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During my searches I've come to a strange position;

The Probit definition in Wikipedia is simple;

$$\operatorname{probit}(p) = \sqrt{2}\,\operatorname{erf}^{-1}(2p-1)$$

Then, I've come to a source that explains using Finley's book

The probit $Y$ of the proportion $P$ is defined by; $$P(Y) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{Y-5} e^{-\frac{1}{2}u^2}du. $$

I've verified both using the Sagemath, but there is no relation that I can see between these two functions. Am I missing something? Are those the same (actually the inverse of each other)?

Note: I'm trying to understand the probits on the concept of the $\text{LD}_{50}$.

kelalaka
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  • Your $P$ function has typographical errors. When you fix them, it should become apparent how closely $P$ is related to erf. – whuber Feb 16 '21 at 20:13
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    @whuber yes, missed the sign, Thanks for the dupe. – kelalaka Feb 16 '21 at 20:24
  • @whuber I've found the relatiom as $\text{Erf}(x) = 2(P(x\sqrt{2}+5) - P(5)) $ – kelalaka Feb 16 '21 at 21:46
  • The appearance of the "-5" in the definition you give is surprising: it does not agree with usual probit definitions. – whuber Feb 16 '21 at 21:49
  • @whuber I agree with this and that is why I wonder. Maybe this is related to the historical development of the $LD_{50}$, I'm looking for it. I'll try to find a copy of the book to see the reasoning. – kelalaka Feb 16 '21 at 21:53
  • @whuber I've tried to validate some works and saw the inconsistency, they link to `Estimation of the ED,, and Its Error by Means of Logarithmic-Probit Graph Paper. ` by Miller and Tainter -1944, and the Table on the source goes back to the book. – kelalaka Feb 16 '21 at 21:57
  • I begin to suspect some typographical error in the source document. It ought to read something like $$p = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\operatorname{probit}(p)}e^{-u^2/2}\,\mathrm{d}u.$$ Maybe some complicated expression at the upper limit of the integral got corrupted? – whuber Feb 16 '21 at 22:01
  • I don't think so, with it I can validate Finney's table ( see on the link [source](http://userwww.sfsu.edu/efc/classes/biol710/probit/ProbitAnalysis.pdf) in the question). – kelalaka Feb 16 '21 at 22:03
  • Sagemath: def proportion(Y): f = (1/(sqrt(2*pi))) * e^(-(1/2)*x^2) res = numerical_integral(f, -Infinity, Y-5) print(res) return res[0] print(proportion(7.33)) – kelalaka Feb 16 '21 at 22:06
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    Thanks for the reference. Finney appears to have felt that people would not like to deal with negative values, and so has added $5$ (by *fiat*) to the usual probits. For most analyses it makes little difference, but the arbitrariness of this offset gives us little incentive to use or recommend it. If you are going to use this definition you need to make sure you explain what you're doing to your audience! – whuber Feb 16 '21 at 22:09
  • That is great, thanks. I'll note that. – kelalaka Feb 16 '21 at 22:12

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