Missing Data and Imputation

Overviews

• Flexible Imputation of Missing Data - Missingness assumptions, multiple imputation, diagnostics, and applications. Online chapters and book source. van Buuren 2018; code, code.
• Missing-data imputation - Chapter 25. Compares deletion, simple replacements, regression, and model-based imputation; explains why treating filled values as observed understates uncertainty. Gelman and Hill 2007.
• Missing Data: Our View of the State of the Art - Reviews missingness assumptions, likelihood inference, and multiple imputation. Pigott (2001) compares these with deletion and single-value methods. Schafer and Graham 2002, Pigott 2001.

Missingness Mechanisms and Assumptions

• Missing completely at random (MCAR) - Missingness probabilities do not depend on observed or unobserved data values. Complete cases then form a random subsample, with reduced information. van Buuren 2018.
• Missing at random (MAR) - Given the observed data, missingness does not additionally depend on the missing values. MAR permits systematic missingness explained by observed variables; it does not mean entries disappear uniformly. van Buuren 2018.
• Missing not at random (MNAR) - Missingness still depends on unobserved values after conditioning on observed data. Distinguishing MAR from MNAR generally requires assumptions beyond the observed data. van Buuren 2018.
• Ignorability - MAR and distinct data/missingness parameters permit ignoring the missingness model in direct-likelihood inference. This does not justify arbitrary deletion or replacement rules. Rubin 1976.
• MNAR sensitivity analysis - Delta-adjusted imputation shifts conditional distributions away from MAR through user-specified sensitivity parameters. Vary plausible departures; the observed data alone do not identify those shifts. docs.
• Graphical missing-data recovery - Uses graphs to express missingness assumptions and derive recoverability and testability conditions, including some MNAR cases. Recovery depends on the graph and target quantity. Mohan and Pearl 2018.

Deletion and Simple Imputation

• Complete-case analysis - Uses rows complete on the variables needed for an analysis. MCAR is a sufficient, not necessary, condition for validity; selection can bias estimates and discarding rows reduces information. van Buuren 2018.
• Available-case / pairwise deletion - Uses all observed values for each statistic, so covariances may use different subsets. The resulting covariance matrix need not be positive semidefinite. This is not deletion of columns. van Buuren 2018.
• Mean, median, and mode replacement - Simple column-wise baselines. Deterministic fills do not represent imputation uncertainty; mean replacement reduces variance and can distort correlations. van Buuren 2018; docs.
• Regression imputation - Predicts a missing variable from observed predictors. Deterministic predictions suppress residual variation; stochastic regression adds residual draws. Multiple imputation also accounts for parameter uncertainty. van Buuren 2018.
• Hot-deck and cold-deck imputation - Hot-deck methods draw observed values from eligible donors in the current sample; cold-deck methods use an external source. Donor selection determines which relationships are preserved. Andridge and Little 2010.
• Predictive mean matching - Selects donors with similar predicted outcomes, then copies an observed donor value. Preserves observed support; results depend on the prediction model and donor pool. van Buuren 2018.
• K-nearest-neighbor imputation - Fills each feature from nearby samples that observed it. Distances use jointly observed coordinates; scaling and sparse overlap affect which donors are selected. docs.

Likelihood and Multiple Imputation

• Expectation-maximization (EM) - Alternates expected complete-data log likelihood and parameter updates. Fits an observed-data model; it is not itself a procedure for producing multiple completed datasets. Dempster et al. 1977.
• Multiple imputation - Generates multiple plausible completions, analyzes each, and combines estimates with within- and between-imputation variance. Pool analysis results, not completed datasets. Rubin 1987; docs.
• Imputation-analysis compatibility - Rubin's rules can fail when imputation and analysis models are uncongenial. Bartlett and Hughes find valid bootstrap-before-imputation variants under consistent point estimation; resampling cannot fix a biased point estimate. Bartlett and Hughes 2019.
• Sequential regression multivariate imputation - Cycles through variable-specific regression models and draws from predictive distributions, accommodating different variable types and logical constraints. Raghunathan et al. 2001.
• Fully conditional specification - Specifies an imputation model for each incomplete variable. Flexible conditional models need not correspond to one compatible joint distribution. van Buuren 2007.
• MICE - Implements chained equations with variable-specific methods, predictor selection, and diagnostics. Azur et al. (2011) explain the workflow and assumptions. van Buuren and Groothuis-Oudshoorn 2011, Azur et al. 2011.
• Joint-model multiple imputation - Develops an alternative computational approach to multiple imputation for incomplete multivariate data. Amelia II uses bootstrap-based EM. King et al. 2001, Honaker et al. 2011.

Tabular and Low-Rank Methods

• missForest - Iteratively fits random forests to incomplete columns, supporting mixed numeric/categorical data. Out-of-bag error estimates imputation error, not inferential confidence intervals. Stekhoven and Bühlmann 2011, Stekhoven and Bühlmann 2011.
• HyperImpute - Extends iterative imputation with automatic selection of each column's prediction model and hyperparameters. Model search is part of fitting and must stay inside evaluation folds. Jarrett et al. 2022.
• SoftImpute - Fits a low-rank matrix through nuclear-norm regularization and repeated soft-thresholded SVD. Suits shared latent structure; low rank alone does not resolve selective missingness. Mazumder et al. 2010.
• Optimal-transport imputation - Optimizes missing entries by matching distributions of imputed batches with Sinkhorn divergences. Can learn an imputation function for new observations. Muzellec et al. 2020; code.

Deep Generative and Pretrained Imputers

• Autoencoders for incomplete records - Evaluates autoencoder imputation on ALS trial data under simulated missingness, measuring reconstruction and progression prediction separately. Beaulieu-Jones et al. 2017.
• MIDA - Uses overcomplete denoising autoencoders for multiple imputation of mixed data. A separate method from Lall & Robinson's MIDAS software. Gondara and Wang 2017.
• Monte Carlo dropout autoencoders - Uses dropout within autoencoder and variational-autoencoder models to generate repeated imputations; evaluates reconstruction error and predictive similarity. Miok et al. 2020.
• GAIN - A generator fills missing entries; a discriminator predicts which entries were observed, aided by a partial missingness hint. The paper's identifiability result assumes MCAR. Yoon et al. 2018.
• MIWAE - Trains a latent-variable model on incomplete data using an importance-weighted bound on observed-data likelihood under MAR. Supports Monte Carlo single and multiple imputations. Mattei and Frellsen 2019.
• not-MIWAE - Jointly models values and missingness with importance-weighted variational inference. MNAR modeling relies on the specified mechanism; neural capacity does not establish identification. Ipsen et al. 2021.
• TabImpute and pattern-specific transformers - Pretrains imputers on synthetic tables with configurable missingness mechanisms. The default TabImpute uses MCAR training; specialists target particular patterns. Feitelberg et al. 2025.
• ReMasker - Randomly masks additional observed entries and trains a masked autoencoder to reconstruct them, then imputes naturally missing values; the training task uses available table entries. Du et al. 2023.
• MissDiff - Trains diffusion models directly on incomplete data by masking the denoising score-matching loss. Avoids first filling every missing entry with a separate imputer. Ouyang et al. 2023.

Time-Series and Sensor Imputation

• Last observation carried forward - Repeats the latest observed value across a gap. Assumes no change during that interval; bias can operate in either direction. Molnar et al. 2008; van Buuren 2018.
• M-RNN - Combines interpolation within temporal streams with imputation across variables using a multidirectional recurrent network. Yoon et al. 2017.
• BRITS - Learns missing values within bidirectional recurrent computation, combining temporal and cross-feature estimates. Uses observations on both sides of a gap, so its full form requires future context. Cao et al. 2018.
• GP-VAE - Places a Gaussian-process prior over a low-dimensional latent trajectory and decodes it into observations. Combines temporal structure with probabilistic imputation. Fortuin et al. 2020.
• CSDI - Trains conditional score-based diffusion models to sample missing time-series values given observations. Self-supervised masking creates training targets from observed entries. Tashiro et al. 2021; code.
• SAITS - Combines two diagonally masked self-attention blocks with learned weights, using reconstruction and masked-imputation objectives. Du et al. 2023; code.
• GRIN - Uses graph message passing and recurrent processing to impute multivariate sensor streams. Exploits relations between sensors as well as temporal dependencies. Cini et al. 2022; code.

Prediction with Missing Inputs

• Missingness indicators - Binary masks preserve which inputs were missing after filling them. They can help prediction; adding indicators does not generally repair biased coefficient estimates. van Buuren 2018; docs.
• RNNs with missingness indicators - Studies diagnosis prediction from irregular clinical measurements and their missingness patterns. The objective is supervised prediction, not recovering every absent measurement. Lipton et al. 2016.
• GRU-D - Incorporates observation masks, elapsed times, and learned decays of inputs and hidden states into a recurrent predictor. Uses informative missingness directly. Che et al. 2018.
• NeuMiss - Derives a prediction architecture from a Neumann-series approximation under specified data models. Multiplication by missingness masks adapts computation to the observed pattern. Le Morvan et al. 2020.
• Trees with native missing-value handling - Histogram gradient boosting learns which child receives missing values at a split. This supports prediction without a separate imputer; it does not reconstruct missing measurements. docs.

Evaluation and Diagnostics

• Imputation for prediction: beware of diminishing returns - Across 19 datasets, better reconstruction had limited predictive benefit with expressive predictors and missingness indicators. Results depend on the prediction setup. Le Morvan and Varoquaux 2025.
• Missingness benchmarks - MissBench evaluates 42 datasets and 11 missingness patterns in the revised preprint. Report the simulated mechanism and rate; random masking alone does not establish performance under MNAR. Feitelberg et al. 2025.
• Probabilistic evaluation - CSDI evaluates distributions with CRPS alongside point errors. For inferential multiple imputation, assess bias and interval coverage of the analysis target; low reconstruction error does not establish valid uncertainty. Tashiro et al. 2021; docs.
• Leakage control - Fit imputers and tune their settings within training folds. Split related records or temporal windows before fitting; held-out outcomes must not become imputation predictors in deployment evaluations. docs.
• Time-series masking - Evaluate isolated gaps and contiguous outages separately. Distinguish retrospective filling from online prediction: bidirectional imputers can use future observations unavailable at decision time. Tashiro et al. 2021, Cao et al. 2018.

Software

• scikit-learn - Simple, KNN, and iterative imputers plus missingness indicators. IterativeImputer produces one completion per call; posterior sampling and repeated calls can generate multiple imputations. docs.
• mice - Chained equations, diagnostics, and pooled inference in R. Amelia implements joint-model multiple imputation; missForest provides iterative forest imputation. Honaker et al. 2011; code, docs.
• miceforest - Multiple imputation with LightGBM-based chained models and mean matching in Python. HyperImpute adds automatic column-wise model selection. code, code.
• MIDASpy - Denoising-autoencoder multiple imputation. Lall and Robinson 2022; code.
• PyPOTS - Implementations for learning from partially observed time series, including BRITS, SAITS, and CSDI. code.
• Impute.jl - Missing-data handling for Julia vectors, matrices, and tables, including interpolation and carry-forward/backward methods. code.

Related topics

• Bayesian Inference
• Neural Networks
• Models