title: Specifying the probability characteristics of funnel plot control limits: an investigation of three approaches.
abstract: BACKGROUND
  Emphasis is increasingly being placed on the monitoring and comparison of clinical outcomes between healthcare providers. Funnel plots have become a standard graphical methodology to identify outliers and comprise plotting an outcome summary statistic from each provider against a specified 'target' together with upper and lower control limits. With discrete probability distributions it is not possible to specify the exact probability that an observation from an 'in-control' provider will fall outside the control limits. However, general probability characteristics can be set and specified using interpolation methods. Guidelines recommend that providers falling outside such control limits should be investigated, potentially with significant consequences, so it is important that the properties of the limits are understood.
  
  
  METHODS
  Control limits for funnel plots for the Standardised Mortality Ratio (SMR) based on the Poisson distribution were calculated using three proposed interpolation methods and the probability calculated of an 'in-control' provider falling outside of the limits. Examples using published data were shown to demonstrate the potential differences in the identification of outliers.
  
  
  RESULTS
  The first interpolation method ensured that the probability of an observation of an 'in control' provider falling outside either limit was always less than a specified nominal probability (p). The second method resulted in such an observation falling outside either limit with a probability that could be either greater or less than p, depending on the expected number of events. The third method led to a probability that was always greater than, or equal to, p.
  
  
  CONCLUSIONS
  The use of different interpolation methods can lead to differences in the identification of outliers. This is particularly important when the expected number of events is small. We recommend that users of these methods be aware of the differences, and specify which interpolation method is to be used prior to any analysis.
authors.affiliation: Department of Health Sciences, University of Leicester, Leicester, United Kingdom. brad.manktelow@le.ac.uk
authors.forename: Bradley N
authors.initials: BN
authors.lastname: Manktelow
authors.forename: Sarah E
authors.initials: SE
authors.lastname: Seaton
mesh_terms: D016012:Poisson Distribution; D011336:Probability