= Complete-case regression under conditional missing at random
{title2=$Y\perp R\mid X\quad\Longrightarrow\quad f(Y\mid X,R=1)=f(Y\mid X)$}
Let $R$ indicate observation of a regression response $Y$, with fully observed predictors $X$. If $Y\perp R\mid X$ and $\mathbb P(R=1\mid X)>0$ on the target support, the observed conditional density equals the full conditional density. Thus a correctly specified <regression model> can be fitted to observed responses under conditional <missing at random>. This does not justify the unadjusted complete-case mean when observation rates and outcome means both vary with $X$, nor does it imply <missing completely at random>.
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