Sequential MAR Across Repeated Follow-Up Visits

Why State MAR Visit by Visit?

In a repeated follow-up study, a participant who misses a later visit may have completed several earlier visits. The information available at each dropout decision therefore changes over time. A single statement that “dropout is MAR” can conceal which outcomes were observed when each decision occurred. Sequential MAR makes the conditioning history explicit at every transition from one visit to the next.

This tutorial concerns monotone dropout: after a participant leaves follow-up, no later study outcomes are observed. Assume that outcomes are collected at a visit before the decision about whether to continue to the next visit. This timing convention matters. It means that a decision made after visit \(t\) can depend on the outcome from visit \(t\), but not on outcomes from later visits under sequential MAR.

The equality is a conditional-independence statement: among participants still observed through visit \(t\), knowing the later outcomes \(Y_{t+1},\ldots,Y_T\) does not further change the dropout probability once \(X,Y_1,\ldots,Y_t\) are given. The condition is about the dropout mechanism, not about whether later outcomes are predictable from earlier ones.

The risk-set qualification is essential. A participant who has already dropped out cannot make a new dropout decision at a later visit. For monotone dropout, \(R_t=1\) means the participant has remained through the earlier visits, so the recorded outcome history available at the transition after visit \(t\) is \(X,Y_1,\ldots,Y_t\).

Reading the Condition at Each Transition

The condition permits dropout probabilities to vary with recorded information. At the transition after visit 1, the available outcome history is \(Y_1\); after visit 2, it is \(Y_1,Y_2\). Baseline information \(X\) can be included at both transitions because it is observed for everyone. The condition rules out an additional dependence on outcomes that have not yet been observed.

For example, if participants with a particular recorded symptom value at visit 2 are more likely to withdraw after that visit, this dependence is compatible with sequential MAR. If the withdrawal decision also depends on the participant’s unobserved visit-3 outcome after conditioning on \(X,Y_1,Y_2\), the visit-2 condition fails. In practice, the complete outcome vector is not observed for people who drop out; the equation defines a statistical assumption, not something that can generally be verified directly from the observed data.

A Short Justification: Combining the Visit-Specific Conditions

Sequential MAR is specified transition by transition, but the transition probabilities combine to describe a whole dropout path. To see this for three visits, let \(q_1=P(D_1=1\mid R_1=1,X,Y_1)\) and \(q_2=P(D_2=1\mid R_2=1,X,Y_1,Y_2)\). The chain rule for conditional probabilities gives the probability of dropping out after visit 2, conditional on the outcome history, as the probability of remaining after visit 1 multiplied by the probability of dropping out after visit 2 among those who remained.

where sequential MAR makes \(q_1\) independent of \(Y_2,Y_3\) given \(X,Y_1\), and makes \(q_2\) independent of \(Y_3\) given \(X,Y_1,Y_2\). Similarly,

The path probabilities therefore contain no additional dependence on outcomes that were unobserved at the corresponding dropout decisions. This is a consequence of the chain rule plus the visit-specific conditions; it does not say that \(Y_3\) is independent of earlier outcomes or of the response pattern in every unconditional analysis.

Worked Example: History Dependence Is Not MCAR

Sequential MAR does not require the dropout decision to be unrelated to all outcomes. It allows dependence on the history available at the decision time. For instance, suppose in a hypothetical study that the probability of dropout after visit 1 is 0.10 for participants with \(Y_1<5\) and 0.30 for those with \(Y_1\geq5\), with no dependence on \(Y_2\) or \(Y_3\) after conditioning on \(Y_1\). Suppose that after visit 2 the dropout probability is 0.20 for each fixed \(X,Y_1,Y_2\), with no dependence on \(Y_3\). These specifications satisfy sequential MAR: each decision depends only on information recorded by that time.

They do not satisfy MCAR, because the first dropout probability varies with the observed outcome \(Y_1\). For a participant with \(Y_1=7\), the visit-1 dropout probability is 0.30; for one with \(Y_1=3\), it is 0.10. MCAR would rule out this dependence on observed outcomes as well as dependence on unobserved outcomes. Thus sequential MAR is less restrictive than MCAR, but it is still an assumption that excludes dependence on future outcomes after conditioning on observed history.

Assumptions, Checks, and Limitations

The condition is meaningful only relative to a defined history and timing of measurement. Specify which outcomes and covariates have actually been recorded when each decision occurs. If a visit outcome is collected at withdrawal, it belongs to the recorded history for the subsequent decision; it should not be treated as missing simply because withdrawal occurred at that visit. The condition also presumes the monotone setting used here. Temporary missed visits followed by later observed outcomes require a different description of the response process.

Useful checks are mostly checks of study design and data definitions: confirm that dropout is permanent under the analysis definition, that each dropout decision is assigned to the correct visit transition, and that the proposed conditioning history contains only information available at that time. Observed dropout rates can be summarized by recorded histories, but such summaries cannot establish that dropout is unrelated to outcomes that remain unobserved. In particular, a model that predicts dropout well from observed history does not by itself demonstrate sequential MAR.

The main alternatives make different assumptions. MCAR says response is independent of outcomes, a stronger restriction. A general MAR condition is also expressed relative to observed information, but sequential MAR emphasizes the visit-specific risk sets and the history available at each transition. An MNAR mechanism allows dependence on outcomes that are unobserved at the relevant decision, as in the preceding example. These labels describe assumptions about the mechanism, not permanent properties of a dataset: the classification depends on the variables and history being conditioned on.

Sequential MAR does not prove that a particular likelihood analysis is valid by itself, nor does it establish that the specified outcome model is correct. Its role is to state precisely which outcome information the dropout decisions may use. Because the relevant future outcomes are missing for dropouts, the key restriction cannot usually be tested from observed data alone; it should be assessed using subject-matter knowledge and sensitivity to plausible departures.