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?
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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.