Missing completely at random (MCAR)

Data are missing completely at random when the missingness indicator is independent of both observed and missing data.

With a fully observed covariate $X$ taking finitely many values of positive probability, response $Y$ of finite variance, and an independent response indicator with probability $\rho$, let $m(X)=E(Y\mid X)$. A covariate-standardized mean has asymptotic variance $[\mathrm{Var}(m)+E\{\mathrm{Var}(Y\mid X)\}/\rho ]/N$, whereas the complete-case mean has variance $\mathrm{Var}(Y)/(N\rho )$. The difference is $(1-\rho )\mathrm{Var}(m)/(N\rho )$. Thus missing completely at random does not justify discarding predictive observed covariates on efficiency grounds.