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I noticed a relationship in my data and was able to get a line to fit reasonably well with nls in R (residual sum-of-squares = 9.652) with the function: y = a * e^(log10(x) * b) a = 120.1 and b - -2.5

I'm super rusty at stats, but it seems like there ought to be something simpler than e^-log10(x) that describes a quick decay followed by a slower decay. Any ideas?

proportion vs. log10(length) graph

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  • $\begingroup$ "Quick decay followed by slower decay" is more complex than "simple exponential decay," which is the model you already have. Could you please clarify your meaning? $\endgroup$
    – whuber ♦
    Commented May 25, 2021 at 23:20
  • $\begingroup$ I didn't call it a "simple exponential decay" intentionally (having Googled that earlier to find a similar shape to no avail), so I'm not sure why you thought that's what I said. I was just wondering if there was another function that has a similar shape that would fit similarly well. If there's not, so be it. But e^-log10(x) has no biological significance so I'm not attached to it if there's something similar. $\endgroup$
    – GenesRus
    Commented May 25, 2021 at 23:37
  • $\begingroup$ Regardless of what you might state in English, your formula is equivalent to $y=ax^\beta$ where $\beta = b/\log(10):$ that is, it's a power law. "Something similar" is too vague. You need to explain more clearly what you're looking for. $\endgroup$
    – whuber ♦
    Commented May 26, 2021 at 12:54
  • $\begingroup$ Yes, as I discovered below with Lys's help. I was looking for an equivalent formula that was simpler (i.e. the power law) or another that might have a similarly good fit for the graph shown. Thank you for also providing the former. I do have one remaining question, though, which base log would it be in the beta? It should be e, right, given my initial function? $\endgroup$
    – GenesRus
    Commented May 26, 2021 at 15:17

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You could just use the exponential decay formula $y = a\cdot e^{kx}$ without the log10 and see if it gives you a better fit. The exponential decay is also easily interpretable.

You might want to check this reference for some non linear equations, and see if any of them would be better for you case (it's in Python, but the important things are the formulas)

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    $\begingroup$ Thanks for the suggestion and the reference! The exponential decay formula didn't work (it throws and infinity apparently), but that's a super helpful reference site and the "Power Regression" Y = a * X^b seems to have been what I was after (using that as the base function arrives at an identical residual sum-of-squares). Much appreciated! $\endgroup$
    – GenesRus
    Commented May 26, 2021 at 1:12

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