Dropout Depending On The Unobserved Current Outcome

Why the Timing of the Outcome Matters

At a follow-up visit, a participant may stop attending because their condition has worsened. If worsening is represented by the outcome at that very visit, the outcome is not recorded for someone who drops out before it is measured. This creates an important distinction: dropout can be related to a past outcome that is available in the data, or to a current outcome that is missing. The two situations can produce similar observed patterns, but they imply different missingness mechanisms.

The earlier material on MNAR through dependence on unobserved outcomes establishes the general idea: missingness is MNAR when its distribution, conditional on the observed data, still depends on an outcome that is missing. Here the key is to locate that dependence in time. We compare dropout before visit \(t\) depending on \(Y_t\), the outcome missed at that visit, with dropout depending on \(Y_{t-1}\), the last recorded outcome.

Definition: For participant \(i\) at visit \(t\), let \(Y_{it}\) be the outcome that would be measured at that visit, and let \(R_{it}=1\) if it is observed and \(R_{it}=0\) otherwise. Under monotone dropout, a participant at risk for dropout at visit \(t\) has \(R_{i,t-1}=1\). Define \(D_{it}=1\) when that participant drops out before \(Y_{it}\) is observed, and \(D_{it}=0\) when the participant remains under observation at visit \(t\). The recorded outcome history before the visit is \(H_{i,t-1}=(Y_{i1},\ldots,Y_{i,t-1})\). Dropout depending on \(Y_{it}\), after conditioning on \(H_{i,t-1}\), depends on an outcome missing when \(D_{it}=1\); it is MNAR relative to that observed history.

This definition concerns the statistical relationship between the dropout indicator and the outcome, not a claim about what causes a participant to leave. It also depends on what is observed. If the outcome at visit \(t\) is collected before withdrawal is recorded, it may not be missing for that participant; the timing and data-collection protocol determine whether it is truly the missed outcome.

Current-Outcome Dependence and Its Consequence

A selection model expresses the probability of dropout conditional on outcome values. For example, among participants still at risk, a visit-specific logistic model might be

P(D_{it}=1\mid R_{i,t-1}=1,H_{i,t-1},Y_{it}) =\operatorname{expit}(\alpha_t+\beta_tY_{it}), \qquad \operatorname{expit}(u)=\frac{1}{1+\exp(-u)}.

If \(\beta_t\ne 0\), participants with different values of the current outcome have different dropout probabilities even after the previous recorded history is held fixed. When \(D_{it}=1\), however, \(Y_{it}\) is unavailable. Consequently, this dropout probability cannot be calculated for each dropout from the observed history alone. The dependence is on the outcome missing at the visit, so the mechanism is MNAR under this conditioning set.

The sign of \(\beta_t\) depends on how the outcome is coded. If larger values represent worse symptoms, \(\beta_t>0\) means higher symptom severity is associated with greater dropout probability. If larger values represent better health, the same sign has a different substantive interpretation. The model describes a conditional association; it does not by itself establish that the outcome caused dropout.

Worked Example: Dropout Probability Depends on the Missed Score

Suppose a hypothetical follow-up study measures symptom severity from 0 to 10, with higher values indicating worse symptoms. Among participants at risk at visit 3, suppose the proposed model is \(\operatorname{logit}P(D_{i3}=1\mid H_{i2},Y_{i3})=-2+0.5Y_{i3}\). The history term is omitted from the equation only because its effect is held fixed for this comparison.

For a current score of 2, the linear predictor is \(-2+0.5(2)=-1\). Therefore, \(P(D_{i3}=1\mid Y_{i3}=2)=1/(1+\exp(1))=0.2689\). For a score of 6, the predictor is \(-2+0.5(6)=1\), giving \(P(D_{i3}=1\mid Y_{i3}=6)=1/(1+\exp(-1))=0.7311\).

Thus, under this hypothetical model, the dropout probability is about 26.9% at a score of 2 and 73.1% at a score of 6, for participants with the same recorded history. But a participant who dropped out before visit 3 has no recorded score to substitute into this model. Dependence on that missed score is what makes this a current-outcome MNAR mechanism, not simply the fact that dropout is predictable in some general sense.

The distinction can be seen directly by considering the probability of an observed dropout. If the outcome distribution given history is \(f(y_t\mid H_{t-1})\), and the dropout probability is \(q(1\mid H_{t-1},y_t)\), then the probability of dropout conditional on the recorded history is obtained by averaging over the unobserved current outcome:

P(D_t=1\mid H_{t-1}) =\int q(1\mid H_{t-1},y_t)f(y_t\mid H_{t-1})\,dy_t.

The expression uses a generic participant index, suppressed for readability. The dropout probability observed for a given history is an average over possible current outcomes. It does not reveal which outcome values belonged to the people who dropped out. In contrast, if dropout depends only on the recorded history, then its probability is already specified by that history and no averaging over the missing current outcome is needed.

Why Dependence on the Previous Outcome Is Different

Suppose instead that dropout before visit \(t\) depends on \(Y_{i,t-1}\), and on no unobserved outcomes once the recorded history is given. For a participant at risk, \(Y_{i,t-1}\) was observed. The mechanism can then be written as \(P(D_{it}=1\mid R_{i,t-1}=1,H_{i,t-1})\), a function of recorded information. This is MAR relative to the history \(H_{i,t-1}\), even if participants with poor last-observed outcomes are more likely to leave.

This is the central contrast with dropout after a poor last-observed outcome: “poor outcome” alone does not determine the label. What matters is whether the outcome governing dropout is recorded for the participant who drops out. If the relevant poor score is at \(t-1\), it is observed; if it is the outcome at the missed visit \(t\), it is not. The earlier material on how mechanism labels depend on the conditioning set is relevant here: these labels must always be read relative to the information being conditioned on.

Worked Example: The Same Score Used at Two Different Times

Consider a hypothetical study of pain severity on a 0-to-10 scale. Among 100 participants still attending before visit 4, 40 had a last-recorded visit-3 score of 7 or higher and 60 had a score below 7. Suppose 16 of the 40 in the first group drop out before visit 4, compared with 6 of the 60 in the second group.

The dropout proportions are \(16/40=0.400\), or 40.0%, for the high last-recorded score group and \(6/60=0.100\), or 10.0%, for the lower-score group. This observed comparison shows that dropout is associated with \(Y_{i3}\) in this hypothetical data. Since \(Y_{i3}\) was recorded before participants left, a mechanism depending only on this score and other recorded history is MAR relative to that history.

Now consider a different proposed mechanism: dropout before visit 4 depends on \(Y_{i4}\), the visit-4 pain score. For those who drop out before that measurement, \(Y_{i4}\) is unavailable. Even if the dropout proportions also differ by the visit-3 score, that observed association does not show that the current score is irrelevant. A mechanism depending on \(Y_{i4}\) remains MNAR relative to the recorded history.

A Short Derivation: How Dropout Can Enrich the Missing Group

A simple binary outcome makes the selection effect explicit. Let \(Y_t=1\) indicate a hypothetical deterioration at the missed visit and \(Y_t=0\) indicate no deterioration. Suppose that, among participants with the same recorded history, \(P(Y_t=1)=0.30\), and dropout probabilities are 0.60 if \(Y_t=1\) and 0.10 if \(Y_t=0\). These are hypothetical model values.

P(D_t=1)= P(D_t=1\mid Y_t=1)P(Y_t=1) +P(D_t=1\mid Y_t=0)P(Y_t=0) =(0.60)(0.30)+(0.10)(0.70)=0.25.

Among those who drop out, the probability of deterioration is therefore \(P(Y_t=1\mid D_t=1)=(0.60)(0.30)/0.25=0.72\). Among those retained, the corresponding probability is \(P(Y_t=1\mid D_t=0)=(0.40)(0.30)/0.75=0.16\). The calculation shows how dropout can select a group with a different distribution of the unobserved current outcome. The observed dropout rate alone, 25%, does not tell us that the deterioration probability among dropouts is 72%; that calculation required assumed outcome and dropout probabilities.

Worked Example: Reading a Hypothetical Selection Model

Suppose a hypothetical study uses the binary setup above and proposes dropout probabilities of 0.60 for participants who deteriorate at the missed visit and 0.10 for participants who do not. The prevalence of deterioration among those at risk is assumed to be 0.30. Find the overall dropout probability and the deterioration prevalence among dropouts.

First average the two dropout probabilities over the assumed outcome distribution: \(P(D_t=1)=(0.30)(0.60)+(0.70)(0.10)=0.18+0.07=0.25\). Thus 25.0% of participants at risk are expected to drop out.

The joint probability of deterioration and dropout is \(P(Y_t=1,D_t=1)=(0.30)(0.60)=0.18\). Dividing by the dropout probability gives \(P(Y_t=1\mid D_t=1)=0.18/0.25=0.72\). Under these assumptions, 72.0% of dropouts would have deteriorated at the missed visit. This is a model-based implication, not a value directly observed in the dropout group.

What Observed Data Can and Cannot Show

Observed data can be used to examine dropout rates across categories of recorded history. Large differences may be consistent with dependence on the last-observed outcome, while little difference does not rule out dependence on the missed current outcome. These comparisons describe associations with available information; they cannot directly test whether dropout depends on \(Y_t\) among participants for whom \(Y_t\) is missing.

This is a key diagnostic limitation, not a reason to ignore the possibility. Evidence about the current outcome among dropouts would require information beyond the ordinary observed history—for example, a measurement collected at withdrawal or a follow-up procedure that obtains outcomes for some people who otherwise leave. Such information can help assess assumptions, but it does not automatically establish that the same mechanism holds for everyone.

A model that includes the current outcome in a dropout probability requires assumptions about both the distribution of the unobserved outcome and its relationship with dropout. Different assumptions can produce the same observed dropout rate while implying different outcome distributions among those who left. Therefore, a model fit to observed outcomes alone cannot verify the assumed current-outcome dependence. The dependence must be motivated and examined through sensitivity analysis or additional information; no particular unobserved-outcome model is confirmed merely because it is mathematically specified.

Common Pitfall and Practical Comparison

Common mistake: “Dropout is predictable from the previous score, so it must be MNAR.” This reasoning confuses prediction with dependence on missing values. If the previous score is recorded and the dropout model depends only on recorded history, the mechanism is MAR relative to that history. Conversely, dropout can be MNAR even when its dependence on the missed current outcome is not visible in a table of dropout rates by previous score.

The practical comparison is about which outcome enters the mechanism, not which outcome sounds clinically important. A previous-outcome model uses an available predictor and can describe differing dropout rates across recorded histories. A current-outcome model describes selection on a value that is missing for dropouts and needs unobserved-outcome assumptions. Both can be plausible in a longitudinal setting, and the observed data may not distinguish them decisively.

When specifying or reviewing an analysis, state the visit at which dropout occurs, identify the outcome available immediately before dropout, and name the outcome the proposed mechanism depends on. Then ask whether that outcome is recorded for the participant with \(D_{it}=1\). This simple timing check prevents the most common misclassification.

Check Your Understanding

Answer each question, then compare with the solutions below.

At visit 5, dropout depends only on the recorded visit-4 outcome and earlier recorded history. How is the mechanism classified relative to that history?
At visit 5, dropout probability increases with the visit-5 outcome, which is not measured for those who drop out beforehand. How is the mechanism classified?
In the binary example, prevalence of deterioration is 0.30 and dropout probabilities are 0.60 after deterioration and 0.10 otherwise. What is the overall dropout probability?
Using the values in question 3, what proportion of dropouts are expected to have deteriorated, under the stated assumptions?
Why does finding similar dropout rates across groups defined by the previous recorded outcome not establish that dropout is independent of the missed current outcome?

Solutions. (1) MAR relative to the stated history, because the outcome used to predict dropout is observed. (2) MNAR relative to that history, because dropout depends on a current outcome missing for dropouts. (3) \(0.30(0.60)+0.70(0.10)=0.25\), or 25.0%. (4) The joint probability of deterioration and dropout is \(0.30(0.60)=0.18\); dividing by \(0.25\) gives \(0.72\), or 72.0%. (5) Those rates condition on observed previous outcomes, not on the unobserved current outcomes of dropouts. The comparison cannot directly reveal dependence on those missing values.