Risk-Adjusted CUSUM (Observed − Expected) calculator

Monitor a pass/fail outcome case by case with an observed-minus-expected chart and a Bernoulli CUSUM. Case-mix adjustment, a signalling limit, and restart after a signal.

What it does

The pair of charts used for sequential monitoring of a pass/fail outcome. The observed − expected chart records the running excess of events; the Bernoulli CUSUM beneath it tests whether the underlying rate has worsened and signals when the evidence crosses a limit.

When to use it

Each row is one case with a yes/no outcome (died within 90 days, complication, readmission) recorded in order, and you want to know whether the rate has deteriorated — the standard approach for monitoring surgical and transplant outcomes.

Alternatives

• Funnel — you are comparing several units at one point in time rather than one unit over time
• CUSUM for a measurement — the outcome is a measured quantity, not an event
• Logistic regression — you want to know WHICH factors predict the outcome, rather than whether the rate has changed
• a risk-adjusted model to produce the expected rate — casemix differs across the period and an unadjusted expected rate would be unfair. This chart takes expected probabilities, it does not estimate them. Fit a logistic model (here, in Regression), save its predicted probabilities as a column, and select that column as the per-row expected rate.

How to read the output

Observed − expected (upper chart)

The running count of excess events. Each non-event steps it DOWN by the expected rate p; each event steps it UP by 1 − p. At p = 10% that is −0.1 and +0.9. A flat line means 'exactly as expected', not 'nothing happened'. Rising is worse than expected, falling is better.

The asymmetric steps

The steps differ because the outcome is rare: one event carries far more information than one non-event, so it moves the chart much further. Do not read the up-step as a penalty and the down-step as a reward. They are the arithmetic of a rare event, not a scoring scheme.

The Bernoulli CUSUM (lower chart)

Accumulates the log-likelihood ratio for the deterioration you asked it to look for. It is floored at zero, so a long clean run banks no credit that a later cluster would have to spend first. Because it cannot go negative, this chart says nothing about performance being BETTER than expected. That question is answered by the upper chart drifting downwards.

Signals (★ and ●)

Marked where the lower chart crossed the limit, and shown on the upper chart so you can see how much excess had accumulated by then. A star is new; a dot is one an earlier report already carried. A signal is a prompt to look, not a verdict. Higher-risk casemix, a short run of bad luck, and a genuinely worse rate all look identical here.

The restart after a signal

The chart restarts partway up rather than at zero, so that if performance does not recover it can signal again quickly instead of climbing the whole way back. The restart makes a second signal easier than the first. Do not read two signals as twice the evidence.

How this calculator is validated · Which statistical test should I use?