Building a Trial Model With Current-Outcome Dropout

Why Build the Outcome and Dropout Models Together?

In a multi-visit randomized trial, outcomes are scheduled over time, but some participants stop providing measurements. If dropout depends on the outcome at the visit where measurement is missed, the selection model must refer to that current outcome even though it is unavailable for a participant who drops out. A useful specification therefore has two connected parts: a model for the outcome trajectory and a model for dropout conditional on the outcome history.

The goal here is to make the model’s ingredients explicit: the treatment assignment, the visit schedule, baseline covariates, observed and latent outcomes, and the sequence of dropout opportunities. A current-outcome selection model is an assumption about how the joint data arise; it does not turn an unobserved outcome into an observed value.

Define the Trial Data and the Dropout History

Suppose participant \(i\) is randomized to treatment \(A_i\), where \(A_i=1\) denotes the active treatment and \(A_i=0\) the control. Let \(Y_{i0}\) be a measured baseline outcome, and let \(Y_{ij}\) be the outcome scheduled at follow-up visit \(j\), for \(j=1,\ldots,J\). The scheduled time of visit \(j\), measured in a specified unit such as weeks, is \(t_j\). Let \(X_i\) be a vector of baseline covariates other than \(Y_{i0}\).

Let \(R_{ij}=1\) if the outcome at visit \(j\) is observed and \(R_{ij}=0\) otherwise. Assume monotone dropout: after a participant stops providing outcomes, no later outcome is observed. Define \(R_{i0}=1\), and define \(D_{ij}=1\) when a participant who was still providing outcomes through visit \(j-1\) drops out before providing \(Y_{ij}\). Thus, among participants with \(R_{i,j-1}=1\), \(D_{ij}=1-R_{ij}\). A participant who already dropped out is not at risk of a new dropout event at later visits.

This setup describes permanent dropout, not every missed visit. If participants can miss one visit and return later, a single monotone dropout indicator is not enough; the observation process must represent that intermittent missingness separately. The outcome and selection equations should also respect temporal order: baseline covariates and treatment assignment precede follow-up, earlier outcomes precede later visits, and the current outcome in the selection equation is the outcome scheduled at the dropout opportunity.

Specify the Outcome Trajectory

One useful continuous-outcome specification is a linear mixed model with a participant-specific random intercept. It allows average outcome levels to vary by treatment, time, and baseline characteristics, while accounting for repeated measurements from the same participant:

Here \(b_i\) is a random intercept, and the residuals \(\varepsilon_{ij}\) are assumed independent across visits and independent of \(b_i\), conditional on the model covariates. The treatment-by-time term allows the average treatment contrast to change over time. The coefficient \(\beta_t\) is the control-arm change in mean outcome per unit of time; the active-arm change per unit of time is \(\beta_t+\beta_{At}\). If the trajectory is not plausibly linear in \(t_j\), visit indicators or a suitable time basis can replace the linear time terms. The time coding and units must be stated so coefficients can be interpreted.

This is a model for the outcome process, including outcomes that may be missing. It does not say that every participant’s observed trajectory follows the population mean exactly: \(b_i\) and \(\varepsilon_{ij}\) represent between-participant and within-participant variation. The distributional assumptions should be chosen to suit the outcome; the displayed Gaussian form is an example for a continuous outcome, not a universal requirement.

Specify Dropout Conditional on Current and Earlier Outcomes

A logistic hazard gives the probability of dropout at visit \(j\), conditional on having remained in follow-up through visit \(j-1\). To make outcome effects comparable across visits, define \(Z_{ij}=(Y_{ij}-c)/s_Y\), where \(c\) and \(s_Y>0\) are fixed centering and scaling constants in outcome units. For the lagged value, set \(Z_{i,j-1}=(Y_{i,j-1}-c)/s_Y\) when \(j\geq2\); at the first follow-up visit, use the baseline outcome \(Y_{i0}\) in place of \(Y_{i,j-1}\). Write that lagged standardized value as \(Z_{i,\mathrm{lag}(j)}\).

Here \(\mathbf{Y}_i=(Y_{i1},\ldots,Y_{iJ})\) denotes the full outcome history, including latent outcomes. The logit of a probability \(p\) is \(\log\{p/(1-p)\}\). A positive \(\alpha_C\) means that, holding the other terms fixed, higher current outcomes correspond to higher dropout odds. A positive \(\alpha_L\) describes an association with the previous outcome after accounting for the current outcome. These are conditional model associations, not causal effects of changing an outcome.

Including \(t_j\) allows the baseline dropout odds to vary over the scheduled follow-up times. Baseline covariates can adjust dropout probabilities for prognostic characteristics, while treatment can account for different dropout odds by randomized arm. The model can be expanded when justified—for example, with treatment-by-time or treatment-by-current-outcome interactions—but each added term requires adequate information and a clear interpretation. At the first follow-up, the lagged value is baseline; later, it is the immediately preceding follow-up outcome.

How the Two Equations Form a Joint Model

The outcome equation supplies a distribution for the full outcome trajectory; the dropout equation supplies conditional probabilities for the response history given that trajectory. A convenient conceptual factorization is the outcome density multiplied by the visit-specific dropout probabilities for participants still at risk. For example, a participant observed at visit 1 and dropping out before visit 2 contributes an outcome density for the observed \(Y_{i1}\), a probability of remaining observed at visit 1, and a visit-2 dropout probability that depends on \(Y_{i2}\) as well as the history. The \(Y_{i2}\) term is latent in this dropout probability.

The reason is the chain rule for conditional probabilities: the model specifies the distribution of outcomes and then specifies dropout at each eligible visit conditional on the outcomes and covariates. For a dropout event at visit \(j\), the selection factor is \(P(D_{ij}=1\mid R_{i,j-1}=1,\mathbf{Y}_i,A_i,X_i)\), not a probability conditioned only on outcomes already observed. Because \(Y_{ij}\) is missing for that event, calculations based on observed data must account for its uncertainty under the joint model. This is a justification for the model structure, not a recipe for replacing a missing current outcome with a guessed value.

Assumptions, Diagnostics, and Alternatives

A coherent specification requires several assumptions. The dropout process must be monotone for the hazard definition above; randomization and baseline covariates must be correctly recorded; the outcome distribution and time structure must represent the repeated measurements adequately; and the logistic equation must capture the chosen dependence on current outcome, earlier outcome, treatment, time, and baseline covariates. There must also be meaningful support for the covariate and outcome combinations used to interpret the model. In particular, a current-outcome association in the dropout equation is difficult to learn from observed data alone because the relevant outcome is missing precisely for some dropout events.

Useful checks begin with the model’s construction and observed implications:

• Confirm that the visit times, time units, treatment coding, and baseline covariate coding match the trial design.
• Verify that each dropout hazard is defined only for participants still at risk, and that the current and lagged outcomes refer to the intended visits.
• Compare the fitted outcome trajectories with observed summaries by treatment and visit, while recognizing that observed summaries can be selected by dropout.
• Inspect fitted dropout probabilities across visits and across plausible covariate and outcome values. Probabilities that are nearly zero or one over ordinary ranges may indicate an implausible specification or sparse information.
• Check whether the assumed linear time effect is credible; systematic departures may motivate visit-specific effects or a more flexible time basis.

A lagged-outcome dropout model uses earlier outcomes but not the current latent outcome; it is a different selection assumption. A model depending only on observed covariates and observed history is another alternative, and a pattern-mixture model instead organizes assumptions by missingness pattern. These are alternative parameterizations and assumptions, not interchangeable labels for the same model. The appropriate choice depends on the scientific question and on how the unobserved outcomes are to be related to dropout.

The central limitation is that observed records do not directly determine how dropout depends on an outcome that is missing at dropout. Different choices for this dependence can be compatible with similar observed data while implying different distributions for missing outcomes. Therefore, current-outcome coefficients should not be presented as established causal effects or as empirically certain merely because a model has been fitted. Their role is to encode a selection assumption, which should be made explicit and examined for scientific plausibility.

Common Mistakes

• Using only the last observed outcome while calling the model current-outcome dependent. A term involving \(Y_{i,j-1}\) is lagged; current-outcome dependence requires \(Y_{ij}\).
• Treating the missing current outcome as observed. In a dropout event, \(Y_{ij}\) enters the joint model as a latent quantity. It is not available just because the selection equation includes it.
• Applying a dropout hazard after dropout. A participant with \(R_{i,j-1}=0\) is not at risk at visit \(j\) under monotone dropout.
• Ignoring the visit schedule or units. A time coefficient has no unambiguous interpretation unless time coding and units are stated.
• Interpreting a selection coefficient causally. A conditional association between outcome level and dropout does not show that changing the outcome would cause a change in dropout.