title: Odds ratios deconstructed: A new way to understand and explain odds ratios as conditional risk ratios.
abstract: OBJECTIVE
  The aim of this analysis was to provide an alternative derivation of the odds ratio (OR) to provide an intuitive meaning, freeing it from any mention of odds, which may make it a more useful concept for clinicians to use when describing treatment effect.
  
  
  METHODS
  By examining the four possible combinations of treatment/control and corresponding outcomes, we considered the conditional risk ratio (RR, also known as relative risk) of an event with the treatment compared with an event with the control for pairs of patients for whom treatment and control would yield different results. Both matched and unmatched studies are considered.
  
  
  RESULTS
  We found that the OR could be derived as the RR of an outcome with treatment compared with an outcome with control conditional on the treatment and control resulting in different outcomes, thus providing a measure of the net benefit of treatment.
  
  
  CONCLUSIONS
  It has been claimed that the OR comparing the effect of treatment vs. control does not have the same clinical interpretability as RR because it involves ratios of odds and so is difficult to explain in terms of patient numbers. This new derivation provides an interpretation of the OR as an RR but conditional on treatment and control resulting in different outcomes. This may help explain the reason ORs cause interpretation difficulties in practice. Moreover, the OR may be a more clinically useful parameter to patients because it deals with only those situations where the outcome differs between the two groups.
authors: Department of Mathematics and Statistics, McMaster University, 1280 Main St. W., Hamilton, Ontario L8S 4K1, Canada. Electronic address: hoppe@mcmaster.ca.
authors: Fred M
authors: FM
authors: Hoppe
authors: Department of Orthopaedic Surgery, Stanford University, 450 Broadway St., Redwood City, CA 94063, USA.
authors: Daniel J
authors: DJ
authors: Hoppe
authors: Department of Clinical Epidemiology and Biostatistics, McMaster University, 1280 Main St. W., Hamilton, Ontario L8S 4K1, Canada.
authors: Stephen D
authors: SD
authors: Walter
mesh_terms: D006801:Humans; D016017:Odds Ratio; D012306:Risk
keywords: Conditional probability; Matched pairs; Number needed to treat; Odds; Odds ratio; Relative risk; Risk; Risk difference; Risk ratio; Unmatched pairs