= Missing-outcome bounds for a binary endpoint
{title2=$r/N\leq p\leq(r+m)/N$}
Suppose an endpoint is binary, $N$ people are in the target cohort, $r$ observed outcomes are successes and $m$ outcomes are missing. With no assumptions about the missing values, the cohort success proportion $p$ satisfies
$$
\frac{r}{N}\leq p\leq\frac{r+m}{N}.
$$
For the proof, let $u$ be the number of successes among the missing outcomes. Since $0\leq u\leq m$ and $p=(r+u)/N$, both <inequalities> follow. Every integer $u$ in that range is a possible completion of the observed data, so the endpoints are sharp. These are <missing data> bounds on a cohort proportion, not <confidence intervals> or bounds on a causal treatment effect.
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