Notes
1. Obesity epidemic in the United States; see https://journals.lww.com/nutritionto-
dayonline/Abstract/2003/03000/The_Supersizing_of_America____ Portion_Size_
and_the.4.aspx.
“Metabolic syndrome is not a disease in itself. Instead, it's a group of risk factors—high blood pressure, high blood sugar, unhealthy cholesterol levels, and abdominal fat”; see https://www.webmd.com/heart/metabolic-syndrome/metabolic-syndrome-what-is-.
2. Doughnut consumption in United States: https://www.statista.com/statistics/ 283198/us-households-consumption-of-donuts--doughnuts-trend/
3. Table of American adult (>20 years old) male body weights: http://www.cdc.gov/ nchs/data/series/sr_11/sr11_252.pdf
4. 'Theoretical physics may extend beyond the bounds of empiricism. See David Deutsch's discussion in Chapter 10.
5. K. S. Button, J. P. loannidis, C. Mokrysz, B. A. Nosek, J. Flint, E. S. Robinson, and M. R. Munafo, “Power Failure: Why Small Sample Size Undermines the Reliability of Neuroscience,” Nature Reviews Neuroscience 14:365-376, 2013; R. Nuzzo, “Scientific Method: Statistical Errors,” Nature 506:150-152, 2014; P. Bacchetti, “Small Sample Size Is Not the Real Problem,” Nature Reviews Neuroscience 14:485, 2013; J. C. Ashton, “Experimental Power Comes from Powerful Theories: The Real Problem in Null Hypothesis Testing,” Nature Reviews Neuroscience 14:585, 2013; and C. Hoppe, “A Test Is Not a Test,” Nature Reviews Neuroscience 14:877, 2013.
6. Gerd Gigerenzer's theories relating to statistics, heuristics and biases, risk and uncertainty, and related matters were the major resource for the history of statistics discussed in this chapter. For convenience, I will reference the collections instead of the original articles, and, where themes in the collections overlap, I will cite one that seems particularly apt without intending to imply that that source is the only, or original, reference.
See G. Gigerenzer, Heuristics: The Foundations of Adaptive Behavior (New York: Oxford University Press; 2011); G. Gigerenzer Simply Rational: Decision Making in the Real World (New York: Oxford University Press; 2015).7. Report in the Baltimore Sun (http://www.baltimoresun.com/business/bs-bz-slots- payouts-20150826-story.html) about the deliberations of state regulators that determine the percentage payout from state-sanctioned slot machines.
8. Central Limit Theorem (CLT); see https://en.wikipedia.org/wiki/Central_limit_the- orem. “In probability theory, the central limit theorem establishes that, in most situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution... even if the original variables themselves are not normally distributed.” The CLT allows us to refer the results of statistical tests to the normal distribution when the underlying distribution is unknown or known to be non-normal.
9. Karl Popper, The Logic of Scientific Discovery (New York: Routledge; 2002).
10. If it seems counterintuitive that a hypothesis entails an infinite number ofpredictions, consider this: your hypothesis predicts that protein Y is a receptor for drug A. You test 10 concentrations of A and find that their binding to Y follows a sigmoidal curve that is consistent with your hypothesis. However, your hypothesis predicts that an infinitely graded series of concentrations of A would fall along the line of the curve. Each concentration is a distinct prediction, and there are an infinite number of points on the line.
11. Moreover, Popper was Austrian by birth and lived in Europe. He spoke and wrote in German until, because he was a Jew, he fled Europe to escape the Nazis and emigrated to New Zealand in 1937. While in New Zealand he resolved never to write in German again, although he continued to follow closely the work ofcolleagues who did. Indeed, his masterwork, Logic der Forschung, was written in German in the 1930s and not published in English until 1959.
Fisher and the other founders of statistics were publishing in English, and it is possible that Popper was simply unaware of their work. Biographical information from the definitive text: Malachi H. Hacohen, Karl Popper: The Formative Years 1902-1945 (New York: Cambridge University Press; 2000). The philosopher of science Deborah Mayo (see Note 12) reports corresponding with Popper and learning that he “regretted not studying statistics,” and she takes him to task for the omission.12. Deborah G. Mayo, Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars (New York: Cambridge University Press; 2018).
13. Gerd Gigerenzer, Zeno Swijtink, Theodore Porter, Lorraine Daston, John Beatty, and Lorenz Kruger, The Empire of Chance (Cambridge: Cambridge University Press; 1989) p. 107. The authors surveyed 30 statistics textbooks and found that none discussed the issues or the authors. In a check of six standard statistics texts, I corroborated their finding in five of them (I can't be sure that our lists didn't overlap); only one book that I reviewed alluded to the divided origins of statistics at all.
14. J. D. Perezgonzalez, “Fisher, Neyman-Pearson or NHST? A Tutorial for Teaching Data Testing,” Frontiers in Psychology 6:223, 2015. http://journal.frontiersin.org/ar- ticle/10.3389/fpsyg.2015.00223/full.
15. J. D. Perezgonzalez, “P-Values as Percentiles. Commentary on: ‘Null Hypothesis Significance Tests. A Mix-Up of Two Different Theories: The Basis for Widespread Confusion and Numerous Misinterpretations,'” Frontiers in Psychology 6:341, 2015. http://journal.frontiersin.org/article/10.3389/fpsyg.2015.00341/full.
16. C. Lamdin, “Significance Tests as Sorcery: Science Is Empirical, Significance Tests Are Not.” http://tap.sagepub.com/content/22/U67; J. Cohen, “The Earth Is Round (p <.05),” American Psychologist, 49:997-1003, 1994.
17. Robert Coe, “It's the Effect Size, Stupid: What Effect Size Is and Why It Is Important. “ Paper presented at the British Educational Association annual conference, Exeter, September 12-14, 2002.
18. See Note 5. We'll re-visit criticisms of statistical power in Chapter 7.
19. Daniel J. Benjamin, James O. Berger, Magnus Johannesson, Brian A. Nosek, E.-J. Wagenmakers, Richard Berk, et al., “Redefine Statistical Significance,” Nature Human Behavior 2:6-10, 2018.
20. Daniel Lakens, Federico G. Adolfi, Casper J. Albers, Farid Anvari, Matthew A. J. Apps, Shlomo E. Argamon, et al., “Justify Your Alpha.” Nature Human Behavior 2:168-171,2018.
21. Lisa Randall, Higgs Discovery: The Power of Empty Space (New York: HarperCollins; 2 012).
22. Not all physicists were elated by the confirmation of the Higgs' appearance; see http://www.newyorker.com/news/news-desk/i-think-we-have-it-is-the-higgs- boson-a-disappointment; http://www.theatlantic.com/technology/archive/2012/07/ why-the-higgs-boson-discovery-is-disappointing-according-to-the-smartest-man- in-the-world/259468/. Because the Higgs was a definite prediction of the Standard Model, its confirmation was expected; yet for that reason it didn't provide new insights into the shortcomings of that model. Physicists were hoping to observe an unpredicted particle that could lead them to exciting new physics.
23. According to newspaper reports, retired mailman Doug Hughes was on his way to deliver letters to members of Congress to protest the existence of corruption in the US government. His gyrocopter landed safely on the West Lawn of the Capitol, and he was eventually sentenced to 4 months in prison (http://www.usatoday.com/story/ news/2016/04/21/gyrocopter-pilot-sentenced-4-months-prison/83271738/; https:// www.washingtonpost.com/news/local/wp/2015/04/15/a-gyrocopter-just-landed- on-the-capitol-lawn/).
24. Jacob Cohen, “The Earth Is Round.”
25. Gerd Gigerenzer, “Statistical Rituals: The Replication Delusion and How We Got There,” Advances in Methods and Practices in Psychological Science 1:198-218, 2018.
26. Paul Meehl, “Theory Testing in Psychology and Physics: A Methodological Paradox,” Philosophy of Science 34:103-115, 1967.
Available at: http://www.jstor.org/stable/ 186099.27. Niels G. Waller, “The Fallacy ofthe Null Hypothesis in Soft Psychology: Commentary,” Applied and Preventive Psychology 11:83-86, 2004.
28. Gerd Gigerenzer, Rationality for Mortals: How People Cope with Uncertainty (New York, Oxford University Press; 2008), p. 161.
0 Q 1 I ZA /"'olc'lll olo Aan VO P1ODCA' A2 I {-1/2 whopA
29. iocaicui.ai.ei. emeanvariance: [{(ii1-1)(s 1) +(ii2-1)(s ^) j/(ii1+ii2-2)jj,w erei^
and n2 are the numbers in each group, and sd1 and sd2 are the standard deviations for the measurements.
30. Coe, “It's the Effect Size, Stupid.”
31. Online calculator for Cohen's d: http://rpsychologist.com/d3/cohend/.
32. http://www.newyorker.com/magazine/2004/04/05/the-height-gap; http://www. newyorker.com/magazine/2004/04/05/the-short-americanhttp://atlanticreview.org/ archives/661-Europeans-are-taller-than-Americans.html; http://www.theatlantic. com/health/archive/2014/05/how-we-get-tall/361881/.
The study of human heights is important because national height is taken as a proxy for all-around well-being; systematic variations in average heights among nations, or within one nation over, say, hundreds of years, offers evidence of significant height-influencing societal forces at work. Although no one interpretation of this phenomenon has emerged, simple averaging of heights across a heterogenous mix of distinct groups can't explain it, because the national trends hold true within even narrowly defined groups within a country.
33. http://www.cdc.gov/nchs/data/series/sr_11/sr11_252.pdf.
34. https://ourworldindata.org/human-height/; http://www.disabled-world.com/ artman/publish/height-chart.shtml. I could only find the mean of the measured heights of a group of Swedish women aged 20-29 years; I made up the group size and standard error of the mean for this example.
35. To calculate the effect size, divide the difference in heights (65.5"- 64.2" = 1.3”) by the mean variance, as in Note 28.
Effect size for women's heights is 1.3/2.51 = 0.52.36. Coe, “It's the Effect Size, Stupid.”
37. For any given interval that has been calculated, the population mean is definitely either in it or not: this is a certainty, not a probability. You can describe your own degree of confidence as to whether the mean is in or out in terms of probability. Statisticians insist on this subtle distinction; common usage is not always as strict. Examples of misinterpretations of confidence intervals can be found at https://en.wikipedia.org/ wiki/ Confidence_interval. As we'll see in Chapter 6, the Bayesian approach of determining probability distributions, rather than single values, for parameters such as the mean can directly tell us whether or not the confidence interval captures the true mean. See https://scholar.google.com/citations?view_op=view_citation&hl=en&use r=ln2kZXIXtMcC&citation_for_view=ln2kZXIXtMcC:u5HHmVD_uO8C.
38. Coe, “It's the Effect Size,” gives, in other notation, the calculation for the 95% confidence interval around the effect size (ES) as
ES ± 1.96(sdES), where
sdES
n1 + n2)/ (n1n2) + (ES)2 / 2(n1 + n2
1/2
39. Gigerenzer, “Statistical Rituals.”