MCAR, MAR, and MNAR¶

Three assumptions about why outcomes are missing, in increasing order of danger. Formal statements live in assumptions.md (A2); this page is the plain-language version.

MCAR — missing completely at random¶

Missingness is unrelated to anything: a coin flip. Under MCAR even the available-case analysis is unbiased (just less precise).

After a lab mix-up, a random 10% of month-3 samples were lost.

MAR — missing at random¶

After conditioning on observed data — baseline covariates, treatment, earlier outcomes — missingness does not additionally depend on the value that is missing.

Participants with high seizure counts at the previous visit are more likely to skip the next one. Dropout depends on data we have.

This is the assumption longmi's imputers work under: an imputation model that conditions on the observed history restores valid inference.

MNAR — missing not at random¶

Missingness depends on the missing value itself, even given everything observed.

Participants skip a visit because of how they feel that day — the very outcome that would have been recorded.

No analysis of the observed data alone can fix this without extra assumptions.

The uncomfortable fact¶

MAR versus MNAR is not testable from the observed data. The data you would need to run the test are exactly the data that are missing. Any software that claims to have "verified MAR" is overreaching — which is why longmi's validity_report() renders the mechanism as [declared], never [verified].

What to do about it¶

1. Make MAR as plausible as possible: condition the imputation model on everything that predicts both the outcome and dropout (auxiliary variables included).
2. Stress-test the assumption with delta adjustment: impute under MAR, then shift the imputed values ("suppose dropouts were 20% worse than MAR predicts") and see whether conclusions survive. A range of deltas traces how far reality would have to deviate from MAR before the conclusion changes.
3. Report the assumed mechanism, the deltas examined, and the fraction of missing information — the reporting guide has the checklist.