Indeed, the choice of e is arbitrary and used for illustration purposes. And the base 6 is simply the choice for which the total burden of CH is larger than that of migraines, so itās also not derived from first principles. This footnote is relevant:
The resulting scaling as 6x would mean that the 0ā10 scale would have to span 4 orders of magnitude. While GĆ³mez-Emilsson & Percy (2023) suggest the scale spans āat least two orders of magnitudeā, private communication with the authors indicates their central estimates might be closer to 4 orders of magnitude, with uncertainty ranging from 2 to 8 OOMs.
The paper cited also mentions the possibility of a linear relationship for lower pain intensities and an exponential relationship at higher intensities (a ākinkedā distribution), highlighting the fact that there are more possibilities beyond a uniform exponential increase.
I personally donāt have a good intuition for what the base should be but might do more work on this specific question.
Iām also not sure what the optimal mapping of intensities for the Russell vs Torelli & Manzoni scales is, also considering the fact that the two studies had different methodologies. I think thereās no correct answer, so that was my best guess (though I could also imagine āVery slightā being more intense than a 1.5). Do let me know if you have other suggestions! (Or feel free to fork the code and play around with the parameters. :) )
Thanks, thatās helpful! I think that footnote may have an error though. 6^10 is 60 million, implying nearly 8 OOMs from 0 to 10. The 1-10 gap would be closer to 4 OOMs if linear from 0-5 and exponential with base 6 from 0-10 though. 2-8 OOMs seems like a reasonable range to me, itās comically broad but highlights our uncertainty about pain magnitude. Iāll definitely give GĆ³mez-Emilsson & Percy (2023) a read, and will fork your cose and play around with the numbers as well!
Thank you for your comment, Tim!
Indeed, the choice of e is arbitrary and used for illustration purposes. And the base 6 is simply the choice for which the total burden of CH is larger than that of migraines, so itās also not derived from first principles. This footnote is relevant:
The paper cited also mentions the possibility of a linear relationship for lower pain intensities and an exponential relationship at higher intensities (a ākinkedā distribution), highlighting the fact that there are more possibilities beyond a uniform exponential increase.
I personally donāt have a good intuition for what the base should be but might do more work on this specific question.
Iām also not sure what the optimal mapping of intensities for the Russell vs Torelli & Manzoni scales is, also considering the fact that the two studies had different methodologies. I think thereās no correct answer, so that was my best guess (though I could also imagine āVery slightā being more intense than a 1.5). Do let me know if you have other suggestions! (Or feel free to fork the code and play around with the parameters. :) )
Thanks, thatās helpful! I think that footnote may have an error though. 6^10 is 60 million, implying nearly 8 OOMs from 0 to 10. The 1-10 gap would be closer to 4 OOMs if linear from 0-5 and exponential with base 6 from 0-10 though. 2-8 OOMs seems like a reasonable range to me, itās comically broad but highlights our uncertainty about pain magnitude. Iāll definitely give GĆ³mez-Emilsson & Percy (2023) a read, and will fork your cose and play around with the numbers as well!
Gee, not sure what happened thereāthanks for pointing that out! Iāve edited the footnote.