# What a negative odd ratio means here

I have a plot from comparing differentially mutated genes between two groups like this

Can you give me an intuition why odd ratio could be negative?

• It's the log odds ratio (i.e. the logarithm of the odds ratio). Exponentiate the log-odds ratio to get the odds ratio. Odds ratios cannot be negative. – COOLSerdash Jan 29 '20 at 12:18
• – Sextus Empiricus Jan 30 '20 at 15:36

## 1 Answer

That’s the logarithm of the odds ratio, not the odds ratio itself. An odds ratio less than zero is nonsense. Looking at the behavior of a logarithm function (the base could be 2, could be 10, could be $$e$$), the function achieves values less than zero when the argument is less than 1, so a negative log of the odds ratio means that the odds ratio is between 0 and 1.

If you want to do calculations with the log of the odds ratio, it will be important to know the base (most likely $$e$$), but for qualitative analysis like checking if the value is less than or greater than zero, the base does not matter. Graph some log functions with different bases if you’re wondering why.

• Thank you. Then in terms of interpretation for a given gene, PML in this plot, what is the different between log odd ratio -4.09 vs 4.09. I mean what do negative and positive values mean? – Angel Jan 29 '20 at 12:27
• $exp(-4.09)$ vs $exp(4.09)$ are your odds ratios. What are you putting into the odds ratio? I can see this as odds of expressing or having the PML gene, but odds ratio has me confused. – Dave Jan 29 '20 at 12:35
• @Angel The positive/negative aspect of the log odds is related to the directionality of your original comparison. Here, it looks like Relapse compared to a baseline of Primary, so the negative log(OR) of -4.09 indicates that PML is less frequently mutated in Relapse than Primary. Of course, you could define the comparison in the opposite way (Primary compared to Relapse), which would yield the opposite log(OR) of 4.09. The log(OR) is sometimes nice since flipping the directionality just means flipping the sign, which isn't the case with raw OR (consider an OR of 2 vs. 0.5, for example). – Nuclear Hoagie Jan 29 '20 at 20:57
• Thank you, but do you think does this make any sense because PML only has mutated in 1 patient among 124 primary while mutated in 11 patients among 58 relapsed patients. Actually I got too puzzled – Angel Jan 31 '20 at 15:55