Jurimetrics
Jurimetrics is the application of quantitative methods, and often especially probability and statistics, to law.[1] In the United States, the journal Jurimetrics is published by the American Bar Association and Arizona State University.[2] The Journal of Empirical Legal Studies is another publication that emphasizes the statistical analysis of law.
The term was coined in 1949 by Lee Loevinger in his article "Jurimetrics: The Next Step Forward".[1][3] Showing the influence of Oliver Wendell Holmes Jr., Loevinger quoted[4] Holmes' celebrated phrase that:
“For the rational study of the law the
The first work on this topic is attributed to Nicolaus I Bernoulli in his doctoral dissertation De Usu Artis Conjectandi in Jure, written in 1709.
Common methods
- Bayesian inference
- Causal inference
- Design of experiments
- Vital for epidemiological studies
- Generalized linear models
- Meta-analysis
- Probability distributions
- Survival analysis
Applications
- Accounting fraud detection[6][7] (Benford's law)
- Airline deregulation[8]
- Analysis of police stops[9] (Negative binomial regression)
- Ban the Box legislation and subsequent impact on job applications[10]
- Calorie labeling mandates and food consumption[11]
- Challenging election results[12] (Hypergeometric distribution)
- Condorcet's jury theorem
- Cost-benefit analysis of renewable portfolio standards for greenhouse gas abatement[13]
- Effect of compulsory schooling on future earnings[14]
- Effect of corporate board size on firm performance[15][16]
- Effect of damage caps on medical malpractice claims[17]
- Effect of a fiduciary standard on financial advice[18][19]
- False conviction rate of inmates sentenced to death[20]
- Legal evidence[21][22][23] (Bayesian network)
- Impact of "pattern-or-practice" investigations on crime[24]
- Legal informatics
- Ogden tables
- Optimal stopping of clinical trials[25][26][27][28]
- Peremptory challenges in jury selection[29]
- Personality predictors of antisocial behavior[30]
- Predictive policing[31]
- Predictors of criminal recidivism[32]
- Prevalence of Caesarean delivery and malpractice claims risk[33]
- Prosecutor's fallacy (People v. Collins)
- Reference class problem
Gender quotas on corporate boards
In 2018, California's legislature passed Senate Bill 826, which requires all publicly held corporations based in the state to have a minimum number of women on their board of directors.[34][35] Boards with five or fewer members must have at least two women, while boards with six or more members must have at least three women.
Using the binomial distribution, we may compute what the probability is of violating the rule laid out in Senate Bill 826 by the number of board members. The probability mass function for the binomial distribution is:
Depending on the number of board members, we are trying compute the cumulative distribution function:
3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|
0.50 | 0.31 | 0.19 | 0.34 | 0.23 | 0.14 | 0.09 | 0.05 | 0.03 | 0.02 |
As Ilya Somin points out,[34] a significant percentage of firms - without any history of sex discrimination - could be in violation of the law.
In more male-dominated industries, such as technology, there could be an even greater imbalance. Suppose that instead of parity in general, the probability that a person who is qualified for board service is female is 40%; this is likely to be a high estimate, given the predominance of males in the technology industry. Then the probability of violating Senate Bill 826 by chance may be recomputed as:
3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|
0.65 | 0.48 | 0.34 | 0.54 | 0.42 | 0.32 | 0.23 | 0.17 | 0.12 | 0.08 |
Bayesian analysis of evidence
Bayes' theorem states that, for events and , the conditional probability of occurring, given that has occurred, is:
Likelihood Ratio | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Prior Odds | 1 | 2 | 3 | 4 | 5 | 10 | 15 | 20 | 25 | 50 |
0.01 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.09 | 0.13 | 0.17 | 0.20 | 0.33 |
0.02 | 0.02 | 0.04 | 0.06 | 0.07 | 0.09 | 0.17 | 0.23 | 0.29 | 0.33 | 0.50 |
0.03 | 0.03 | 0.06 | 0.08 | 0.11 | 0.13 | 0.23 | 0.31 | 0.38 | 0.43 | 0.60 |
0.04 | 0.04 | 0.07 | 0.11 | 0.14 | 0.17 | 0.29 | 0.38 | 0.44 | 0.50 | 0.67 |
0.05 | 0.05 | 0.09 | 0.13 | 0.17 | 0.20 | 0.33 | 0.43 | 0.50 | 0.56 | 0.71 |
0.10 | 0.09 | 0.17 | 0.23 | 0.29 | 0.33 | 0.50 | 0.60 | 0.67 | 0.71 | 0.83 |
0.15 | 0.13 | 0.23 | 0.31 | 0.38 | 0.43 | 0.60 | 0.69 | 0.75 | 0.79 | 0.88 |
0.20 | 0.17 | 0.29 | 0.38 | 0.44 | 0.50 | 0.67 | 0.75 | 0.80 | 0.83 | 0.91 |
0.25 | 0.20 | 0.33 | 0.43 | 0.50 | 0.56 | 0.71 | 0.79 | 0.83 | 0.86 | 0.93 |
0.30 | 0.23 | 0.38 | 0.47 | 0.55 | 0.60 | 0.75 | 0.82 | 0.86 | 0.88 | 0.94 |
If we take to be some criminal behavior and a criminal complaint or accusation, Bayes' theorem allows us to determine the conditional probability of a crime being committed. More sophisticated analyses of evidence can be undertaken with the use of Bayesian networks.
Screening of drug users, mass shooters, and terrorists
In recent years, there has been a growing interest in the use of screening tests to identify drug users on welfare, potential mass shooters,[36] and terrorists.[37] The efficacy of screening tests can be analyzed using Bayes' theorem.
Suppose that there is some binary screening procedure for an action that identifies a person as testing positive or negative for the action. Bayes' theorem tells us that the conditional probability of taking action , given a positive test result, is:
- Sensitivity is equal to the statistical power, where is the type II error rate
- Specificity is equal to , where is the type I error rate
Therefore, the form of Bayes' theorem that is pertinent to us is:
- We screen welfare recipients for cocaine use. The base rate in the population is approximately 1.5%,[38] assuming no differences in use between welfare recipients and the general population.
- We screen men for the possibility of committing mass shootings or terrorist attacks. The base rate is assumed to be 0.01%.
With these base rates and the hypothetical values of sensitivity and specificity, we may calculate the posterior probability that a positive result indicates the individual will actually engage in each of the actions:
Drug Use | Mass Shooting |
---|---|
0.6012 | 0.0098 |
Even with very high sensitivity and specificity, the screening tests only return posterior probabilities of 60.1% and 0.98% respectively for each action. Under more realistic circumstances, it is likely that screening would prove even less useful than under these hypothetical conditions. The problem with any screening procedure for rare events is that it is very likely to be too imprecise, which will identify too many people of being at risk of engaging in some undesirable action.
Jurimetrics and law and economics
The difference between jurimetrics and
See also
- Bayesian inference
- Computational criminology
- Disparate impact#Statistical criticism of disparate impact
- Forensic statistics
- Law and economics
- Quantitative methods in criminology
- Rules of evidence for expert testimony
- Simpson's paradox#UC Berkeley gender bias
- Survival analysis
References
- ^ ISBN 0195142365.
- ^ "Jurimetrics". American Bar Association. Retrieved 2015-02-06.
- ^ Loevinger, Lee (1949). "Jurimetrics--The Next Step Forward". Minnesota Law Review. 33: 455.
- ^ Loevinger, L. "Jurimetrics: Science and prediction in the field of law". Minnesota Law Review, vol. 46, HeinOnline, 1961.
- ^ Holmes, The Path of the Law, 10 Harvard Law Review (1897) 457.
- ^ Nigrini, Mark J. (1999-04-30). "I've Got Your Number: How a mathematical phenomenon can help CPAs uncover fraud and other irregularities". Journal of Accountancy.
- ^ Durtschi, Cindy; Hillison, William; Pacini, Carl (2004). "The Effective Use of Benford's Law to Assist in Detecting Fraud in Accounting Data". Journal of Forensic Accounting. 5: 17–34.
- S2CID 153646501.
- S2CID 8505752.
- S2CID 18615965.
- PMID 24760208.
- JSTOR 1121228.
- ^ Greenstone, Michael; McDowell, Richard; Nath, Ishan (2019-04-21). "Do Renewable Portfolio Standards Deliver?" (PDF). Energy Policy Institute at the University of Chicago, Working Paper No. 2019-62.
- S2CID 153718259.
- ISSN 0304-405X.
- S2CID 3868815.
- .
- SSRN 3101426.
- ^ Van Doren, Peter (2018-06-25). "The Fiduciary Rule and Conflict of Interest". Cato at Liberty. Cato Institute. Retrieved 2019-12-14.
- PMID 24778209.
- PMID 23110576.
- S2CID 12449479.
- ISBN 9780387849270.
- ^ Devi, Tanaya; Fryer, Roland G. Jr. (2020). "Policing the Police: The Impact of "Pattern-or-Practice" Investigations on Crime" (PDF). NBER Working Paper Series. No. 27324.
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has extra text (help) - PMID 16592839.
- PMID 26123873.
- PMID 27660563.
- ^ "Adaptive Designs for Clinical Trials of Drugs and Biologics: Guidance for Industry". U.S. Department of Health and Human Services/Food and Drug Administration. 2019.
- S2CID 120315268.
- ISSN 0047-2352.
- ^ Perry, Walter L.; McInnis, Brian; Price, Carter C.; Smith, Susan; Hollywood, John S. (2013). "Predictive Policing: The Role of Crime Forecasting in Law Enforcement Operations". RAND Corporation. Retrieved 2019-08-16.
- S2CID 144566819.
- PMID 8418343.
- ^ a b Somin, Ilya (2018-10-04). "California's Unconstitutional Gender Quotas for Corporate Boards". Reason.com. The Volokh Conspiracy. Retrieved 2019-08-13.
- ^ Stewart, Emily (2018-10-03). "California just passed a law requiring more women on boards. It matters, even if it fails". Vox. Retrieved 2019-08-13.
- ^ Gillespie, Nick (2018-02-14). "Yes, This Is a Good Time To Talk About Gun Violence and How To Reduce It". Reason.com. Retrieved 2019-08-17.
- ^ "Terrorist Screening Center". Federal Bureau of Investigation. Retrieved 2019-08-17.
- ^ "What is the scope of cocaine use in the United States?". National Institute on Drug Abuse. Retrieved 2019-08-17.
- ^ Buscaglia, Edgardo (2001). "The Economic Factors Behind Legal Integration: A Jurimetric Analysis of the Latin American Experience" (PDF). German Papers in Law and Economics. 1: 1.
- .
Further reading
- Angrist, Joshua D.; Pischke, Jörn-Steffen (2009). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton, NJ: Princeton University Press. ISBN 9780691120355.
- Borenstein, Michael; Hedges, Larry V.; Higgins, Julian P.T.; Rothstein, Hannah R. (2009). Introduction to Meta-Analysis. Hoboken, NJ: John Wiley & Sons. ISBN 9780470057247.
- Finkelstein, Michael O.; Levin, Bruce (2015). Statistics for Lawyers. Statistics for Social and Behavioral Sciences (3rd ed.). New York, NY: Springer. ISBN 9781441959843.
- Hosmer, David W.; Lemeshow, Stanley; May, Susanne (2008). Applied Survival Analysis: Regression Modeling of Time-to-Event Data. Wiley-Interscience (2nd ed.). Hoboken, NJ: John Wiley & Sons. ISBN 9780471754992.
- McCullagh, Peter; Nelder, John A. (1989). Generalized Linear Models. Monographs on Statistics and Applied Probability (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC. ISBN 9780412317606.
External links
- Bernoulli (1709). The use of the Art of conjecturing in Law.
- Kadane, J.B. (2006). Misuse of Bayesian Statistics in Court, CHANCE, 19, 2, 38-40.
- Stern & Kadane (2014). Compensating for the loss of a chance. Department of Statistics, Carnegie Mellon University.
- Jurimetrics, The Journal of Law, Science, and Technology
- Journal of Empirical Legal Studies