# 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.