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Likelihood ratio interpretation of the relative risk
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• Published on:
Claims about the main claim
• Suhail A Doi, Clinical Endocrinologist and Professor of Clinical Epidemiology Qatar University
• Other Contributors:
• Polychronis Kostoulas, Associate Professor of Epidemiology
• Paul Glasziou, Director of the Institute for Evidence-Based Healthcare

Title: “Claims about the main claim”
Author: Suhail A, Doi, Polychronis, Kostoulas, Paul, Glasziou
In response to the published article "Likelihood ratio interpretation of the relative risk"

Rapid response :
September 16, 2022

The problem in evidence-based medicine arises when we port relative risks derived from one study to settings with different baseline risks. For example, a baseline risk of 0.2 and treated risk of 0.4 for an event in a trial gives a RR of 2 (0.4/0.2) and the complementary cRR of 0.75 (0.6/0.8). Thus the ratio of LRs (RR/cRR) is 2/0.75 = 2.67. If applied to a baseline risk of 0.5 the predicted risk under treatment with the RR “interpretation” is 1.0 but with the ratio of LRs “interpretation” is 0.73. Here, the interpretation of the risk ratio as a likelihood ratio, using Bayes’ theorem, clearly gives different results, and solves the problem of impossible risks as clearly depicted in the manuscript and the example.
If, in our effort to highlight the need of this correct interpretation, we have used strong wording that annoyed the commentator we feel the need to express regret. We hope that the commentator could also feel similarly for his scientifically unbecoming choice of wording that culminated with “Doi’s Conjecture”.
Conflict of Interest
None declared

Conflict of Interest:
None declared.
• Published on:
The main claim in this paper is not even wrong

Dear Prof. Franco,

I am writing to request further clarification on the paper “Likelihood ratio interpretation of the relative risk”. The “key messages” section of this paper states that the study adds the following to the literature:

⇒ It is demonstrated that the conventional interpretation of the relative risk is in conflict with Bayes’ theorem.
⇒ The interpretation of the relative risk as a likelihood ratio connecting prior (unconditional) intervention risk to outcome conditional intervention risk is required to avoid conflict with Bayes’ Theorem

I will refer to the first bullet point as “Doi’s Conjecture”. Doi’s Conjecture is also stated in the second section of the main text, where it is claimed that “the usual interpretation (33% increase in the +ve outcome under treatment) contravenes Bayes Theorem”.

No attempt is made within the text to prove Doi’s Conjecture. But perhaps more worryingly, no attempt is made to define the term “interpretation”, a term which is not defined in standard probability theory. The meaning of Doi’s Conjecture is therefore at best ambiguous. Moreover, the manuscript relies substantially on claims about how effect measures are “perceived”, another term which is defined neither in probability theory not in the manuscript.

The relative risk is defined as the risk of the outcome under treatment, divided by the risk of the outcome under the control condition; that is, as a ratio of two probabilities. Thi...