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Missing at Random
A missing-data mechanism in which the probability that a value is missing may depend on other observed variables in the dataset, but not on the unobserved value itself after conditioning on those observed variables -- the standard working assumption behind most modern imputation methods.
In Plain English
Data are missing at random (MAR) when whether a value is missing can depend on other things you did measure, but not on the missing value itself, once you've accounted for those other measured things. For example, if older respondents are less likely to answer an income question, but within any given age group, missingness has nothing to do with the actual income amount, that's MAR: missingness depends on age (observed), but not on income itself (unobserved) once age is accounted for. This is a weaker, more realistic assumption than MCAR, and it's the standard working assumption behind most modern missing-data methods, including multiple imputation, which is why choosing a rich set of predictors for the imputation model, ones likely to explain both the missingness and the outcome, is so important in practice.
Definition
Data are missing at random (MAR) when the probability that a variable's value is missing may depend on other, fully observed variables in the dataset, but does not depend on the value that would have been observed itself, after conditioning on those observed variables, formally . MAR is a strictly weaker and more realistic assumption than MCAR (MCAR is the special case where missingness depends on nothing at all, not even observed variables), and it is the assumption under which standard multiple imputation and full-information maximum likelihood methods produce valid, unbiased results, provided the imputation or likelihood model conditions on a sufficiently rich set of the observed variables actually driving the missingness; crucially, MAR is fundamentally not testable from the observed data alone, since verifying it would require knowing something about the unobserved values themselves, which is precisely the information that's missing, so in practice MAR is a plausibility judgment based on subject-matter knowledge of why data are likely missing, not an assumption that can be empirically confirmed.
Properties
- MAR is the pivotal, working assumption underlying essentially all standard modern missing-data methods, multiple imputation and full-information maximum likelihood both produce valid, unbiased results under MAR (given a correctly specified model), which is exactly why these methods have become the default recommended practice over simpler approaches like complete-case analysis or mean imputation.
- Because whether MAR truly holds cannot be verified directly from the observed data, careful missing-data practice emphasizes including a rich, well-chosen set of auxiliary variables in the imputation or analysis model, ones subject-matter knowledge suggests are related to why data are missing, to make the MAR assumption as plausible as possible even though it can never be proven.
- Under MAR, complete-case analysis (simply discarding incomplete cases) can produce biased estimates, unlike under MCAR, since the retained complete cases are no longer a representative random subsample once missingness depends on observed variables correlated with the outcome, this is a key reason MAR-aware methods like multiple imputation are preferred over simple case deletion whenever MCAR cannot be safely assumed.
At a Glance