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On this page I present my unstructured thoughts about statistics and epidemiology from time to time.

Finding evidence in the context of uncertainty is challenging.

Surprisal S-value: let’s toss a coin to understand the value of a p-value

February 4, 2022

Function S=log2(p)

Web ressources

Fraundorf P. Examples of Surprisal. http://www.umsl.edu/~fraundorfp/egsurpri.html
Rafi Z. P-Values Are Tough and S-Values Can Help. https://lesslikely.com/statistics/s-values
Zobeck, M. S-Values - Part 1: Calibrate your intuition about p-values. https://www.thedoctorsdialectic.com/2021/11/20/s-values

Article

Cole SR, Edwards JK, Greenland S. Surprise! Am J Epidemiol. 2021 Feb 1;190(2):191-193. doi: 10.1093/aje/kwaa136.
Greenland S. Invited Commentary: The Need for Cognitive Science in Methodology. Am J Epidemiol. 2017 Sep 15;186(6):639-645. doi: 10.1093/aje/kwx259.
Greenland S. Valid P-Values Behave Exactly as They Should: Some Misleading Criticisms of P-Values and Their Resolution With S-Values. The American Statistician. 2019;73(sup1):106-114. doi: 10.1080/00031305.2018.1529625.
Rafi Z, Greenland S. Technical Issues in the Interpretation of S-values and Their Relation to Other Information Measures. arXiv. 2020 Aug 29. https://arxiv.org/abs/2008.12991.
Rafi Z, Greenland S. Semantic and cognitive tools to aid statistical science: replace confidence and significance by compatibility and surprise. BMC Med Res Methodol. 2020 Sep 30;20(1):244. doi: 10.1186/s12874-020-01105-9.
Rothman KJ. Taken by Surprise. Am J Epidemiol. 2020 Jul 10:kwaa137. doi: 10.1093/aje/kwaa137.

**S-value interpretation in relation to examples of p-value**
p		Surprisal value	Bits of information (nearest integer of S)
as decimal	as fraction	S = -log2(p)	narrative	x times in a row head H
1	1	0	0 bits of information	-
0.75	³⁄₄	0.42	0 bits of information	-
0.50	¹⁄₂	1	1 head on a toss of 1 coin	H
0.25	¹⁄₄	2	2 heads on a toss of 2 coins	H-H
0.10	¹⁄₁₀	3.32	3 heads on a toss of 3 coins	H-H-H
0.05	¹⁄₂₀	4.32	4 heads on a toss of 4 coins	H-H-H-H
0.01	¹⁄₁₀₀	6.64	6 heads on a toss of 6 coins	H-H-H-H-H-H
0.005	¹⁄₂₀₀	7.64	8 heads on a toss of 8 coins	H-H-H-H-H-H-H-H
0.001	¹⁄_1,000	9.97	10 heads on a toss of 10 coins	H-H-H-H-H-H-H-H-H-H
0.0001	¹⁄_10,000	13.29	13 heads on a toss of 13 coins	H-H-H-H-H-H-H-H-H-H-H-H-H
0.00001	¹⁄_100,000	16.61	17 heads on a toss of 17 coins	H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H
0.000001	¹⁄_1,000,000	19.93	20 heads on a toss of 20 coins	H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H-H
0	-	infinity	Infinity bits of information	∞

Function S=log2(p)