Modeling Dropout Driven by Prior Outcomes

Why Prior Outcomes Belong in a Dropout Model

In a longitudinal study, a participant’s decision or ability to remain under observation may depend on how their health has been changing. For example, participants whose symptoms were severe at the previous visit may be more likely to miss the next visit. If the censoring model ignores that observed symptom history, it can assign the same retention probability to people whose follow-up prospects are quite different.

The key modeling question is therefore not simply whether dropout is associated with the outcome. It is whether the variables used to model retention include the outcome information that was observed before each retention decision. A lagged outcome can be a valid predictor even when the outcome at the visit being missed is unavailable.

A discrete-time model for \(q_t\) can include the most recent outcome, a summary of earlier outcomes, and other predictors known before the decision. For example, a pooled logistic model might specify \(\operatorname{logit}\{q_t(H_{t-1})\}=\alpha_t+\beta^\top X+\gamma Y_{t-1}\), where \(X\) is a baseline covariate vector and \(\alpha_t\) allows the baseline retention rate to vary by visit. The form should reflect the study’s observation schedule and substantive knowledge; the equation is a model, not a guarantee that the true retention probability has that form.

The timing rule is essential. If the decision to attend visit \(t\) occurs before \(Y_t\) is measured, \(Y_t\) cannot be a predictor in the model for retention at \(t\). It is unavailable for participants who do not attend. By contrast, \(Y_{t-1}\) can be included if it was recorded before that decision. A model that uses information measured after dropout, or treats an unobserved outcome as though it were observed, does not define a usable observed-history censoring model.

Conditional Independence and the Weighting Rationale

The relevant identifying assumption is sequential conditional independence: among people still observed at the start of a visit, retention at that visit is independent of the future full-data outcomes once the measured past is given. More formally, for each \(t\), assume \((Y_t,\ldots,Y_K)\perp R_t\mid R_{t-1}=1,H_{t-1}\). This assumption says that, within levels of the recorded history, those retained at \(t\) represent those who would have been retained or lost with respect to their future outcomes. Including a prior outcome is important when it helps make this conditional statement plausible.

Also require positivity: for histories occurring in the target population, \(q_t(H_{t-1})>0\). To estimate a population quantity using inverse retention probabilities, the relevant conditional probabilities must be correctly specified or consistently estimated, and the target expectations must be finite. Positivity is not secured merely by fitting a logistic model: fitted probabilities close to zero indicate that some histories have little support among those retained.

Proof. For a participant retained through \(t-1\), condition on the history \(H_{t-1}\). By the definition of \(q_t\) and sequential conditional independence, the conditional expectation of \(R_t g(H_t)\) is \(q_t(H_{t-1})\) times the conditional expectation of \(g(H_t)\). Dividing by \(q_t(H_{t-1})\) therefore removes the selection at visit \(t\). Repeating this argument backward, one visit at a time, cancels each factor \(q_j(H_{j-1})\) in the product and gives \(E[g(H_t)]\). Participants with \(R_t=0\) contribute zero to the weighted expression; histories needed to evaluate its nonzero terms are observed. Integrability and positivity permit the conditional expectations and divisions used in this argument. \(\square\)

For \(g(H_t)=Y_t\), the identity explains how weighting retained participants can recover the full-data mean at visit \(t\). The lagged outcome is not added to redefine that mean. It helps model who remains observed, so that the weighted observed histories can represent the histories that dropout would otherwise remove.

Building and Interpreting the Model

A practical data set for a discrete-time retention model has one row per participant-visit at which the participant is at risk of dropout. The response indicates whether the participant remains observed at the next visit; the predictors are measured before that retention decision. A model may include visit indicators, baseline covariates, the last observed outcome, and a small number of justified functions of the outcome history. The exact specification depends on the measurement schedule and the hypothesized dropout process.

The choice of lagged-outcome terms should reflect how prior health could drive follow-up. If current health status is predictive, the last value may be appropriate. If deterioration or improvement matters, a recent change such as \(Y_{t-1}-Y_{t-2}\) may carry information beyond the level. If the relationship is not plausibly linear on the log-odds scale, categories, splines, or interactions with visit can be considered. These choices should be based on available pre-decision information, not on whichever specification yields the most convenient weights.

Model Checks, Alternatives, and Limits

The model should be checked for problems that undermine useful adjustment. Review whether each predictor was actually measured before the relevant retention decision, whether the risk set and visit indicators are coded correctly, and whether predicted probabilities are plausible across important histories. Very small probabilities imply large inverse weights and weak empirical support for some retained histories. Sparse combinations of visit, prior outcome, and covariates can also make fitted probabilities unstable. These are warnings about the data and model, not proof that a particular assumption has failed or succeeded.

A model using baseline covariates alone is simpler, but it can leave dropout associated with prior health within baseline strata. A richer observed-history model can better represent that association, though unnecessary terms can increase variance and estimation instability. Outcome regression or imputation are alternative ways to address missing outcomes, but they require their own assumptions and model choices. They do not make the censoring model’s conditional-independence assumption automatically true.

No observed-data diagnostic can establish that dropout is independent of unobserved future outcomes conditional on observed history. If people with the same recorded history still differ in retention because of an unmeasured impending health decline, the assumption may fail. This is dependence on information unavailable at the decision point, not a problem solved by inserting the missing \(Y_t\) into the retention model. In that setting, weighting alone cannot identify the full-data distribution without additional assumptions or information.

• Common mistake: “The outcome predicts dropout, so I should include the outcome at the visit that was missed.” That outcome is unavailable for dropouts and may occur after the retention decision.
• Correction: Include outcomes and other variables observed before the decision, such as \(Y_{t-1}\). State the resulting conditional-independence assumption explicitly and assess whether the measured history plausibly captures the factors linking dropout to future outcomes.
• Another pitfall: Treating a good fit among retained participants as evidence that conditional independence holds. Model fit assesses the observed retention process, not its dependence on unobserved future outcomes.

The central limitation is thus substantive as well as statistical: conditional modeling can address dropout driven by recorded prior outcomes only to the extent that those outcomes and other observed history capture the relevant selection. It cannot recover information about dropout mechanisms that remain unmeasured.