Super Crunchers (31 page)

Possible flaws with Summers's methodology:
The difference in standard deviations may not be as great as 20 percent and the bell curves of intelligence may not follow a normal distribution—especially when you go so far into the tail.

The Naegele rule:
Janelle Durham, “Calculating Due Dates and the Impact of Mistaken Estimates of Gestational Age,” Jan. 2002, http://www.transitionto parenthood.com/ttp/birthed/duedatespaper.htm.

Other methods of predicting due dates:
R. Mittendorf et al., “The Length of Uncomplicated Human Gestation,” 341
N. Engl. J. Med.
461 (1999). For more on probabilities and predicting pregnancy terms, see W. Casscells et al., “Interpretation by Physicians of Clinical Laboratory Results,” 299
N. Engl. J.
Med.
999 (1978); David M. Eddy and Jacquis Casher, “Usefully Interpreting the Triple Screen Assay to Detect Birth Defects,” working paper, Dept. of Statistics and of Biostatistics and Medical Informatics, Aug. 3, 2001; “Probabilistic Reasoning in Clinical Medicine: Problems and Opportunities,” in
Judgment Under
Uncertainty: Heuristics and Biases
(D. Kahneman et al., eds., 1982); Gerd Gigerenzer, “Ecological Intelligence: An Adaptation for Frequencies,” in
The
Evolution of Mind
(D. D. Cummins and C. Allen, eds., 1998), pp. 9–29; David J. Weiss, “You're Not the Only One Who Is Confused About Probability…,” http://instructional1.calstatela.edu/dweiss/Psy302/Confusion.htm.

Predicting contribution to the competitive bottom line:
Alan Schwarz, “Game Theory Posits Measure of Baseball Players' Value,”
N.Y. Times
, Nov. 7, 2004.

Predicting the probability of Down syndrome:
N. J. Wald et al., “Integrated Screening for Down's Syndrome Based on Tests Performed During the First and Second Trimesters,” 341
N. Engl. J. Med.
1935 (1999); Women's Health Information, Down syndrome, http://www.womens-health.co.uk/downs.asp; Miriam Kuppermann et al., “Preferences of Women Facing a Prenatal Diagnostic Choice: Long-Term Outcomes Matter Most,” 19
Prenat. Diag.
711 (1999).

The Bayes' theorem:
Thomas Bayes, “An Essay Towards Solving a Problem in the Doctrine of Chances,” 53
Phil. Trans.
370 (1763). See also Eliezer Yudowsky, “An Intuitive Explanation of Bayesian Reasoning,” 2003, http://yudkowsky.net/ bayes/bayes.htm; Gerd Gigerenzer and Ulrich Hoffrage, “How to Improve Bayesian Reasoning Without Instruction: Frequency Formats,” 102
Psych. Rev.
684(1995).

Quantifying the probability of Down syndrome:
The tendency of some insurance companies to cover amniocentesis if the probability of Down syndrome is greater than the probability of miscarriage has a more sinister foundation. Instead of promoting their patient's welfare, the miscarriage probability rule minimizes the insurance companies' expected “cost-per-case detected.”

Bayes' theorem can be stated:
The posterior probability of cancer given a positive test is equal to the prior probability of cancer multiplied by a likelihood ratio, where the likelihood ratio is the probability of a positive test given that person has cancer divided by the probability of a positive test. Applied to these facts, the likelihood ratio equals 7.5 because the probability of cancer given a positive test is .8 and the probability of a positive test is .107 (107/1000). So Bayes' theorem says that we update the prior probability of 1 percent by multiplying by a likelihood ratio of 7.5 to yield the posterior probability of 7.5 percent.

Suggested readings:
Ray C. Fair,
Predicting Presidential Elections and Other Things
(2002). Steven Levitt and Stephen J. Dubner,
Freakonomics: A Rogue Economist Explores the
Hidden Side of Everything
(2005). John Allen Paulos,
Innumeracy: Mathematical Illiteracy and Its Consequences
(1989). John Donohue,
Beautiful Models, and Other Threats to Life, Law, and Truth
(forthcoming).

Also by Ian Ayres

Responsive Regulation: Transcending the Deregulation Debate
(1992)

Studies in Contract Law
(1997)

Voting with Dollars: A New Paradigm for Campaign Finance
(2002)

Pervasive Prejudice?: Unconventional Evidence of Race and Gender Discrimination
(2002)

Why Not?: How to Use Everyday Ingenuity to Solve Problems Big and Small
(2003)

Straightforward: How to Mobilize Heterosexual Support for Gay Rights
(2005)

Insincere Promises: The Law of Misrepresented Intent
(2005)

Optional Law: The Structure of Legal Entitlements
(2005)

FOOTNOTES

*1 The method is named after Genichi Taguchi, who devised it more than fifty years ago to help manufacturers test multiple aspects of a manufacturing process using just a subset of the tests that had been traditionally required.
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*2
The correct answers
: (1) 39 years. (2) 4,187 miles. (3) 13 countries. (4) 39 books. (5) 2,160 miles. (6) 390,000 pounds. (7) 1756. (8) 645 days. (9) 5,959 miles. (10) 36,198 feet.
Return to text.

*3 “Overfitting” refers to fitting a statistical model that contains too many parameters.
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*4 To use the 2SD rule, you need to estimate two things. First, what do you think is the average height of adult men in the U.S.? If you said 5'9", you're doing well. Now here's the harder question. Ninety-five percent of adult males fall between what two heights? Try to have your range of heights centered on the average male height of 5' 9". Forget about standard deviations and just answer the question based on what you know about the world.
The adult male height is distributed almost normally so that if your answer is correct, there will be 2.5 percent of men who are below your lower height and 2.5 percent of men who are taller than your upper height. Don't worry about being precise. Force yourself to write down a lower and an upper height. The answer is in the endnotes.
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*5 By the way, the standard deviation for the margin of victory was 10.9 points. Can you apply the 2SD rule and say something intuitive about the variability of the actual victory margin relative to the Las Vegas line? It means that 95 percent of the time the actual margin of victory will be within about 21 points of the Las Vegas point spread.
Return to text.

*6 The likelihood ratio measures how likely it is that you would have seen this AFP level if the baby in fact had Down syndrome. Technically, the likelihood ratio is the probability that a mother with a Down syndrome child would have this AFP score divided by the probability that the mother would have this AFP level (regardless of whether her baby has Down syndrome).
Return to text.

*7 Two out of the 893 negative tests will have cancer—so the posterior probability falls to just .2 percent (one-fifth of the prior probability).
Return to text.

SUPER CRUNCHERS

A Bantam Book / September 2007

Published by

Bantam Dell

A Division of Random House, Inc.

New York, New York

All rights reserved

Copyright © 2007 by Ian Ayres

Bantam Books is a registered trademark of Random House, Inc., and the colophon is a trademark of Random House, Inc.

Library of Congress Cataloging-in-Publication Data

Ayres, Ian.

Super crunchers: why thinking-by-numbers is the new way to be smart/Ian Ayres.

p. cm.

Includes bibliographical references.

1. Statistics.                           2. Regression analysis.                           3. Sampling (Statistics) 4. Standard deviations.                           I. Title.

HA29.A86 2007

519.5—dc22

2007013804

www.bantamdell.com

eISBN: 978-0-553-90413-0

v3.0