Sensitivity to Missing-Data Assumptions in Exposure-Response

Why Missing-Data Assumptions Matter for Exposure-Response Decisions

An exposure-response analysis uses measured or modeled exposure to predict a clinical outcome. If some outcomes are missing because participants discontinue, miss visits, or become too unwell to provide measurements, the observed participants may not represent everyone whose outcomes matter to the decision. A prediction based only on observed outcomes can therefore depend on an implicit assumption about what would have happened after dropout.

This tutorial focuses on sensitivity of decision-relevant exposure-response predictions to those assumptions. The central comparison is between missing at random (MAR), where missingness can be explained by information already observed, and informative dropout beyond observed information, where missingness also depends on unobserved outcomes. The latter is often described as missing not at random (MNAR). Neither label is a diagnosis that can usually be established from the observed data alone.

A decision target must be stated before comparing assumptions. For example, it may be the mean week-12 response predicted at a specified exposure in a target population, or the probability that this mean exceeds a development threshold. An exposure-response prediction describes an association unless the design and analysis also support a causal interpretation. Missing-data sensitivity analysis addresses uncertainty about unobserved outcomes; it does not by itself resolve confounding, exposure measurement error, or misspecification of the exposure-response relationship.

What MAR Allows—and What It Does Not

Under MAR, participants with the same observed exposure, covariates, and response history provide information about outcomes for participants who dropped out with that same observed history. A likelihood analysis can use this information without explicitly modeling the dropout process, provided the outcome model is adequate and the usual ignorability conditions hold. In particular, the parameters governing the outcome model and missingness mechanism must be distinct.

The factorization explains the point. Write \(Y_{\mathrm{obs}}\) and \(Y_{\mathrm{mis}}\) for observed and missing outcomes. Under MAR, the probability of the observed missingness pattern, conditional on outcomes and covariates, depends on observed outcomes but not on \(Y_{\mathrm{mis}}\). If the outcome parameter \(\theta\) and missingness parameter \(\psi\) are distinct, the joint likelihood can be written as:

The second factor contains no \(\theta\), so it does not change likelihood-based estimation of \(\theta\). This is why an ignorable likelihood analysis need not fit a dropout model under MAR and parameter distinctness. This does not mean the outcome model can be careless: it must represent the relevant exposure, covariate, and observed response-history relationships. Any missingness model used by a chosen method, such as inverse-probability weighting, must also be adequate. The covariates needed for prediction must be observed for the participants to whom the analysis applies, and there must be sufficient overlap in observed exposure and history patterns.

MAR is an assumption about dependence, not a claim that dropout is unrelated to prognosis. If participants with poor observed responses are more likely to discontinue, that can still be MAR when the analysis conditions appropriately on those observed responses. In contrast, if participants discontinue because of an unobserved worsening that is not captured by their recorded history, MAR may fail.

Representing Informative Dropout With Sensitivity Parameters

When outcomes after dropout may be worse or better than MAR predicts, a sensitivity analysis can shift the predicted missing-outcome distribution. A simple pattern-mixture specification adds a parameter \(\delta\) to the MAR-predicted mean for missing outcomes. In a group with a common shift:

Here \(\delta\) is measured in the outcome's units. A negative value represents outcomes lower than the MAR prediction; a positive value represents higher outcomes. The data generally cannot estimate \(\delta\) from the observed outcomes alone, so analysts examine a range justified by clinical knowledge, external information, or explicit decision stress tests. The pattern-mixture approach makes the departure transparent, but a constant shift may be too simple if the departure varies by exposure, response history, or time.

A tipping point is the value of a sensitivity parameter at which a specified decision changes. It is a useful summary of decision robustness, not evidence that the tipping-point value is the true missing-data mechanism.

Comparing Assumptions and Checking the Analysis

A useful analysis reports the decision prediction under a primary assumption and under alternatives that represent plausible departures. For example, a likelihood analysis under MAR can be compared with one or more MNAR pattern-mixture shifts. A selection model instead specifies dropout probability conditional on observed information and the potentially unobserved outcome. These approaches parameterize the missingness assumption differently; neither makes MNAR departures identifiable merely by fitting the observed data. Complete-case analysis is a simpler alternative, but it estimates the target mean without bias only under additional conditions, such as missingness being independent of the outcome within the analyzed groups. It is not a general substitute for a missing-data assumption.

Diagnostics can reveal weaknesses in the observed-data analysis, but cannot verify MAR against MNAR using observed outcomes alone. Check whether dropout rates vary with exposure, baseline covariates, and recorded response history; whether predicted outcomes are supported by participants with similar observed histories; and whether model fit, convergence, and uncertainty are acceptable. Sparse histories or exposure ranges with few observed outcomes signal extrapolation. A highly influential dropout pattern or unstable predictions across reasonable outcome models also warrants caution.

For the primary outcome model, examine whether its exposure-response form and relevant covariate and history effects describe the observed data adequately. For methods that explicitly model dropout, examine that missingness model as well. Do not require a dropout model solely because the likelihood analysis assumes MAR: under MAR with parameter distinctness, an adequate likelihood for the outcome can be ignorable without one. Equally, do not mistake adequate fit among observed outcomes for evidence that the unobserved outcomes follow MAR predictions.

The practical limitation is fundamental: observed data alone typically cannot distinguish MAR from MNAR. Sensitivity results are therefore conditional statements: if the missing outcomes differ from MAR predictions by the stated amount, the decision prediction is as reported. They should be presented with the assumptions, units, and uncertainty, rather than as a single assumption-free answer.

Common Pitfall

Wrong reasoning: “Dropout is related to prognosis, so a missingness model must be fitted for a valid MAR likelihood analysis.” Prognosis can affect dropout through observed response history, while MAR still holds conditional on that history. Under MAR and parameter distinctness, the likelihood for the outcome parameters can be ignorable without modeling dropout.

Correction: Adequately model the outcome relationships relevant to the target prediction, and ensure the required covariates and histories are observed with sufficient overlap. Assess a missingness model when the chosen method uses one. Then evaluate how the decision changes under explicit departures from MAR, while recognizing that observed-data diagnostics cannot establish which missingness assumption is true.