Selection From Missing Mediator or Outcome Data

Why Missing Mediators and Outcomes Can Change a Genetic Mediation Analysis

A genetic mediation analysis may have genotype, exposure, mediator, and outcome information for different subsets of participants. For example, molecular measurements may be available only for people with usable tissue, while outcome follow-up may be incomplete. An analysis restricted to participants with every variable recorded can then describe a selected subset rather than the target population. The central question is not simply how many values are missing, but what determines which values are observed.

Selection can alter mediator and outcome distributions and can also induce or distort associations among genotype, exposure, mediator, and outcome. This tutorial distinguishes three missingness mechanisms, states the conditions under which inverse-probability weighting can address selection, and shows how a sensitivity analysis can represent outcome-dependent selection that the observed data cannot identify by themselves.

These definitions concern a specified set of variables and a specified missingness pattern. They are not properties of a data set in isolation. For instance, outcome availability may be MAR if it depends on a fully observed baseline covariate, but MNAR if it depends on the unrecorded outcome itself after conditioning on available information. Selection into a study based on an observed variable is not automatically MNAR; the relevant issue is whether selection depends on information unavailable for the selected analysis.

Reading the Mechanism in a Mediation Setting

For mediator data, let \(R_M=1\) indicate that \(M\) is measured. If mediator measurement depends on observed genotype, exposure, outcome, and covariates, then the missing mediator may be MAR conditional on those variables, provided they are observed for everyone in the relevant analysis. If the chance that a mediator is measured also depends on the unobserved mediator value after conditioning on those observed variables, the mechanism is MNAR.

The same logic applies to \(R_Y\), the indicator that the outcome is observed. If outcome follow-up depends on observed baseline variables, MAR may be plausible. If people with more severe, unrecorded outcomes are more or less likely to remain in follow-up, outcome missingness is MNAR unless the relevant severity information is observed and included in the conditioning set. Complete-case analysis is not automatically protected by calling a mechanism MAR: the assumptions must match the actual variables available and the analysis being performed.

Inverse-Probability Weighting Under a Stated MAR Model

Inverse-probability weighting (IPW) gives greater weight to observed individuals who represent groups with lower observation probabilities. To use a simple complete-case version, let \(O\) contain the relevant variables that are fully observed for everyone, and let \(\pi(O)=P(R=1\mid O)\). For the weighting identity below, \(O\) must include all relevant observed components of \(W\), and the required assumption is stronger and more precise than merely saying that the missing values do not predict response:

The conditional independence condition matters. If response can depend on an observed component of \(W\) omitted from \(O\), then the response probability in the weights need not be the response probability conditional on the full data. In that case, weighting by \(\pi(O)\) may fail even if response is independent of the missing component. MAR in a general pattern-specific analysis can condition on variables observed for that pattern; a simple complete-case weight based only on common predictors requires the more explicit condition just stated.

For any integrable function \(\psi(W)\), the population expectation can be recovered by weighting observed individuals under these assumptions:

To see why, condition on the complete data \(W\). Since \(O\) is included in, or determined by, \(W\), the assumption gives \(E(R\mid W)=P(R=1\mid W)=\pi(O)\). Therefore,

and taking expectations over \(W\) proves the identity. The proof uses the response probability conditional on the full data; assuming only independence from missing values while omitting relevant observed components does not justify the step.

For a parameter \(\theta\) defined by an estimating equation \(E\{\psi(W;\theta)\}=0\), the weighted estimating equation replaces the population expectation with its observed-data counterpart:

Here \(\hat{\pi}(O_i)\) estimates the probability that individual \(i\)'s required data are observed. The estimating function must correspond to the target parameter, and the observation model must match the missingness pattern. When mediator and outcome availability have different patterns, a single complete-case indicator and one propensity model may not suffice; each pattern needs an appropriate observed-data model and assumptions. Weighting cannot recover information in covariate strata with zero probability of observation.

Outcome-Dependent Selection Requires Sensitivity Analysis

When missingness depends on the unobserved mediator or outcome, the response probability \(P(R=1\mid W)\) cannot generally be learned from the observed data alone. An ordinary MAR-based propensity model omits the very dependence at issue. A principled response is to state a sensitivity parameter, examine results across plausible values, and report how strongly conclusions depend on it. Such an analysis does not turn an unidentified mechanism into an identified one; it makes the unverifiable assumption visible.

For a simple binary outcome illustration, let \(Y\) equal 1 or 0, let \(q=P(Y=1\mid R=1)\) be the prevalence among respondents, and define the selection ratio \(\lambda=P(R=1\mid Y=1)/P(R=1\mid Y=0)\). This parameter describes how much more or less likely people with \(Y=1\) are to be observed than people with \(Y=0\). If \(p=P(Y=1)\) is the population prevalence, then Bayes' rule gives:

The observed prevalence \(q\) does not determine both \(p\) and \(\lambda\). Different plausible selection ratios imply different population prevalences. In mediation, selection may depend on a continuous outcome or mediator and may vary with genotype or exposure; a single binary sensitivity parameter is then only a teaching illustration. A practical analysis should define a parameterization that matches the outcome scale and selection process, and examine how the mediation estimate changes across specified values.

Choosing and Reporting an Analysis

In genetic mediation, the consequences are not limited to a shifted mean. If selection depends on variables associated with genotype and mediator or outcome, conditioning on observation can change their joint distribution. The weighted estimating equation may address this under its assumptions, but it does not repair other identification failures, such as unmeasured confounding or an invalid genetic instrument. Likewise, a sensitivity analysis for selection is distinct from sensitivity analysis for mediator–outcome confounding: the parameters describe different uncertainties.

Common Pitfalls

• Calling a mechanism MAR without naming the conditioning variables. “Missingness is MAR” is incomplete unless the observed information used to condition on it is specified for each pattern.
• Leaving relevant observed components out of the weights. If \(R\) depends on an observed component of \(W\) omitted from \(O\), then \(P(R=1\mid W)\) need not equal \(\pi(O)\), and the IPW identity need not hold.
• Treating positive estimated probabilities as proof of positivity. A model that returns nonzero values does not establish that every target stratum has a genuine chance of observation. Extremely small probabilities can also create unstable, high-variance weights.
• Claiming that weighting solves MNAR selection. IPW based on observed predictors cannot identify dependence on an unobserved outcome without additional assumptions or information. State and vary a sensitivity parameter instead.
• Interpreting weighted estimates causally by default. Correcting one selection mechanism does not establish all assumptions needed for a causal mediation interpretation.