Timing the Dropout Decision and Latent Current Outcomes

Why the Timing of Dropout Matters

A dropout record tells us that an outcome was not collected; by itself, it does not say when the decision to stop occurred. That timing determines which outcome values could have influenced the decision. If a participant completes a visit’s assessment and then withdraws, the outcome at that visit is observed: withdrawal can affect later outcomes, but cannot make that already recorded outcome missing. If withdrawal occurs before the assessment, the outcome scheduled for that visit is unobserved. A model may nevertheless represent the decision as depending on the value that would have been measured.

This distinction is central to a selection model with dependence on the current outcome. “Current” means the outcome at the visit where dropout occurs, not the last outcome previously recorded. The current outcome is latent precisely for participants who leave before its measurement. Thus the model specifies a relationship involving a value that is unavailable for those participants, rather than a directly observed association.

The phrase “would be measured” requires care. It assumes the outcome has a coherent meaning for a participant who drops out, at least under the protocol being modeled. If an outcome is undefined after an event such as death, treating it as an ordinary missing value may be inappropriate; a different estimand or model may be needed. Here, the discussion concerns outcomes that remain conceptually defined but are not collected.

Writing the Timing Into the Model

For a participant still responding before visit \(t\), let \(H_{i,t-1}\) denote the observed history available before that visit, including relevant baseline covariates and outcomes through visit \(t-1\). Define the conditional dropout probability

This is a hazard among participants at risk of dropping out at visit \(t\). The current-outcome dependence is present when \(q_t(y,h)\) varies with \(y\) after conditioning on the prior history \(h\). A logistic specification, for example, can be written

The parameter \(\gamma_t\) describes how the conditional log odds of dropout change with a one-unit increase in the current outcome, holding prior history fixed. Its meaning depends on the outcome’s scale and the chosen model. The earlier material on Logistic Dependence on the Current Outcome introduces this type of response equation; the present issue is how to interpret it when the decision precedes measurement.

The equation is not a claim that the investigators observe \(Y_{it}\) for dropouts. For a participant who leaves before the visit-\(t\) assessment, the model’s conditioning value is latent. The selection model pairs an outcome distribution, such as \(f_t(y\mid h)\), with a dropout probability \(q_t(y,h)\). The first describes possible current outcomes given history; the second describes selection at each possible outcome value.

How Selection Changes the Outcome Distribution

A useful consequence of the timing-based model is that outcomes among dropouts need not have the same distribution as outcomes among participants who remain. Conditional on history \(h\), suppose the current outcome has density \(f_t(y\mid h)\) in the at-risk population. The dropout probability at value \(y\) is \(q_t(y,h)\). Then the distribution of the latent current outcome among participants who drop out is proportional to the population outcome distribution multiplied by the dropout probability:

The denominator is the total conditional probability of dropout, obtained by averaging \(q_t(Y_{it},h)\) over possible current outcomes. For participants who remain, the corresponding factor is \(1-q_t(y,h)\):

These expressions follow from conditional probability: the joint contribution at a given \(y\) is the probability of that outcome multiplied by the probability of the response pattern at that outcome, and division by the marginal probability of the response pattern normalizes the result to integrate to one. Thus, when high values make dropout more likely, high values receive more weight in the dropout group and less weight in the continuing group, all else equal.

Assumptions, Interpretation, and What Can Be Checked

The interpretation above depends on several assumptions. First, dropout is correctly placed relative to measurement: the recorded response indicator must represent whether the scheduled current outcome was obtained. Second, the latent \(Y_{it}\) must be defined for those who drop out. Third, the outcome model \(f_t(y\mid h)\) and the response model \(q_t(y,h)\) must describe the relevant conditional distributions adequately. In applications, the functional form, included history, and scale of \(Y_{it}\) matter; misspecification in either part can distort the implied selection tilt.

Observed records can help assess whether the timing and response coding are plausible. Investigators can check visit dates, withdrawal dates, whether assessments were completed before withdrawal, and whether missingness is monotone as assumed. Among participants whose outcomes are measured, they can examine whether observed outcomes and prior history show patterns consistent with the proposed model. Such checks can reveal coding errors, poor fit in observable portions of the model, or implausible predictions.

They cannot directly verify the relationship between dropout and \(Y_{it}\) among people who dropped out before measuring it: those are exactly the outcomes unavailable in that group. Without additional identifying information or assumptions, the current-outcome dependence is generally not determined by the observed data alone. The earlier material on Identifiability and Untestable Assumptions explains why plausible sensitivity assumptions should be distinguished from empirically verified facts.

Alternatives, Limitations, and a Common Mistake

A model in which dropout at visit \(t\) depends only on the observed history \(H_{i,t-1}\) sets aside direct dependence on the latent \(Y_{it}\), conditional on that history. It is a different assumption, not a conclusion established merely because the current outcome is unavailable. A model where the decision is made after the current assessment instead uses an observed \(Y_{it}\) in the history for a later dropout event. Pattern-mixture models organize outcome distributions by response pattern rather than writing dropout probabilities conditional on outcomes. These approaches describe missingness from different conditional directions; none resolves untestable assumptions by notation alone.

A practical limitation of current-outcome selection is that the relevant association may be weakly identified or unidentified without external information, validation measurements, or additional assumptions. The model can still be useful for sensitivity analysis: specify a range of plausible current-outcome effects and examine how conclusions change. But a numerical coefficient should not be presented as directly estimated from the missing outcomes unless the data and identification assumptions genuinely support that claim.

Common mistake: “The participant dropped out at visit 2, so the visit-2 outcome is just the last observed outcome.” This confuses the last recorded value with the current value whose measurement dropout prevented. The visit-1 outcome is observed history; the visit-2 outcome is latent under premeasurement dropout. A second mistake is to say that a model cannot depend on \(Y_{i2}\) because \(Y_{i2}\) is missing. A probability model can condition on a latent value, but its implications then rely on assumptions that observed data alone may not test.