Hospital mortality funnel plot ($(n_i,{\hat{p}}_i)$)

A hospital mortality funnel plot places each hospital's observed mortality proportion against its case volume. Under an independent binomial distribution model with common target p, the standard error is $\sqrt{p(1-p)/n_i}$, so uncertainty bands contract as volume grows. This is a provider-performance plot, distinct in purpose and axes from a meta-analysis funnel plot. A signal requires investigation of data quality and patient case mix before drawing a causal conclusion.

The displayed pointwise control limits follow from the normal approximation to an observed binomial proportion. Clip plotted limits to the probability range. Their nominal coverage is approximate and can be poor for small expected counts; exact binomial tails give a discrete alternative. Pointwise coverage does not provide simultaneous coverage of all hospitals; use Bonferroni correction or another familywise error rate procedure when that is the required guarantee.