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Likelihood ratio interpretation of the relative risk
  1. Suhail A R Doi1,
  2. Polychronis Kostoulas2,
  3. Paul Glasziou3
  1. 1 Department of Population Medicine, College of Medicine, QU Health, Qatar University, Doha, Qatar
  2. 2 Faculty of Public Health, University of Thessaly, Volos, Greece
  3. 3 Institute for Evidence-Based Healthcare, Bond University, Gold Coast, Queensland, Australia
  1. Correspondence to Professor Suhail A R Doi, Department of Population Medicine, College of Medicine, QU Health, Qatar University, Doha, Qatar; sardoi{at}

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Key messages

What is already known on this topic

  • The risk ratio (relative risk) is a ratio of two risks that is interpreted as connecting the intervention conditional risks in a clinical trial.

What this study adds

  • It is demonstrated that the conventional interpretation of the risk ratio is in conflict with Bayes’ theorem.

  • The interpretation of the risk ratio as a likelihood ratio connecting prior (unconditional) intervention risk to outcome conditional intervention risk is required to avoid conflict with Bayes’ theorem.

How this study might affect research, practice or policy

  • The interpretation of the risk ratio as an effect measure in a clinical trial is naïve and better replaced by its interpretation as a likelihood ratio.

  • The ratio of the complementary risk ratio's (or likelihood ratio's) is what should actually be interpreted as an effect measure connecting the intervention conditional risks in a clinical trial.

Interpreting diagnostic test results in medicine

The likelihood ratio (LR) is today commonly used in medicine for diagnostic inference. Historically, it was preceded by introduction, in 1966, of the predictive value of a diagnostic test in Medicine1 and within a decade of the latter, it was realised that the true-positive to false-positive ratio2 3 also then called the likelihood value4 was the main driver of the change from prior probabilities to posterior predictive values. The latter were also called post-test likelihoods and this ratio became known as the LR in Medicine. The change from prior probabilities to posterior predictive values was formulated using Bayes’ theorem5 and represented a more versatile approach to predictive values. The reason this is considered more versatile is that Bayes’ theorem allows a physician to compute the predictive value (probability) of a diagnosis conditional on a specific test result. For example, if we denote test status as +ve (positive) and –ve (negative) and the gold standard (eg, underlying diagnosis) as D (diagnosed) and nD (not diagnosed), respectively, then from Bayes’ theorem,5 the posterior …

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  • Contributors All authors contributed to the conceptualisation and methods described and critically reviewed and revised the manuscript. SARD is responsible for the overall content as guarantor.

  • Funding This work was made possible by Programme Grant number NPRP-BSRA01-0406-210030 from the Qatar National Research Fund (a member of Qatar Foundation).

  • Competing interests None declared.

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

  • Supplemental material This content has been supplied by the author(s). It has not been vetted by BMJ Publishing Group Limited (BMJ) and may not have been peer-reviewed. Any opinions or recommendations discussed are solely those of the author(s) and are not endorsed by BMJ. BMJ disclaims all liability and responsibility arising from any reliance placed on the content. Where the content includes any translated material, BMJ does not warrant the accuracy and reliability of the translations (including but not limited to local regulations, clinical guidelines, terminology, drug names and drug dosages), and is not responsible for any error and/or omissions arising from translation and adaptation or otherwise.