Article Text

Download PDFPDF
Estimating risk when zero events have been observed
  1. John Quigley1,
  2. Matthew Revie1,
  3. Jesse Dawson2
  1. 1Department of Management Science, University of Strathclyde, Glasgow, UK
  2. 2Institute of Cardiovascular and Medical Sciences, University of Glasgow, Glasgow, UK
  1. Correspondence to Dr Matthew Revie, Department of Management Science, University of Strathclyde, 40 George Street, Glasgow G1 0QU, UK; matthew.j.revie{at}

Statistics from

Request Permissions

If you wish to reuse any or all of this article please use the link below which will take you to the Copyright Clearance Center’s RightsLink service. You will be able to get a quick price and instant permission to reuse the content in many different ways.

Assessing the risk of complications or adverse events following an intervention presents challenges when they have not yet occurred. Suppose, for instance, a chronic shortage of cardiac telemetry beds has prompted a hospital to implement a new policy that places low-risk patients admitted to ‘rule out myocardial infarction’ in regular ward beds (ie, with no telemetry). After 6 months and the admission of 100 such patients, no cardiac arrests or other untoward events have occurred. This absence of harm (ie, zero adverse events) indicates a low risk, but clearly we cannot infer a risk of zero on the basis of only 100 patients. But, what can we say about the true underlying risk?

The Rule of Three (Ro3), first proposed by Hanley and Hand,1 provides an estimate for the upper bound of the underlying risk when zero events of interest have occurred through considering the lack of occurrences as moderately rare. The Ro3 estimates the risk as no greater than 3/n, where n is the number of opportunities or exposures to the risk. Technically, 3/n is obtained through assigning the probability 0.95 to measure the likelihood of realising more events than those observed. Note that the Ro3 estimate applies only when one has at least 30 observations. In the above example, the Ro3 would estimate the upper limit for the risk of adverse events associated with the new policy as 3/100 or 3%.

This estimate does not sound as reassuring …

View Full Text


  • Contributors All three authors contributed in the following way: contributed to the paper's concept including structuring and collating data, contributed to the writing and editing of the final article, approved the final version, take ownership for the content and integrity of the paper.

  • Competing interests None.

  • Provenance and peer review Not commissioned; internally peer reviewed.