outcome-independent dropout process plausible. A genuinely random follow-up schedule may support that argument. Second, compare dropout rates across outcome information available before dropout, using the relevant at-risk participants at each visit. Third, check whether the data show other patterns inconsistent with the proposed mechanism. A clear association between an observed outcome and subsequent dropout is evidence against MCAR; a lack of a detected association is not proof of MCAR.

This asymmetry matters because observed-data diagnostics cannot generally check independence from values that were never recorded. Even if dropout rates look similar across groups defined by observed outcomes, unobserved outcomes may still differ between dropout patterns. A test that fails to detect an association may also reflect limited sample size, noisy outcomes, or sparse groups. Report the comparisons and their uncertainty rather than treating a nonsignificant result as confirmation.

The main alternatives in practice are to justify MCAR from design, to investigate observed predictors of dropout, or to use a model that allows dropout to depend on outcome information rather than assuming independence. The first is a design-based argument; the second is a diagnostic that can expose violations; the third changes the assumptions used for analysis. None makes missing outcomes become observed, and each requires its own justification. MCAR’s advantage is conceptual simplicity. Its limitation is that it can be implausible when health, function, treatment response, or other outcomes affect continued participation.

Common pitfall: “Dropout rates did not differ significantly by the last observed outcome, so dropout is MCAR.” This conclusion is too strong. The comparison concerns only the observed outcome information tested, a failure to detect a difference may reflect limited precision, and independence from unobserved outcomes remains unverified. The defensible conclusion is narrower: the analysis did not detect the particular observed-outcome association it examined.

MCAR is a statement about a mechanism, not a label inferred solely from the percentage of participants who dropped out. A high dropout rate can still be MCAR if the pattern is independent of outcomes; a low dropout rate can violate MCAR if dropout depends on an outcome. State which outcomes the independence claim concerns, which evidence supports it, and what the available data cannot assess.

Check Your Understanding

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

In one sentence, state the MCAR condition using $R$ and $Y$.
A hypothetical study uses a random number generator to select participants for withdrawal, without using any outcome values. What feature of the rule supports MCAR?
At an observed visit, later dropout is more common among participants with worse outcomes. If this difference reflects the population mechanism, can MCAR hold with respect to outcomes?
Why does observing no association between dropout and the last measured outcome fail to establish MCAR?
In a validation sample, 12 of 30 dropouts and 15 of 50 retained participants have a high complete outcome. Calculate and compare the two high-outcome proportions. What would a difference imply if it reflected the population?

Solutions. (1) MCAR requires $P(R=r\mid Y=y)=P(R=r)$ for every possible pattern $r$ and trajectory $y$; the full observation pattern is independent of the complete outcome trajectory. (2) The withdrawal rule does not use outcomes, so its pattern probabilities are the same for every trajectory, as long as the randomization is implemented as specified. (3) No. If subsequent dropout probability truly varies with an observed outcome, the pattern depends on a component of $Y$, contradicting MCAR. (4) The comparison checks only the observed outcome tested; it may have low power and says nothing conclusive about dependence on unobserved outcomes. (5) The proportions are $12/30=0.400$ among dropouts and $15/50=0.300$ among retained participants, a difference of $0.100$, or 10 percentage points. If that difference represented the population distributions, it would contradict the implication of MCAR that complete outcomes have the same distribution across dropout patterns.

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