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The ubiquity of the Simpson’s paradox. (English) Zbl 1382.35275

Summary: The Simpson’s Paradox is the phenomenon that appears in some datasets, where subgroups with a common trend (say, all negative trend) show the reverse trend when they are aggregated (say, positive trend). Even if this issue has an elementary mathematical explanation, it has a deep statistical significance. In this paper, we discuss basic examples in arithmetic, geometry, linear algebra, statistics, game theory, gender bias in university admission and election polls, where we describe the appearance or absence of the Simpson’s Paradox. In the final part, we present our results concerning the occurrence of the Simpson’s Paradox in Quantum Mechanics with focus on the Quantum Harmonic Oscillator and the Nonlinear Schrödinger Equation. We discuss how likely it is to incur in the Simpson’s Paradox and give some concrete numerical examples. We conclude with some final comments and possible future directions.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
62H20 Measures of association (correlation, canonical correlation, etc.)
62H17 Contingency tables
62P35 Applications of statistics to physics
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[1] Bickel, PJ, Hammel, EA, O’Connell, JW: Sex Bias in Graduate Admissions: Data From Berkeley. Science. 187(4175), 398-404 (1975). · doi:10.1126/science.187.4175.398
[2] Berestycki, H, Lions, PL: Nonlinear Scalar field equations. Arch. Rational Mech. Anal. 82(3), 313-345 (1983). · Zbl 0533.35029
[3] Berestycki, H, Lions, PL, Peletier, LA: An ODE approach to the existence of positive solutions for semilinear problems in RN. Indiana Univ. Math. J. 30(1), 141-157 (1981). · Zbl 0522.35036 · doi:10.1512/iumj.1981.30.30012
[4] Berezin, FA, Shubin, MA: The Schrödinger Equation. Translated from the 1983 Russian edition by Yu. Rajabov, DA Leı̆tes and NA Sakharova and revised by Shubin. With contributions by G. L. Litvinov and Leı̆tes. Mathematics and its Applications (Soviet Series), 66. Kluwer Academic Publishers Group, Dordrecht (1991). ISBN:0-7923-1218-X 81-01 (35J10 35P05 46N50 47F05 47N50). · Zbl 0749.35001
[5] Blyth, CR: On Simpson’s Paradox and the Sure-Thing Principle. J. Am. Stat. Assoc. 67(338), 364-366 (1972). · Zbl 0245.62008 · doi:10.1080/01621459.1972.10482387
[6] Cialdi, S, Paris, MGA: The data aggregation problem in quantum hypothesis testing. Eur. Phys. J. D. 69, 7 (2015). doi:10.1140/epjd/e2014-50425-7. · doi:10.1140/epjd/e2014-50425-7
[7] Good, IJ, Mittal, Y: The Amalgamation and Geometry of Two-by-Two Contingency Tables. Ann. Stat. 15(2), 694-711 (1987). · Zbl 0665.62058 · doi:10.1214/aos/1176350369
[8] Gidas, B, Ni, WM, Nirenberg, L: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209-243 (1979). · Zbl 0425.35020 · doi:10.1007/BF01221125
[9] Goltz, HH, Smith, ML: Yule-Simpson’s Paradox in Research. Pract. Assess. Res. Eval. 15(15), 1-9 (2010).
[10] Kwong, MK: Uniqueness of positive solutions of Δu−u+up=0 in Rn. Arch. Rational Mech. Anal. 105(3), 243-366 (1989). · Zbl 0676.35032 · doi:10.1007/BF00251502
[11] Martel, Y, Merle, F: Multi solitary waves for the nonlinear Schrödinger Equations. Ann. I. H. Poincaré. 23(6), 849-864 (2006). · Zbl 1133.35093 · doi:10.1016/j.anihpc.2006.01.001
[12] Norris, F: Can Every Group Be Worse Than Average? Yes (2013). https://economix.blogs.nytimes.com/2013/05/01/can-every-group-be-worse-than-average-yes/. Accessed 1 May 2013. · Zbl 0676.35032
[13] Paris, MGA: Two quantum Simpson’s Paradoxes. J. Phys. A. 45, 132001 (2012). · doi:10.1088/1751-8113/45/13/132001
[14] Pavlides, MG, Perlman, MD: How likely is Simpson’s Paradox?Am. Stat. 63, 226-233 (2009). · doi:10.1198/tast.2009.09007
[15] Pearson, K, Lee, A, Bramley-Moore, L: Genetic (reproductive) selection: Inheritance of fertility in man, and of fecundity in thoroughbred racehorses. Phil. Trans. R. Soc. A. 192, 257-330 (1899). · JFM 30.0223.01 · doi:10.1098/rsta.1899.0006
[16] Selvitella, A: The Simpson’s Paradox in quantum mechanics. J. Math. Phys. 58(3), 37 (2017). 032101. · Zbl 1359.81031 · doi:10.1063/1.4977784
[17] Shi, Y: Quantum Simpson’s Paradox and High Order Bell-Tsileron Inequalities (2012). preprint available at arxiv.org/pdf/1203.2675.
[18] Simpson, EH: The Interpretation of Interaction in Contingency Tables. J. R. Stat. Soc. Ser. B. 13, 238-241 (1951). · Zbl 0045.08802
[19] Yule, GU: Notes on the Theory of Association of Attributes in Statistics. Biometrika. 2(2), 121-134 (1903). · doi:10.1093/biomet/2.2.121
[20] Yule, GU, Kendall, MG: An Introduction to the Theory of Statistics. Griffin, London (1937). · JFM 63.1128.03
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