Can you provide a worked example of how risk-adjustment works?

A worked example is given below:

Observed events = number of patients at the Trust with the event of interest.

Expected events = sum of predicted risks over all patients at the Trust, not just those with the event (from multivariable prediction model).

Adjusted rate = Overall national rate x (Observed events / Expected events)

In a rather artificial example, a Trust operates on 10 patients, 1 of whom dies within 90 days, and so their observed mortality is 10%. These patients are predicted to die with the following predicted risks (from the prediction model):

0.02, 0.05, 0.05, 0.10, 0.10, 0.10, 0.20, 0.25, 0.30, 0.35.

Observed deaths = 1

Expected deaths = 0.02 + 0.08 + 0.05 + 0.10 + 0.10 + 0.10 + 0.10 + 0.15 + 0.20 + 0.25 = 1.15

The overall 90-day mortality across England and Wales is 3.8%

The hospital’s risk-adjusted mortality is therefore 3.8x(1/1.15) = 3.3%.

This Trust was operating on particularly high-risk patients. The number of deaths observed at this hospital was a little less than the number of deaths expected for patients with these characteristics and therefore their risk-adjusted mortality is a little lower than the overall national rate, and much lower than their observed rate.