MNAR Through Dependence On Unobserved Outcomes

Why Unobserved Outcomes Matter

In longitudinal follow-up, dropout can be associated with a participant’s outcome history. That association alone does not tell us whether dropout depends on information already recorded or on an outcome that is unavailable when the dropout decision is made. The distinction matters because an analysis using recorded outcomes can condition on the former, but cannot directly condition on the latter.

The earlier material on MAR given the recorded outcome history defines the key contrast: under MAR, dropout probabilities may vary with recorded information, but not with outcomes that remain unobserved after conditioning on that information. Here the focus is the alternative. A mechanism is MNAR when dropout still depends on an outcome that is unavailable at the decision, after accounting for the recorded information used to describe the mechanism.

To express the definition, let \(R_{it}=1\) mean participant \(i\)'s outcome is observed at visit \(t\), and \(R_{it}=0\) mean it is not observed. For a participant still under follow-up at the decision point, let \(D_{it}=1\) indicate dropout at that point. Write \(H_i^{\mathrm{obs}}\) for the recorded information available then, including the relevant observed outcome history. Let \(Y_{iu}\) be an outcome component not available at that decision. The MNAR condition is that, for some values \(y\) and \(y'\),

The conditioning history is held fixed in this comparison. The difference is specifically whether the unavailable outcome changes dropout probability. If the probabilities are equal across unavailable-outcome values for every fixed recorded history, then dropout is MAR relative to that history. This is a statement about the data-generating mechanism, not simply a description of a pattern in one observed sample.

How Outcome-Dependent Observation Changes the Data

Suppose \(Y\) is an outcome of interest, \(R=1\) means it is observed, and \(H\) is the recorded history at the dropout decision. Among participants with a given history \(H=h\), the distribution of outcomes among those observed follows directly from conditional probability:

The numerator weights each outcome value by both its prevalence in that history group and its probability of being observed. The denominator is the total probability of observation in that group. Therefore, when observation probabilities vary with \(Y\), the observed outcomes need not have the same distribution as the outcomes in the full group. With a continuous outcome, the same argument applies using densities and an integral in place of the sum.

This calculation explains a central consequence of MNAR: the observed outcome distribution can be shifted toward outcome values with higher observation probabilities. It does not, by itself, say which direction a real study’s estimate will shift. That depends on the relationship between outcome values and dropout probabilities, as well as on the outcome measure and analysis.

Dependence on Recorded History Is Not Enough

A dropout rate can be strongly associated with recorded outcomes and still be MAR. For example, participants with worse recorded scores may be more likely to withdraw, while participants with the same recorded history have equal dropout probabilities regardless of an outcome that is not observed. The recorded score predicts dropout, but the mechanism in this example does not depend on the unobserved outcome after conditioning on that score.

What Observed Data Can and Cannot Establish

The probabilities in an MNAR definition refer to the mechanism that generated the data. In a study, the outcomes of participants who dropped out are not available for directly comparing dropout probabilities across those outcomes. As a result, observed data can reveal patterns worth investigating, but cannot generally establish MNAR dependence on the missing outcomes without additional assumptions or information.

Diagnostics, Comparisons, and Limitations

Useful checks begin with the recorded data: describe dropout rates by recorded history, inspect whether those rates vary across groups, and check the timing and completeness of outcome recording. Such checks can identify inconsistencies, data-quality problems, and observed predictors of dropout. They cannot test whether dropout also depends on an outcome that is missing. A table showing different dropout rates across recorded score groups is evidence of dependence on recorded history; it is not evidence, by itself, of MNAR.

The main mechanism labels make different claims:

The labels are defined relative to the information specified in the conditioning history. A history that omits a recorded variable can make a dependence appear to involve unobserved information when it would not after conditioning on the fuller recorded history. Conversely, variables that are not recorded cannot be conditioned on as observed history. Careful definition of what is available at each decision is therefore essential.

A common mistake is to reason, “Dropout differs across observed score groups, so it is MNAR.” That conclusion does not follow: differences by recorded scores are compatible with MAR, as the second worked example shows. Another mistake is to infer MNAR because a sample has more high outcomes among dropouts in a simulated or observed data set. Unequal realized proportions can occur by chance even if the underlying dropout probabilities are equal. To describe a stipulated mechanism as MNAR, specify probabilities that differ across unavailable-outcome values at fixed recorded history; to describe data alone, say the observed pattern is consistent with, but does not establish, MNAR.

MNAR does not automatically imply that a particular estimate is biased upward or downward, nor does it identify the magnitude of any bias. Such conclusions require assumptions about the missing outcomes and the analysis being used. The mechanism label is a statement about how observation relates to the full outcomes, not a numerical correction or a guarantee about an estimator. In practice, conclusions that rely on unavailable outcomes should make those assumptions explicit and assess how results change under plausible alternatives.