 Missingness overlap from marginal and complete-case counts

For $N$ observations, let $m_X$ and $m_Y$ count missing values of two variables, and let $n_{XY}$ count observations on which both are available. The number missing at least one variable is $N-n_{XY}$. Applying the inclusion-exclusion principle to the two missing-value sets gives the displayed number missing both. The disjoint counts are consequently $m_X-m_{XY}$ missing only $X$, $m_Y-m_{XY}$ missing only $Y$, $m_{XY}$ missing both, and $n_{XY}$ complete cases. Their nonnegativity is a necessary consistency check on the supplied totals.

This identifies an aggregate missingness pattern without reconstructing individual records. It does not identify a missing-data mechanism, establish missing completely at random, or justify treating a substantive response such as an explicit unknown category as a missing observation. Counts from different cohorts cannot be inserted into the identity without first establishing a common denominator.