Missing-Data Assumptions in Bayesian Evidence Packages

Why Missing-Data Assumptions Belong in the Evidence Package

A Bayesian analysis can represent uncertainty about treatment effects, parameters, and missing outcomes. It cannot recover outcomes that were never observed without assumptions. Those assumptions can influence the posterior distribution, the probability that treatment provides clinically important benefit, and ultimately the decision supported by the evidence. A submission should therefore make the missing-data model visible, explain why it is relevant to the clinical question, and show how conclusions change under plausible alternatives.

The first task is to specify the target quantity, or estimand: for example, response by week 12 among all randomized participants, regardless of treatment discontinuation. Missingness is not itself an estimand. It is a feature of the data that requires a strategy for estimating the target. The same missing-data assumptions can have different implications for different estimands, endpoints, and decisions.

These terms describe assumptions about the data-generating and observation processes, not conclusions established by a test. For example, if participants with poorer recorded symptom scores are more likely to discontinue, a model including those recorded scores may make a MAR assumption more plausible. If discontinuation also depends on the unrecorded symptom score at the time of discontinuation, that dependence is MNAR under this definition. The variables used to make a MAR assumption plausible must be included and modeled appropriately.

Under MAR, with distinct parameters for the outcome and missingness processes, the missingness mechanism can often be ignored when making inference about the outcome model. That result does not mean missing values can be ignored: the outcome model must still use the observed covariates, histories, and outcomes appropriately. Under MNAR, the observed data alone generally do not identify the dependence on unobserved outcomes. Additional assumptions are necessary, so the submission should show how conclusions behave across clinically credible choices.

Specify the Model Before the Sensitivity Analysis

A clear evidence package states the endpoint, time point, analysis population, estimand, amount and pattern of missingness, primary missing-data assumption, and how missing outcomes enter the Bayesian model. It should also explain how covariates, repeated measures, treatment discontinuation, and intercurrent events are handled when relevant. State the prior distributions and computational approach for the outcome model as well; missing-data sensitivity results can depend on those choices.

For a sensitivity analysis, define the assumption that changes and the assumptions held fixed. A useful scenario has a clinical interpretation, such as missing participants in one arm having a lower response probability than otherwise comparable participants with observed outcomes. A grid of arbitrary parameter values is less informative than scenarios connected to reasons for missingness and the decision at stake. Earlier material on sensitivity analyses for regulatory robustness gives the broader rationale for examining alternative assumptions; here the emphasis is on making the missing-data assumptions explicit.

Binary Outcomes: Keep Observed Outcomes Fixed

For a binary endpoint, a sensitivity scenario can assign a response probability \(q_T\) to missing outcomes in the treatment arm and \(q_C\) to missing outcomes in the control arm. If these probabilities are fixed for a scenario, the number of missing responses is uncertain. The expected all-randomized response proportion uses the observed response count as known and adds the expected responses among participants with missing outcomes. It does not replace known outcomes with a model-based response rate.

Here \(y_T\) and \(y_C\) are the observed numbers of responses, \(m_T\) and \(m_C\) are the numbers with missing outcomes, and \(N_T\) and \(N_C\) are the numbers randomized to the respective arms. These expressions give expected proportions under the scenario. A full Bayesian analysis can propagate uncertainty by treating the missing outcomes as latent variables, sampling them under the specified model, and calculating the target quantity for each posterior draw.

Continuous Outcomes: Use Clinically Interpretable Shifts

A common MNAR sensitivity approach for a continuous outcome is a pattern-mixture model. It describes the distribution of outcomes separately by missingness pattern and specifies how the expected outcome among participants with missing measurements differs from an appropriate observed-outcome prediction. A shift parameter, often denoted \(\delta\), expresses that difference in the outcome's units. Its sign must be interpreted relative to the outcome scale: a negative shift is unfavorable when higher scores are better, but may be favorable when lower scores are better.

For a simple illustration, suppose a model predicts an observed-outcome mean \(\mu_{\mathrm{obs}}\) for the missing subgroup and a scenario sets its mean to \(\mu_{\mathrm{obs}}+\delta\). If the endpoint is defined for all randomized participants, the expected arm mean is the weighted combination of the known observed outcomes and the assumed missing-subgroup outcomes. For an actual Bayesian submission, the analogous calculation is made across posterior draws, allowing uncertainty in model parameters and predictions to flow into the treatment comparison.

Connect Scenarios to the Decision

A sensitivity analysis is most useful when it answers whether a conclusion changes under credible missing-data assumptions. Report the treatment effect and its uncertainty for each scenario, together with a decision quantity that was defined for the clinical question. Such a quantity might be the posterior probability that benefit exceeds a prespecified clinically important threshold. The threshold and the decision rule should be stated rather than selected after seeing which scenario favors treatment.

Diagnostics, Alternatives, and Limitations

Useful diagnostics describe the observed missingness rather than claiming to prove a missingness assumption. Examine missingness by arm, visit, site, baseline risk, recorded outcome history, and reason for discontinuation. Check whether the covariates used in the MAR model are available and have adequate overlap across relevant groups. For longitudinal outcomes, inspect when participants stop contributing measurements and whether the model represents the observed history before that point. A model that omits a strong predictor of both outcome and missingness may make its MAR assumption difficult to defend.

Convergence checks and posterior predictive checks can reveal problems in the fitted outcome model, but neither can verify an MNAR assumption from observed data alone. Sensitivity analyses should also be checked for numerical stability and for whether the chosen ranges cover meaningful adverse departures. A tipping point should be reported with its assumptions and clinical interpretation; it is not a substitute for explaining why the range is plausible.

Alternative approaches answer related but not identical questions. Complete-case analysis uses only participants with observed outcomes and can be biased when the analyzed subset differs systematically from the target population. Likelihood-based or multiple-imputation analyses under MAR can use observed covariates and outcome history, but their validity still depends on the model and MAR assumption. MNAR approaches, such as delta-adjusted pattern-mixture models or selection models, make the unidentifiable dependence on missing outcomes explicit and explore its consequences. No approach is automatically best for every endpoint or estimand.

A frequent mistake is to say, “The Bayesian model accounts for missing data, so missingness cannot bias the result.” The correction is that Bayesian computation propagates uncertainty conditional on a specified model; it does not make that model assumption-free. Another common error is to apply a posterior response probability to outcomes that were already observed. For an all-randomized response proportion, known observed responses remain fixed in the numerator, as in the binary example. If the target is instead a modeled subgroup response rate or another estimand, define it explicitly and use the corresponding formula.