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Missing Not at Random
A missing-data mechanism in which the probability that a value is missing depends on the value itself, even after conditioning on every observed variable -- the most difficult mechanism to handle, since standard imputation methods cannot correct for it without additional external assumptions or data.
In Plain English
Data are missing not at random (MNAR) when the very reason a value is missing is tied to what that missing value actually would have been, and no amount of conditioning on other observed variables can explain that connection away. A classic example is income reporting, people with very high or very low incomes may be more likely to skip the income question specifically because of what their income is, not because of their age, education, or anything else you happened to measure. MNAR is the hardest missing-data mechanism to handle correctly, because the standard toolkit (multiple imputation, maximum likelihood) all assume MAR and can produce genuinely biased results under MNAR, correcting for MNAR generally requires either strong, often unverifiable modeling assumptions about the missingness mechanism itself, or additional external data (like a follow-up validation study) that can shed light on the values that went missing.
Definition
Data are missing not at random (MNAR) when the probability that a value is missing depends on the unobserved value itself, even after conditioning on every observed variable, formally , the missing-data mechanism is said to be nonignorable. Standard missing-data methods, multiple imputation, full-information maximum likelihood, all rely on the MAR assumption and can produce biased results under MNAR, since no adjustment based solely on observed variables can correct for a dependence on the unobserved value itself; addressing MNAR generally requires explicit selection models (jointly modeling the outcome and the missingness mechanism together, in the spirit of a Heckman-style selection correction) or pattern-mixture models (modeling the outcome distribution separately within each missingness pattern), both of which require strong, typically unverifiable assumptions about the specific form of the missingness mechanism, or, where feasible, supplementary data collection (a smaller, more intensive follow-up or validation sample specifically targeting nonrespondents) that can provide direct external information about the values that went missing.
Properties
- MNAR is the mechanism against which no standard, purely statistical adjustment based on observed data alone can fully protect, because the very information needed to correct for it, the relationship between missingness and the unobserved value, is by definition not available in the observed data, any correction necessarily rests on an unverifiable assumption about that relationship.
- Income and sensitive behaviors (substance use, illegal activity, socially stigmatized attitudes) are classic real-world MNAR candidates, in each case there is a plausible, well-documented reason to suspect the act of nonresponse is tied to the true, unreported value itself, rather than being fully explained by other measured characteristics.
- Sensitivity analysis, examining how conclusions change under a range of plausible MNAR assumptions rather than committing to a single unverifiable model, is standard best practice when MNAR is suspected, since MNAR can never be fully ruled out or corrected with certainty from the observed data alone, honest reporting of this residual uncertainty is preferred over false confidence in a single correction.
At a Glance