# What Is Meant by "Missing at Random"?

arXiv: https://arxiv.org/abs/1306.2812 · HTML: https://mlsanity.com/paper/1306.2812/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2013.

## Built on by (most cited first, 16 of 16)

- [HyperImpute: Generalized Iterative Imputation with Automatic Model Selection](https://mlsanity.com/paper/2206.07769/) (2022)
- [MIRACLE: Causally-Aware Imputation via Learning Missing Data Mechanisms](https://mlsanity.com/paper/2111.03187/) (2021)
- [Transformed Distribution Matching for Missing Value Imputation](https://mlsanity.com/paper/2302.10363/) (2023)
- [Learning from data with structured missingness](https://mlsanity.com/paper/2304.01429/) (2023)
- [Doubly robust nearest neighbors in factor models](https://mlsanity.com/paper/2211.14297/) (2022)
- [DAEMA: Denoising Autoencoder with Mask Attention](https://mlsanity.com/paper/2106.16057/) (2021)
- [Sharing pattern submodels for prediction with missing values](https://mlsanity.com/paper/2206.11161/) (2022)
- [Multiple imputation and test-wise deletion for causal discovery with incomplete cohort data](https://mlsanity.com/paper/2108.13331/) (2021)
- [Adaptive Optimization for Prediction with Missing Data](https://mlsanity.com/paper/2402.01543/) (2024)
- [Improving Variational Autoencoder Estimation from Incomplete Data with Mixture Variational Families](https://mlsanity.com/paper/2403.03069/) (2024)
- [Missing Data in Signal Processing and Machine Learning: Models, Methods and Modern Approaches](https://mlsanity.com/paper/2506.01696/) (2025)
- [CountsDiff: A Diffusion Model on the Natural Numbers for Generation and Imputation of Count-Based Data](https://mlsanity.com/paper/2604.03779/) (2026)
- [Data is missing again -- Reconstruction of power generation data using k-Nearest Neighbors and spectral graph theory](https://mlsanity.com/paper/2409.00300/) (2024)
- [High-Dimensional Gaussian Mean Estimation under Realizable Contamination](https://mlsanity.com/paper/2603.16798/) (2026)
- [Extending Model-x Framework to Missing Data](https://mlsanity.com/paper/2202.13054/) (2022)
- [Generative Modeling under Non-Monotone MAR Missingness via Approximate Wasserstein Gradient Flows](https://mlsanity.com/paper/2604.04567/) (2026)

## Abstract (arXiv)

The concept of missing at random is central in the literature on statistical analysis with missing data. In general, inference using incomplete data should be based not only on observed data values but should also take account of the pattern of missing values. However, it is often said that if data are missing at random, valid inference using likelihood approaches (including Bayesian) can be obtained ignoring the missingness mechanism. Unfortunately, the term "missing at random" has been used inconsistently and not always clearly; there has also been a lack of clarity around the meaning of "valid inference using likelihood". These issues have created potential for confusion about the exact conditions under which the missingness mechanism can be ignored, and perhaps fed confusion around the meaning of "analysis ignoring the missingness mechanism". Here we provide standardised precise definitions of "missing at random" and "missing completely at random", in order to promote unification of the theory. Using these definitions we clarify the conditions that suffice for "valid inference" to be obtained under a variety of inferential paradigms.