Informative Dropout In Longitudinal PKPD Data

Why Dropout Can Change the PKPD Conclusion

Longitudinal PKPD analyses use repeated measurements to describe how exposure, efficacy, or toxicity changes over time. If participants stop treatment or leave follow-up, later outcomes may be missing. The key issue is not simply how many values are missing, but why they are missing. Participants with poor response may be more likely to discontinue for lack of efficacy; participants with high toxicity may stop even when their response is good. Either process can make those still observed at later visits unrepresentative of the original cohort.

First distinguish treatment discontinuation from dropout from outcome follow-up. A participant may stop study treatment but continue to provide efficacy and safety measurements. Those measurements can be relevant to an analysis of outcomes under a treatment-policy strategy, which includes outcomes after discontinuation. If follow-up also ends, later outcomes are missing. An analysis asking what would have happened had everyone continued treatment targets a different, hypothetical estimand; observations after stopping treatment do not directly supply that counterfactual.

This tutorial focuses on a scheduled longitudinal outcome, such as a response score. The same reasoning applies to PKPD outcomes, but the target and the way the outcome is defined must be stated. For example, a concentration measured after treatment stops is not interchangeable with the concentration that would have occurred under continued treatment.

Start With the Estimand and Missingness Assumptions

A missing-data method cannot decide what scientific question to answer. State the target first: for example, the mean week-12 response score among all randomized participants under a strategy that continues outcome collection even after treatment discontinuation. If follow-up ends at treatment discontinuation, that week-12 outcome is unobserved for some participants. A complete-case mean then describes participants with observed week-12 outcomes, not automatically the original cohort.

A common working assumption is missing at random (MAR): conditional on recorded information available before the outcome goes missing, whether the outcome is observed does not additionally depend on its unobserved value. The recorded history could include baseline severity, earlier response, prior toxicity, exposure, treatment assignment, and recorded reason for stopping. MAR is a conditional independence assumption, not a claim that dropout is unrelated to outcomes. By contrast, if observation still depends on the unobserved outcome after conditioning on that history, the process is missing not at random (MNAR).

MAR cannot generally be confirmed from observed outcomes alone: values that were never measured cannot be compared with the values that were measured. Recorded reasons for discontinuation and earlier trajectories can make a proposed model more credible, but they do not establish the assumption. A dropout-adjusted analysis should therefore be accompanied by diagnostics and, when plausible, sensitivity analyses for departures from MAR.

Inverse-Probability Weighting

Inverse-probability weighting (IPW) gives more weight to observed participants who resemble people with similar recorded histories but are less likely to remain observed. At a single target visit, let \(H_i\) denote the recorded history used to model retention and let \(\pi_i=P(R_i=1\mid H_i)\) be the estimated probability that participant \(i\)'s outcome is observed, conditional on that history. A normalized, or Hájek, weighted mean is

Only participants with observed outcomes contribute to the sums. A participant with a small estimated observation probability receives a larger weight, representing similar participants who are missing. In monotone longitudinal dropout, the probability of being observed at a later visit can be modeled from the sequence of conditional retention probabilities at earlier intervals. The weight for remaining observed through visit \(t\) is the reciprocal of that cumulative probability.

Why the Weighting Formula Works

The basic justification is an inverse-probability identity. Suppose the probability of observation is \(\pi(H)=P(R=1\mid H)\), with \(\pi(H)>0\), and assume \(R\) is conditionally independent of \(Y\) given the recorded history \(H\). Conditional on \(Y\) and \(H\), the expected weighted contribution is

The equality \(E(R\mid Y,H)=\pi(H)\) uses the conditional independence assumption. Averaging over participants therefore recovers the target outcome mean in expectation when the assumptions hold. Replacing the population quantities with estimated probabilities and dividing by the sum of weights yields the normalized estimate above. This derivation explains both the appeal and the vulnerability of IPW: if the recorded history omits a factor that links dropout to the outcome, or if a retention probability is near zero, the result can be biased or unstable.

Likelihood Models and Sensitivity Analyses

IPW is not the only approach. A likelihood-based longitudinal model, such as a mixed-effects model fitted to the observed outcomes, can use incomplete outcome histories without first filling in every missing value. Under a correctly specified outcome model and an appropriate MAR assumption, likelihood inference can be valid. It relies on a different model specification from IPW: the outcome model describes the longitudinal outcomes, while IPW models observation. Comparing the two can reveal sensitivity to modeling choices, but agreement does not prove that either assumption is correct.

When dropout may depend on unobserved outcomes, a sensitivity analysis makes that uncertainty explicit. For example, a pattern-mixture analysis can shift the outcomes predicted for participants who drop out by a chosen amount relative to predictions under MAR. The shift is a sensitivity parameter, not a value learned from the observed outcomes alone. A joint model can instead specify a shared association between longitudinal outcomes and time to dropout, but its conclusions also depend on assumptions about the joint distributions and association structure.

Diagnostics, Comparisons, and Failure Modes

Before interpreting a dropout-adjusted estimate, examine the observation process itself. Summarize retention by visit, treatment group, recorded response history, toxicity history, and reason for discontinuation. Compare observed earlier trajectories between participants who remain and those who later leave. Such comparisons can reveal measured predictors of dropout and help specify the retention or outcome model. They cannot rule out dependence on the unobserved outcome.

For IPW, inspect the distribution of estimated probabilities and weights. Very small observation probabilities produce large weights, so a few participants may dominate the estimate. One simple diagnostic is the effective sample size, calculated from the observed weights \(w_i=1/\hat{\pi}_i\):

This quantity is no greater than the number of observed outcomes and becomes small when a few large weights dominate. Also check whether histories associated with dropout have any observed participants at all. If a history has zero probability of observation, the positivity assumption fails; reweighting cannot recover its outcomes from the data without further modeling or external information. Weight truncation can limit the influence of extreme weights, but it changes the estimator and may introduce bias. Report the rule and compare results with and without truncation rather than treating it as a guaranteed fix.

A common mistake is to say, “The complete cases are the patients who tolerated treatment, so their mean is the treatment effect.” That reasoning confuses a selected observed group with the target population, and it may also confuse outcomes under continued treatment with outcomes under a strategy that includes discontinuation. The correction is to define the estimand, describe which outcomes are missing and why, then state the assumptions that connect observed outcomes to the target. A related mistake is to regard an MAR fit or a set of finite weights as proof that dropout is ignorable. Diagnostics can expose problems, but no observed-data diagnostic can verify what unobserved outcomes would have been.

Informative dropout matters because treatment discontinuation and loss of follow-up can selectively remove participants with clinically important response or toxicity histories from later analyses. A complete-case estimate remains a useful description of observed participants, but it is not automatically an estimate for everyone enrolled. A credible PKPD conclusion makes the target explicit, presents the dropout assumptions, and shows whether the conclusion is robust to reasonable alternatives.