Home Statistical Dictionary Missing Completely at Random
Missing Completely at Random
The strongest and most restrictive missing-data mechanism, in which the probability that a value is missing is entirely unrelated to both the observed and the unobserved data -- missingness is essentially a coin flip, independent of everything.
In Plain English
Data are missing completely at random (MCAR) when whether a value is missing has absolutely nothing to do with anything, not the value itself, and not any other variable you've measured. A classic (if artificial) example is a lab sample accidentally getting physically lost or a survey page getting spilled on with coffee, purely mechanical accidents unrelated to what the actual measurements would have shown. MCAR is the strongest, most convenient assumption to have, because under MCAR, the cases with complete data are essentially a random subsample of the full dataset, which means even the simplest approach, just dropping every case with any missing value (complete-case analysis), remains unbiased, though still wasteful of information, it's just rarely a realistic assumption for real survey nonresponse, which is almost always related to something.
Definition
Data are missing completely at random (MCAR) when the probability that a value is missing is unrelated to both the observed data and the unobserved (missing) data itself, formally , missingness is statistically independent of every variable in the dataset, observed or not. MCAR is the strongest of the three classical missing-data mechanisms (alongside missing at random and missing not at random), and under MCAR, complete-case analysis (discarding every case with any missing value) remains unbiased, since the retained complete cases constitute an unbiased random subsample of the full dataset, though still statistically inefficient, discarding partial information that other methods could recover; MCAR can be partially tested empirically (for example, Little's MCAR test compares the observed-data patterns of cases with and without missingness on a given variable to check whether they differ systematically on other observed variables), but true MCAR is a strong assumption rarely fully satisfied by real-world survey nonresponse, which is typically at least somewhat related to observed respondent characteristics.
Properties
- MCAR is a strictly stronger (more restrictive) assumption than missing at random, every MCAR mechanism is also MAR, but not every MAR mechanism is MCAR, in practice, true MCAR is rare for survey and observational data, since nonresponse is usually related to at least some observed characteristics (age, education, prior responses).
- Complete-case analysis (listwise deletion) is only statistically unbiased, though never fully efficient, under the MCAR assumption, if the true mechanism is instead MAR or MNAR, dropping incomplete cases can introduce genuine bias, not just lost efficiency, which is why verifying or at least reasoning carefully about the missingness mechanism matters before choosing a missing-data strategy.
- Little's MCAR test provides one empirical, if imperfect, check on the MCAR assumption, by comparing whether cases with and without missing values on a given variable differ systematically on other, fully observed variables, though passing this test does not prove MCAR, it can only rule out certain violations detectable from the observed data.
At a Glance