Statistical Inference for Nonignorable Missing Data

Statistical inference for nonignorable missing data addresses situations in which the probability that an observation is missing depends on the unseen value itself. Unlike data missing completely at random or missing at random, nonignorable missingness (also known as missing not at random) poses fundamental identifiability challenges because the mechanism driving absence cannot be fully verified from the observed data alone. Researchers have developed two principal frameworks to confront these challenges. Selection models factorise the joint distribution of outcomes and missingness indicators, explicitly modelling the missingness mechanism, while pattern‐mixture models stratify the analysis by observed missingness patterns and impose assumptions on the unobserved strata. A third class, shared‐parameter models, introduces latent variables to capture dependence between missingness and measurement processes.

Estimation techniques range from full‐likelihood approaches, which integrate over unobserved responses under parametric assumptions, to semiparametric and nonparametric methods that rely on instrumental variables or sensitivity parameters to restore identifiability. Bayesian frameworks have gained traction by embedding prior beliefs or shrinkage penalties, thereby regularising inference when data are sparse or high‐dimensional. In parallel, weighting methods such as inverse probability weighting (IPW) and its doubly robust extensions offer computationally tractable alternatives. Across fields from epidemiology to survey analysis, these methods enable valid effect estimates and improve the reliability of conclusions drawn from incomplete data, informing policy decisions and scientific understanding on a global scale.

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Recent studies have advanced computational and methodological tools to handle nonignorable missingness. One line of work leverages quantile regression to capture complex conditional distributions when responses are missing in a nonrandom fashion. By combining a sampling importance resampling algorithm with augmented inverse probability weighting, researchers have devised smoothed estimating equations that ensure consistency and asymptotic normality under both correctly specified and misspecified working models. Simulation studies demonstrate the approach’s robustness across different quantiles and real‐world applications, such as treatment effect analysis in clinical cohorts.

Another contribution extends Bayesian adaptive lasso techniques to linear regression with nonignorable missing responses. By specifying a logistic model for the missingness mechanism and employing a hybrid Gibbs sampler and Metropolis–Hastings procedure, this method simultaneously estimates regression coefficients and shrinkage parameters, while offering goodness‐of‐fit diagnostics via posterior predictive checks. Simulation experiments and applied examples illustrate improved variable selection and parameter recovery compared to standard imputation.

In longitudinal and panel data contexts, innovative grouping strategies address item nonresponse that correlates with unobserved outcomes. By utilising auxiliary instruments uncorrelated with the missingness probability, a modified generalised method of moments yields consistent estimation of both the missingness parameters and population quantities. The resulting estimators achieve asymptotic normality and demonstrate increased efficiency over complete‐case analyses in simulations and real data applications.

Statistical Inference for Nonignorable Missing Data publication trend

The graph below shows the total number of articles in statistical inference for nonignorable missing data across all publications each year (not limited to Nature Index journals).

Technical terms

Nonignorable missing data: Data are nonignorable when the chance of an observation being missing depends on its unobserved value.

Missing not at random (MNAR): A missing‐data mechanism in which missingness is directly related to the unseen measurement.

Augmented inverse probability weighting (AIPW): A doubly robust estimation technique that combines weighting by inverse missingness probabilities with outcome modelling to reduce bias.

Sampling Importance Resampling (SIR): A Monte Carlo algorithm that generates samples from a target distribution by weighting and resampling from a proposal distribution.

Quantile regression: A regression method modelling specified conditional quantiles, useful for capturing heterogeneity in outcome distributions under missingness.

Bayesian adaptive lasso: A Bayesian regularisation approach that applies lasso‐type shrinkage through hierarchical priors for variable selection and parameter estimation.

References

1. Sampling Importance Resampling Algorithm with Nonignorable Missing Response Variable Based on Smoothed Quantile Regression. Mathematics (2023).
2. Bayesian Adaptive Lasso for Regression Models with Nonignorable Missing Responses. Journal of Mathematics (2022).
3. Nonignorable item nonresponse in panel data. Statistical Theory and Related Fields (2020).