Imputing Monotone Longitudinal Dropout

Why monotone dropout calls for a sequential model

In a longitudinal study, a participant may provide outcomes at early visits and then stop contributing outcome measurements. When no later outcome is observed after the first missed visit, the missingness pattern is monotone dropout. The task is to predict each missing visit using the information available before that visit, including earlier outcomes and relevant covariates. Because later predictions can use earlier imputed outcomes, the order of imputation matters.

This setting differs from intermittent missingness, where an outcome may be missing at one visit but observed again later. Here the observed records form a sequence: participants still observed at visit \(t\) have their earlier outcomes, while participants who have dropped out contribute no outcomes at later visits. A visit-specific conditional model naturally follows this sequence.

Let \(X_i\) denote observed covariates, which may include baseline predictors and visit-specific covariates known at the time an outcome is imputed. A sequential factorization writes the conditional distribution of the repeated outcomes as a first-visit model followed by one model for each later visit:

This factorization motivates the procedure: impute the first missing visit from its conditional model, then use that completed value, together with the observed history, to impute the next visit. Continue until all scheduled visits have been filled. The conditional models should include the outcome history and covariates needed to represent the longitudinal relationships and the substantive analysis.

Assumptions and the observed-data fitting sets

A key assumption is a sequential form of missing at random (MAR). In words, among participants still at risk of providing an outcome at visit \(t\), whether that visit is observed may depend on recorded history and covariates, but not on the unobserved current or future outcomes after conditioning on that information. One way to state the condition is:

This condition describes dropout before the visit-\(t\) outcome is observed. If dropout is driven by the unobserved value at that visit, such as a participant being more likely to leave because of an unrecorded worsening outcome, the condition may fail. That is a missing-not-at-random concern, not something the sequential regressions resolve by themselves.

Under this MAR condition, the conditional distribution of \(Y_{it}\) among participants observed at visit \(t\), given their history, is the same as the target conditional distribution. In particular, the visit-\(t\) model can be estimated using participants with \(R_{it}=1\); their prior outcomes are observed because dropout is monotone. Estimation also requires positivity: for histories relevant to the target population, there must be a nonzero chance of remaining observed at the visit. Otherwise the conditional outcome distribution for those histories cannot be learned directly from observed data.

The conditional models must be suitably specified for the outcome type and design. For a continuous outcome, this may be a regression with an appropriate residual distribution; for a binary outcome, it may be a logistic regression. Time, nonlinear trends, interactions, and stable or time-varying covariates may matter. For clustered or repeated-measures data, the model may also need to represent dependence through suitable participant-level effects or other longitudinal structure. A conditional mean model alone is not a complete imputation model: a predictive distribution is needed to generate plausible values.

Generating imputations in order

For each imputed data set, uncertainty should enter both through the model parameters and through the outcome draw. A typical implementation draws parameters from an estimated sampling or posterior distribution, then draws an outcome from the predictive distribution conditional on those parameters and the participant’s current history. When the next visit is imputed, the previously generated outcome is treated as part of that participant’s history. Repeating this process with fresh draws produces multiple completed data sets.

The following calculation uses illustrative parameter draws to show the mechanics. It is not a model fit to study data. Assume a continuous outcome and a visit-3 model of the form shown below. Suppose the displayed coefficients and residual standard deviation are one draw from the fitted model’s uncertainty distribution, and use a standard normal predictive error for this particular draw.

For a binary outcome, the conditional distribution must instead respect the two possible values. A logistic model gives a conditional probability, followed by a Bernoulli draw. The same sequential principle applies: the generated binary outcome becomes a predictor for a later visit where appropriate.

Derivation: why observed cases identify the conditional model under MAR

The central justification for fitting a visit-specific model among observed participants is a conditional-independence argument. Write \(H_{it}=(Y_{i1},\ldots,Y_{i,t-1},X_i)\) for the observed history and covariates available before visit \(t\). Under the sequential MAR condition, \(Y_{it}\) and \(R_{it}\) are conditionally independent given \(H_{it}\) among those still at risk. Therefore:

The first equality follows from conditional independence of the current outcome and response indicator given the history; the second expresses the target conditional distribution for the at-risk population. Consequently, observed outcomes at visit \(t\), together with their histories, can inform the visit-\(t\) conditional model. This reasoning does not guarantee accurate extrapolation: the model form must be adequate, and the relevant histories must be represented among observed participants.

Diagnostics, alternatives, and limitations

Diagnostics should assess both the sequential models and the implications of generating outcomes repeatedly. Examine observed-versus-imputed distributions by visit, conditional relationships with prior outcomes and covariates, residual patterns for continuous models, and predicted probabilities or calibration for binary models. Compare imputed values across dropout times and inspect whether late-visit predictions are driven by a small, unusual set of observed participants. Implausible ranges, overly narrow distributions, or abrupt changes in trajectories can indicate misspecification, omitted predictors, or unsupported extrapolation.

A key limitation is that later-visit models may be estimated from progressively fewer participants. If dropout is substantial, positivity may be weak, standard errors may be large, and estimates can depend heavily on model assumptions. A sparse late-visit fitting set is not repaired simply by generating many imputations. Also, MAR is not testable from the observed outcomes alone; sensitivity analysis may be needed when dropout could depend on unobserved outcomes.

Sequential conditional models are one approach, not the only one. A joint longitudinal model specifies the repeated outcomes together, while sequential models specify a conditional distribution at each visit. Either can be suitable when its assumptions match the data and analysis. For monotone dropout, the sequential approach makes the imputation order explicit and can accommodate outcome types with different visit-specific models. In contrast, carrying the last observed value forward is not a predictive model for the missing outcome distribution and generally fails to represent uncertainty. Complete-case analysis discards participants with any missing visit and can lose substantial information; it also does not by itself solve bias from informative dropout.

A common mistake is to fit the visit-4 model using only a baseline outcome and then use that same baseline-only model to impute every missing visit. This ignores the information in observed intermediate outcomes and does not implement the sequential factorization. The correction is to model each visit conditional on an appropriate observed history, then impute in chronological order so generated intermediate outcomes inform later draws. A second mistake is to use conditional means at every step; predictive draws, including parameter uncertainty, are needed for proper multiple imputation.