Conditional MAR With Baseline And Time-Varying Covariates

Why the conditioning set matters

Dropout can be associated with characteristics that are recorded before a participant leaves a longitudinal study. For example, participants with more severe symptoms at the previous visit may be more likely to withdraw. Their withdrawal is not independent of the recorded data, but that association does not by itself make the mechanism nonignorable: the relevant question is whether dropout still depends on outcomes that are unobserved, after accounting for what was observed.

Baseline characteristics and time-varying covariates can both be part of that observed information. Baseline treatment group, for instance, may affect both later outcomes and the chance of withdrawal. A current symptom score or a recorded adverse event may also predict withdrawal. To assess MAR, however, it is essential to respect timing: a covariate belongs in the conditioning history only if it was actually recorded and available when the dropout decision occurred.

Definition: Let \(Y_{it}\) denote participant \(i\)'s outcome at scheduled visit \(t\), and let \(R_{it}=1\) if that outcome is observed and \(R_{it}=0\) otherwise. Let \(X_i\) be a fully observed baseline covariate and \(Z_{it}\) a time-varying covariate, when recorded. Under monotone dropout, once \(R_{it}=0\), later outcomes are also unobserved. Conditional MAR given baseline and time-varying covariates means that, for each possible response pattern, the probability of that pattern given the full data depends only on the outcomes and covariates observed under that pattern. In symbols, \(P(R=r\mid Y,X,Z)\) is a function only of \(Y_{\mathrm{obs}(r)}, X,\) and \(Z_{\mathrm{obs}(r)}\). Here \(Y_{\mathrm{obs}(r)}\) and \(Z_{\mathrm{obs}(r)}\) denote the values observed under pattern \(r\).

This definition treats the baseline covariate \(X_i\) as fully observed. For a time-varying covariate, only its values recorded under the response pattern can be used in the observed-data conditioning set. If \(Z_{it}\) is measured before the visit-\(t\) dropout decision, it may be available at that decision. If the decision happens before the measurement, it is not. Likewise, a covariate value from after dropout is not part of the observed history simply because it could have existed in a hypothetical complete data set.

In a visit-specific description, suppose a participant is still under follow-up just before visit \(t\). Write \(H_{it}\) for the information actually recorded and available by the dropout decision at that visit. It can include \(X_i\), observed earlier outcomes, and time-varying covariates measured by then. Conditional MAR says that, after conditioning on \(H_{it}\), the dropout probability does not further depend on the unobserved outcome trajectory. The exact contents of \(H_{it}\) depend on the study’s measurement and decision timing.

What belongs in the observed history?

A covariate belongs in the conditioning set when it is both observed and relevant to the proposed description of the dropout process. Relevance is not limited to variables that predict dropout: a covariate may help define the information available at the decision, even if its association with dropout is weak. In practice, attention should focus on plausible common causes or predictors of dropout and later outcomes, such as baseline severity, treatment assignment, recent symptoms, or a recorded adverse event.

Including a covariate does not automatically make a mechanism MAR. A recorded symptom score can explain part of the association between dropout and outcome, yet dropout may still depend on the participant’s unrecorded current health or future outcome. Nor does a statistical association establish that a covariate caused withdrawal. The assessment concerns dependence in the mechanism, not causal interpretation.

Worked Example: A Recorded Symptom Score Predicts Dropout

Hypothetical data: In a study with 100 participants still under follow-up at visit 2, a symptom score from visit 1 was recorded for everyone. Among 40 participants with a high score, 12 withdrew before visit 2; among 60 with a low score, 6 withdrew. Assume for this example that the withdrawal probability depends on the high/low score, but not on visit-2 or later outcomes once that score and the observed history are held fixed. Determine what this implies about MAR and calculate the observed withdrawal rates.

For the high-score group, the withdrawal rate is \(12/40=0.30\), or 30%. For the low-score group, it is \(6/60=0.10\), or 10%. The observed rate difference is \(0.30-0.10=0.20\), or 20 percentage points.

The hypothetical mechanism makes withdrawal more likely for participants with a high recorded score, so dropout is not independent of the observed history. But under the stated assumption, withdrawal does not additionally depend on unobserved outcomes after conditioning on the score and the other recorded information. That is consistent with conditional MAR. The rate comparison alone would not establish this assumption; the absence of further dependence on unobserved outcomes is part of the stipulated mechanism, not something these observed rates can prove.

A short justification using dropout probabilities

The connection between a visit-specific dropout rule and a response-pattern probability can be seen by considering monotone dropout. Define \(D_{it}=1\) when a participant still under follow-up withdraws at visit \(t\). Let \(A_{it}\) be the event that the participant remains under follow-up just before that decision. For a participant who withdraws at visit \(k\), the probability of that dropout path can be written as a product of the conditional probabilities of remaining and then withdrawing:

P(R=r\mid Y,X,Z) = \left\{\prod_{t<k} P(D_{it}=0\mid A_{it},\text{full data})\right\} P(D_{ik}=1\mid A_{ik},\text{full data}).

If each factor depends on the full data only through information observed by its decision time, then every factor in this product depends only on observed outcomes and covariates for the eventual response pattern. Consequently, the product also depends only on the values observed under that pattern, which is the pattern-level conditional MAR requirement. This argument relies on the specified dropout process and the correct timing of the covariates; it does not show that the condition can be verified from observed data alone.

Worked Example: Checking a Visit-Specific Rule

Hypothetical mechanism: At visit 3, among participants still under follow-up, the probability of dropout is \(0.05\) for participants assigned to treatment A and with a low visit-2 symptom score, and \(0.20\) for those assigned to treatment A and with a high visit-2 score. For treatment B, the corresponding probabilities are \(0.10\) and \(0.30\). Treatment assignment and the visit-2 score are recorded before the visit-3 dropout decision. Assume these probabilities do not vary with the unobserved visit-3 outcome or later outcomes once those recorded values are fixed.

Consider a participant assigned to treatment B with a high visit-2 score. The specified conditional dropout probability is \(0.30\), so the probability of remaining under follow-up at that decision is \(1-0.30=0.70\). Both quantities are functions of recorded values: treatment B and the high visit-2 score.

Because the visit-3 decision is assumed not to depend further on the unobserved visit-3 or later outcomes, this rule is compatible with conditional MAR. It is not MCAR: the dropout probability varies with treatment and the recorded score. The conclusion also depends on the timing assumption. If the score had been measured only after the dropout decision, it could not serve as an observed predictor for that decision.

When covariate adjustment does not suffice

A common error is to reason that dropout is MAR simply because it is predictable from recorded variables. Predictability is not the condition. The question is whether the recorded history accounts for all dependence of the dropout process on the missing outcomes (and, under the stated formulation, on the covariate information relevant to the response pattern). An observed predictor can be useful while leaving an unobserved driver in the mechanism.

Worked Example: An Unobserved Outcome Still Matters

Hypothetical mechanism: Among participants with the same baseline group and the same recorded visit-1 symptom score, suppose the chance of withdrawal before visit 2 is 0.10 for those whose unobserved visit-2 outcome would be low and 0.40 for those whose unobserved visit-2 outcome would be high. Assume the observed covariates do not distinguish these two groups.

Conditional MAR would require the dropout probability to be the same across those unobserved outcome values after conditioning on the observed history. Instead, the probabilities differ by \(0.40-0.10=0.30\), or 30 percentage points. The mechanism therefore depends on an unobserved outcome even after accounting for the recorded baseline group and symptom score, so it is not MAR under this conditioning set.

Adding the observed score to an analysis cannot remove that remaining dependence, because participants with the same recorded score still have different dropout probabilities according to an outcome that is missing for them. This hypothetical mechanism illustrates why a plausible set of covariates can support an MAR argument but cannot guarantee it.

Assessment, comparisons, and limitations

MAR is an assumption about how response patterns relate to full data. Observed records can show whether dropout rates differ across observed baseline or time-varying covariate values, and can reveal that a model omitting those variables fails to describe recorded patterns. They cannot directly show whether people who dropped out had different unobserved outcomes from people with the same observed history. Thus, data summaries and dropout models can inform a reasoned assessment, but they cannot confirm conditional MAR.

The conditioning set also depends on what is being treated as observed. If a time-varying covariate is itself missing, or is only recorded for participants who remain in follow-up, it cannot simply be treated as a fully observed adjustment variable. Its missingness and the timing of its measurement must be accounted for in the description of the observed information. Similarly, if withdrawal depends on a covariate value that would have been recorded after withdrawal, that value cannot be used as though it were available at the decision.

The main alternatives differ in their conditioning assumptions:

Mechanism description	Dependence allowed	Key distinction
MCAR	Dropout does not depend on the full data.	Stronger than conditional MAR; observed history does not explain variation in dropout.
MAR given outcomes alone	Dropout may depend on observed outcomes but not unobserved outcomes after conditioning on them.	May be inadequate when recorded baseline or time-varying covariates also help determine dropout.
Conditional MAR given outcomes and covariates	Dropout may depend on the outcomes and covariates observed under the response pattern, but not further on unobserved outcomes.	Requires the covariates used in the conditioning set to be genuinely observed at the relevant time.
MNAR	Dropout depends on unobserved outcomes even after conditioning on the observed history.	Adding observed covariates does not remove the remaining dependence.

Conditional MAR is therefore a substantive modelling assumption, not a label determined by the dataset or by finding a strong predictor of dropout. A useful assessment identifies a plausible decision-time history, explains why it is observed, and considers whether unrecorded severity or future outcomes could still affect withdrawal after conditioning on that history. The assumption may be more credible with rich, well-timed measurements, but it remains untestable from the observed outcomes alone.

Common pitfall: “Dropout is predictable from baseline treatment and the last recorded symptom score, so dropout is MAR.” Predictability from observed covariates is not enough. The correction is to ask whether dropout still depends on an unobserved outcome after conditioning on those covariates and all other information available at the decision. If it does, the mechanism is not MAR under that conditioning set.

Check Your Understanding

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

A study records baseline treatment group and a symptom score immediately before each visit’s withdrawal decision. Can both be included in the observed history? What timing condition matters?
Among participants with the same recorded history, hypothetical dropout probabilities are 0.12 for a low unobserved current outcome and 0.28 for a high unobserved current outcome. Is the mechanism conditionally MAR?
Why does observing that withdrawal rates differ by baseline severity not, by itself, rule out MAR?
A time-varying covariate is measured only after the visit-3 withdrawal decision. Can it be included as an observed predictor of that decision? Explain.
What is the main distinction between MCAR and conditional MAR given observed covariates?

Solutions. (1) Yes, if both were recorded and available before the decision; a measurement made after the decision is not in its observed history. (2) No. The probabilities differ by \(0.28-0.12=0.16\), or 16 percentage points, across unobserved outcome values despite the same observed history. (3) MAR allows dropout to depend on observed baseline severity; it requires no further dependence on unobserved outcomes after conditioning on observed information. (4) No. It was not available at the decision, so it cannot explain that decision as an observed predictor. (5) MCAR allows no dependence on the full data, whereas conditional MAR allows dependence on observed outcomes and covariates while excluding further dependence on unobserved outcomes.