Explaining Odds Ratio and Relative Risk to the statistically challenged I'm peer-reviewing a manuscript for a psychology journal in which I believe the authors have mixed up odds-ratio and risk-ratio. They are being so stubborn in their insistence that they have not mixed them up that I feel like I need a sanity check. I have written a paragraph intended to encapsulate the whole issue. Could anyone please simply endorse the following paragraph or explain how it needs improving? I have edited some words (e.g. factor name) to preserve anonymity of the manuscript.
Quoting directly from the manuscript:

Inspection of the regression model revealed that the mean population odds ratio is estimated to be 1.90 (95% CI: 1.00, 3.58), suggesting that [participants] are 1.90 times more likely to [do X in] the [A condition] than the [B condition].

This is from a straightforward binary logistic regression with a categorical factor [A vs. B] which can be [A] or [B]. Let's look at a standard definition of Risk Ratio (https://en.wikipedia.org/wiki/Relative_risk): "relative risk or risk ratio (RR) is the ratio of the probability of an event occurring (...) in an exposed group to the probability of the event occurring in a comparison, non-exposed group". In this context, "times more likely" (manuscript) is the same as "ratio of the probability of the event occurring in a ... group to the probability of the event occurring in a comparison ... group" (wikipedia's definition of RR). The first worked example of RR in wikipedia in fact uses the exact same phrase "times more likely". This clearly demonstrates that the authors are referring to an OR as if it was an RR. The internet is full of easily found accurate pages explaining that OR is not the same thing as RR (but that they are easily confused). QED.
 A: You are of course right and it is a common mistake to describe an odds ratio like a relative risk ratio. I would suggest that it would be helpful to propose a more appropriate phrasing to them such as "suggesting that the odds of [participants] [doing X in] [A condition] is 1.90 times higher than in [B condition]." Once the authors realise that is all they would have to do, hopefully this is not too much of a discussion.
A: You are right and they are wrong. This is SUCH a common error that there is a whole genera of papers where the author just catalogues all of the times odds ratios are misinterpreted as changes in likelihood in published papers in a particular field. Here's just one example. Here is a JAMA article discussing the same issue. Just type "odds ratio risk ratio" into Google scholar and cite as many of the articles that come up as you need to to convince the author that they are wrong.
A: I believe you were correct with your response to the authors, but that a source other than Wikipedia would have strengthened your response.
Much has been published about odds vs risk in reliable journals, which would likely be more convincing than the Wikipedia article.
Here is one from 1998 that gives a balanced answer to when and how odds ratios can be misinterpreted and how to avoid that. And here from 2006 is another that perhaps takes a stronger position but also perhaps better demonstrates the difficulty of rendering odds ratios into plain language.
