= MCAR auxiliary-variable efficiency
{c}
{title2=$V_{\mathrm{CC}}-V_{\mathrm{std}}=(1-\rho)\operatorname{Var}(m)/(N\rho)$}
With a fully observed covariate $X$, response $Y$, and an independent response indicator with <probability> $\rho$, let $m(X)=E(Y\mid X)$. A covariate-standardized mean has asymptotic <variance> $[\operatorname{Var}(m)+E\{\operatorname{Var}(Y\mid X)\}/\rho]/N$, whereas the complete-case mean has <variance> $\operatorname{Var}(Y)/(N\rho)$. The difference is $(1-\rho)\operatorname{Var}(m)/(N\rho)$. Thus <missing completely at random> does not justify discarding predictive observed covariates on efficiency grounds.
Back to article page
