Missing Not At Random (MNAR)

In MNAR data, the missingness depends on the unobserved value itself after controlling for observed variables. For example, patients with severe depression are more likely to skip mental health questionnaires—the missingness is directly correlated with the severity score that would have been recorded. This violates the ignorability assumption required by standard imputation methods like multiple imputation by chained equations (MICE). Unlike Missing Completely At Random (MCAR) or Missing At Random (MAR) , MNAR requires explicit modeling of the missing data mechanism using techniques such as pattern-mixture models or selection models.

MNAR patterns are pervasive in clinical data:

• Lab test avoidance: Patients with dangerously high blood glucose levels skip follow-up HbA1c tests, creating missing values correlated with poor glycemic control
• Income non-response: High-income individuals systematically refuse to disclose earnings in health surveys
• Dropout in longitudinal studies: Patients experiencing rapid disease progression withdraw from clinical trials, biasing results toward healthier populations
• Pain scale omission: Patients with extreme pain may be unable to complete self-reported pain scales, removing the most severe cases from analysis

MNAR cannot be definitively proven from observed data alone—it requires domain expertise and sensitivity analysis. Key diagnostic approaches include:

• Pattern-mixture modeling: Comparing parameter estimates under different assumptions about the missing data distribution
• Tipping point analysis: Identifying how extreme the missing values must be to overturn study conclusions
• Proxy variable analysis: Using auxiliary variables correlated with the missing values to assess bias magnitude
• Heckman selection models: Jointly modeling the outcome and the selection process when a theoretical basis for MNAR exists

Standard imputation methods produce biased estimates under MNAR. Specialized approaches include:

• Selection models: Explicitly model the probability of missingness as a function of the unobserved value using maximum likelihood estimation
• Pattern-mixture models: Stratify data by missingness pattern and estimate separate distributions for observed and missing groups
• Delta-adjustment methods: Apply a clinically meaningful offset to imputed values to reflect the expected difference between observed and missing data
• Federated MNAR imputation: In decentralized settings, each institution models its own missingness mechanism locally before contributing to a federated imputation model that accounts for site-specific MNAR patterns

MNAR data poses unique challenges in federated healthcare networks:

• Non-ignorable site heterogeneity: Different hospitals may have distinct MNAR mechanisms—a tertiary care center's missing lab values reflect different clinical processes than a community clinic's
• Bias amplification: Federated averaging can propagate and amplify MNAR bias across nodes if local imputation models do not account for the missingness mechanism
• Federated sensitivity analysis: Collaborative sensitivity analyses across institutions can bound the impact of MNAR on global model performance without sharing raw patient data
• Privacy-preserving MNAR modeling: Techniques like federated Heckman models allow sites to jointly estimate selection equations while keeping individual-level data local

Understanding the taxonomy of missing data mechanisms is essential:

• MCAR: Missingness is completely random and unrelated to any variable (observed or unobserved). Example: a blood sample is accidentally dropped in transit
• MAR: Missingness depends on observed variables but not the missing value itself. Example: older patients are less likely to complete digital surveys, but age is recorded
• MNAR: Missingness depends on the unobserved value. Example: patients with substance use disorders skip addiction screening questions

Only MNAR requires explicit modeling of the missingness mechanism; MCAR and MAR allow likelihood-based methods to produce unbiased estimates without such modeling.

The probability of a value being missing is independent of both observed and unobserved data. This is the most benign mechanism.

• Example: A lab sample is accidentally dropped, destroying the result randomly
• Complete-case analysis yields unbiased estimates
• Rarely holds in real-world clinical settings
• Testable using Little's MCAR test

The probability of missingness depends on observed data only, not the missing value itself. Once conditioned on observed variables, the missingness becomes ignorable.

• Example: Older patients are less likely to report pain scores, but age is recorded
• Multiple imputation and maximum likelihood methods are valid
• Requires careful modeling of the missingness mechanism
• Most clinical missing data is assumed MAR

A modeling framework that stratifies the population by missing data patterns and fits separate models within each stratum. The overall estimate is a weighted average.

• Explicitly models differences between observed and unobserved groups
• Requires untestable assumptions about the missing data distribution
• Useful for sensitivity analysis in MNAR scenarios
• Contrasts with selection models that model the missingness probability directly

A two-stage econometric approach that jointly models the outcome equation and the selection equation (probability of being observed), connected through a correlation parameter.

• Originated in labor economics for wage modeling
• Identifies and corrects for selection bias when missingness is non-ignorable
• Requires an exclusion restriction: a variable that predicts missingness but not the outcome
• The inverse Mills ratio is computed from the selection equation and included as a regressor

An imputation approach that systematically varies assumptions about the departure from MAR to assess sensitivity. Delta-adjustment and pattern-mixture imputation are common implementations.

• Imputes missing values under a range of plausible MNAR scenarios
• Produces a distribution of inferences rather than a single point estimate
• Enables principled tipping-point analysis
• Recommended by regulatory agencies for clinical trial sensitivity analyses

A diagnostic approach that tests whether observed outcomes differ systematically across patterns of intermittent missingness in longitudinal data.

• Examines if current outcomes predict future dropout
• Graphical methods: mean profiles stratified by dropout time
• If outcomes diverge before dropout, MNAR is plausible
• Cannot definitively prove MNAR, but strengthens the case for sensitivity analyses