Dropout Driven by Unobserved Current Health

When Current Health Affects Attendance

A participant can miss a scheduled biomarker measurement because their health has worsened, even when that current health is not recorded at the visit. For example, worsening symptoms might make travel difficult, or a participant may feel too unwell to attend. If the relevant health status or biomarker value is unobserved precisely because the visit was missed, dropout may depend on information missing from the data.

The important distinction is not simply whether health affects attendance. It is whether dropout still depends on a value that is unobserved after conditioning on the information actually recorded. If a recorded symptom score fully captures the health information affecting attendance, dropout may be conditionally independent of the missing biomarker given that recorded score. If the decision also depends on current health not captured in the observed history, the mechanism is MNAR.

Here, “current” means the health status at the time of the missed scheduled measurement. The value \(Y_{ij}\) is a well-defined underlying value for the participant at that time, although it was not collected. A biomarker can be coded so that larger values indicate worse health, but that direction must be stated; the sign of a model coefficient has no fixed interpretation without it.

Writing the Mechanism Precisely

Let the observed history just before visit \(j\) include baseline covariates, prior observed biomarker values, and prior attendance. Denote it by \(\mathcal H^{\mathrm{obs}}_{i,j-1}\). A current-health-dependent dropout model can be written as

This is an illustrative model, not a required form for every MNAR mechanism. If larger \(Y_{ij}\) means worse health, \(\gamma>0\) says that higher current biomarker values are associated with greater dropout probability, conditional on the stated history. The corresponding odds ratio for a one-unit increase is \(\exp(\gamma)\). If a recorded history variable explains all the health-related attendance difference, the model should condition on it; dependence that remains on an unobserved current value is the MNAR feature.

The timing matters. If a biomarker is measured at visit \(j\) and its recorded value predicts whether the participant attends visit \(j+1\), dropout at \(j+1\) can depend on observed history. By contrast, if the underlying value at the missed visit \(j\) affects whether that measurement is obtained, that current value is missing and the dependence can be MNAR. This classification concerns what is observed when dropout occurs, not merely whether health is relevant.

Why the Missing Value Matters to the Data Model

A short calculation shows why an ordinary analysis of observed values cannot, by itself, reveal the dropout dependence. Consider one scheduled visit, with biomarker density \(f_\theta(y)\), where \(\theta\) describes the biomarker distribution. Let \(q_\gamma(y)=P(D=1\mid Y=y)\). A participant who drops out contributes no measured \(Y\), so the probability of dropout in the observed data is

This follows by summing over every possible value of the missing \(Y\): the joint density of \(Y=y\) and dropout is \(q_\gamma(y)f_\theta(y)\), and integration removes the unobserved value. The resulting dropout probability depends on both the biomarker distribution and the dropout rule. A participant who remains observed contributes information involving both \(f_\theta(y)\) and \(1-q_\gamma(y)\); their observed biomarker values are therefore drawn from a group selected partly by the dropout mechanism.

If dropout is independent of \(Y\) conditional on observed history, the response mechanism can often be separated from inference about the biomarker distribution under appropriate conditions. With MNAR dropout, that separation generally fails: the distribution of values among those measured need not identify the distribution among those missing. This is a reason to state assumptions about the missingness mechanism rather than treating a standard analysis of observed measurements as automatically sufficient.

The calculation is a justification for concern, not a recipe for identifying \(\gamma\). Since \(Y\) is unavailable for the dropout cases, many combinations of a biomarker distribution and a dropout rule can produce similar observed data. Additional assumptions, auxiliary measurements, or sensitivity analyses are needed to assess how conclusions change under plausible MNAR mechanisms.

Sequential Dropout and Retention

With monotone dropout, a participant who has dropped out does not return for later scheduled measurements. At each visit, the relevant quantity is the conditional dropout probability among those still under follow-up. Overall retention across several visits is obtained by multiplying the conditional probabilities of remaining observed, provided each probability is defined conditional on the participant having remained under observation to that point. Those probabilities may depend on each visit’s unobserved current health.

What Can Be Checked, and What Cannot

Observed data can show whether dropout is associated with recorded history. For example, one can compare dropout frequencies across prior biomarker values, recorded symptoms, or baseline characteristics. Such patterns help assess whether an MCAR assumption is plausible and whether observed predictors of missingness should be included in an analysis. They do not establish whether dropout also depends on a current value that was never measured.

In particular, a lack of association between dropout and recorded variables does not prove MCAR: an unobserved current health factor may still govern attendance. Conversely, an association with a recorded prior biomarker does not by itself prove MNAR. If conditioning on that recorded value removes the association with missingness, the mechanism may be MAR with respect to the recorded data. These distinctions follow the standard definitions of MCAR, MAR, and MNAR.

A practical diagnostic is to define the time ordering and missingness indicator carefully, then examine whether observed predictors of dropout have been represented in the analysis. If current health is measured by a separate symptom score even when the biomarker visit is missed, that score may be useful auxiliary information. But it only changes the classification if it captures the relevant dependence; a noisy proxy need not remove all dependence on unobserved health.

There are limits to what diagnostics can settle. The missing current values cannot be directly compared between dropouts and attendees, so the MNAR dependence is not generally testable from the observed outcomes alone. A defensible analysis therefore needs explicit assumptions and, where relevant, sensitivity analyses that vary the strength or form of dependence on the missing current value. This tutorial’s focus is that missingness mechanism, not a particular joint model or an analysis of future event risk.

Common Pitfall

Incorrect reasoning: “Dropout is related to health, so it must be MNAR.” This is not enough. If the relevant health information is observed and included in the conditioning history, dropout may be MAR. The reverse mistake is also common: “The dropout rate is similar across recorded groups, so it must be MCAR.” Unobserved current health can still distinguish dropouts from attendees within those groups.

Correction: State what is observed at the time of dropout, define the history being conditioned on, and ask whether dropout still depends on an unobserved current value. MNAR is a statement about conditional dependence on missing information, not a synonym for “health-related,” “nonrandom,” or “difficult to diagnose.”