# Jurimetrics

Short description: Application of quantitative metrics to law

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 blackletter man may be the man of the present, but the man of the future is the man of statistics and the master of economics.”[5]

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.

## Applications

### 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:$\displaystyle{ \mathbb{P}(X=k) = {n\choose{k}}p^{k}(1-p)^{n-k} }$where $\displaystyle{ p }$ is the probability of getting $\displaystyle{ k }$ successes in $\displaystyle{ n }$ trials, and $\displaystyle{ {n\choose{k}} }$ is the binomial coefficient. For this computation, $\displaystyle{ p }$ is the probability that a person qualified for board service is female, $\displaystyle{ k }$ is the number of female board members, and $\displaystyle{ n }$ is the number of board seats. We will assume that $\displaystyle{ p = 0.5 }$.

Depending on the number of board members, we are trying compute the cumulative distribution function:$\displaystyle{ \begin{cases} \mathbb{P}(X\leq 1) = (1-p)^{n} + np(1-p)^{n-1}, \quad &n\leq 5 \\ \mathbb{P}(X \leq 2) = \mathbb{P}(X\leq 1) + {n(n-1)\over{2}}p^{2}(1-p)^{n-2}, \quad &n \gt 5 \end{cases} }$With these formulas, we are able to compute the probability of violating Senate Bill 826 by chance:

Probability of Violation by Chance (# of board members)
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:

Probability of Violation by Chance (# of board members)
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

Main page: Evidence under Bayes theorem

Bayes' theorem states that, for events $\displaystyle{ A }$ and $\displaystyle{ B }$, the conditional probability of $\displaystyle{ A }$ occurring, given that $\displaystyle{ B }$ has occurred, is:$\displaystyle{ \mathbb{P}(A|B) = {\mathbb{P}(B | A )\mathbb{P}(A)\over{\mathbb{P}(B)}} }$Using the law of total probability, we may expand the denominator as:$\displaystyle{ \mathbb{P}(B) = \mathbb{P}(B | A )\mathbb{P}(A) + \mathbb{P}(B | \sim A )[1 - \mathbb{P}(A)] }$Then Bayes' theorem may be rewritten as:\displaystyle{ \begin{aligned} \mathbb{P}(A|B) &= {\mathbb{P}(B | A )\mathbb{P}(A)\over{ \mathbb{P}(B | A )\mathbb{P}(A) + \mathbb{P}(B | \sim A )[1 - \mathbb{P}(A)] }} \\ &= {1\over{1 + {1 - \mathbb{P}(A)\over{\mathbb{P}(A)}} {\mathbb{P}(B | \sim A )\over{\mathbb{P}(B | A )}} }} \end{aligned} }This may be simplified further by defining the prior odds of event $\displaystyle{ A }$ occurring $\displaystyle{ \eta }$ and the likelihood ratio $\displaystyle{ \mathcal{L} }$ as:$\displaystyle{ \eta = {\mathbb{P}(A)\over{1- \mathbb{P}(A)}}, \quad \mathcal{L} = {\mathbb{P}(B|A)\over{\mathbb{P}(B|\sim A)}} }$Then the compact form of Bayes' theorem is:$\displaystyle{ \mathbb{P}(A|B) = {1\over{ 1 + (\eta \mathcal{L})^{-1} }} }$Different values of the posterior probability, based on the prior odds and likelihood ratio, are computed in the following table:

Likelihood Ratio Prior Odds 1 2 3 4 5 10 15 20 0.01 0.02 0.03 0.04 0.05 0.09 0.13 0.17 0.20 0.33 0.02 0.04 0.06 0.07 0.09 0.17 0.23 0.29 0.33 0.50 0.03 0.06 0.08 0.11 0.13 0.23 0.31 0.38 0.43 0.60 0.04 0.07 0.11 0.14 0.17 0.29 0.38 0.44 0.50 0.67 0.05 0.09 0.13 0.17 0.20 0.33 0.43 0.50 0.56 0.71 0.09 0.17 0.23 0.29 0.33 0.50 0.60 0.67 0.71 0.83 0.13 0.23 0.31 0.38 0.43 0.60 0.69 0.75 0.79 0.88 0.17 0.29 0.38 0.44 0.50 0.67 0.75 0.80 0.83 0.91 0.20 0.33 0.43 0.50 0.56 0.71 0.79 0.83 0.86 0.93 0.23 0.38 0.47 0.55 0.60 0.75 0.82 0.86 0.88 0.94

If we take $\displaystyle{ A }$ to be some criminal behavior and $\displaystyle{ B }$ 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 $\displaystyle{ V }$ that identifies a person as testing positive $\displaystyle{ + }$ or negative $\displaystyle{ - }$ for the action. Bayes' theorem tells us that the conditional probability of taking action $\displaystyle{ V }$, given a positive test result, is:$\displaystyle{ \mathbb{P}(V | +) = {\mathbb{P}(+|V)\mathbb{P}(V)\over{\mathbb{P}(+|V)\mathbb{P}(V) + \mathbb{P}(+|\sim V)\left[1-\mathbb{P}(V)\right]}} }$For any screening test, we must be cognizant of its sensitivity and specificity. The screening test has sensitivity $\displaystyle{ \mathbb{P}(+|V) }$ and specificity $\displaystyle{ \mathbb{P}(-|\sim V) = 1 - \mathbb{P}(+|\sim V) }$. The sensitivity and specificity can be analyzed using concepts from the standard theory of statistical hypothesis testing:

• Sensitivity is equal to the statistical power $\displaystyle{ 1-\beta }$, where $\displaystyle{ \beta }$ is the type II error rate
• Specificity is equal to $\displaystyle{ 1-\alpha }$, where $\displaystyle{ \alpha }$ is the type I error rate

Therefore, the form of Bayes' theorem that is pertinent to us is:$\displaystyle{ \mathbb{P}(V | +) = {(1-\beta)\mathbb{P}(V)\over{(1-\beta)\mathbb{P}(V) + \alpha \left[1-\mathbb{P}(V)\right]}} }$Suppose that we have developed a test with sensitivity and specificity of 99%, which is likely to be higher than most real-world tests. We can examine several scenarios to see how well this hypothetical test works:

• 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:

Posterior Probabilities
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 law and economics is that jurimetrics investigates legal questions from a probabilistic/statistical point of view, while law and economics addresses legal questions using standard microeconomic analysis. A synthesis of these fields is possible through the use of econometrics (statistics for economic analysis) and other quantitative methods to answer relevant legal matters. As an example, the Columbia University scholar Edgardo Buscaglia published several peer-reviewed articles by using a joint jurimetrics and law and economics approach.[39][40]

## References

1. Garner, Bryan A. (2001). "jurimetrics". A Dictionary of Modern Legal Usage. Oxford University Press. p. 488. ISBN 0195142365.
2. "Jurimetrics". American Bar Association.
3. Loevinger, Lee (1949). "Jurimetrics--The Next Step Forward". Minnesota Law Review 33: 455.
4. Loevinger, L. "Jurimetrics: Science and prediction in the field of law". Minnesota Law Review, vol. 46, HeinOnline, 1961.
5. Holmes, The Path of the Law, 10 Harvard Law Review (1897) 457.
6. 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.
7. Moore, Thomas Gale (1986). "U. S. Airline Deregulation: Its Effects on Passengers, Capital, and Labor". The Journal of Law & Economics 29 (1): 1–28. doi:10.1086/467107. ISSN 0022-2186.
8. Gelman, Andrew; Fagan, Jeffrey; Kiss, Alex (2007). "An Analysis of the New York City Police Department's "Stop-and-Frisk" Policy in the Context of Claims of Racial Bias". Journal of the American Statistical Association 102 (479): 813–823. doi:10.1198/016214506000001040. ISSN 0162-1459.
9. Agan, Amanda; Starr, Sonja (2018-02-01). "Ban the Box, Criminal Records, and Racial Discrimination: A Field Experiment" (in en). The Quarterly Journal of Economics 133 (1): 191–235. doi:10.1093/qje/qjx028. ISSN 0033-5533.
10. Kiszko, Kamila M.; Martinez, Olivia D.; Abrams, Courtney; Elbel, Brian (2014). "The influence of calorie labeling on food orders and consumption: A review of the literature". Journal of Community Health 39 (6): 1248–1269. doi:10.1007/s10900-014-9876-0. ISSN 0094-5145. PMID 24760208.
11. Finkelstein, Michael O.; Robbins, Herbert E. (1973). "Mathematical Probability in Election Challenges". Columbia Law Review 73 (2): 241. doi:10.2307/1121228.
12. Greenstone, Michael; McDowell, Richard; Nath, Ishan (2019-04-21). "Do Renewable Portfolio Standards Deliver?". Energy Policy Institute at the University of Chicago, Working Paper No. 2019-62.
13. Angrist, Joshua D.; Krueger, Alan B. (1991). "Does Compulsory School Attendance Affect Schooling and Earnings?". The Quarterly Journal of Economics 106 (4): 979–1014. doi:10.2307/2937954. ISSN 0033-5533.
14. Eisenberg, Theodore; Sundgren, Stefan; Wells, Martin T. (1998). "Larger board size and decreasing firm value in small firms". Journal of Financial Economics 48 (1): 35–54. doi:10.1016/S0304-405X(98)00003-8. ISSN 0304-405X.
15. Guest, Paul M. (2009). "The impact of board size on firm performance: evidence from the UK". The European Journal of Finance 15 (4): 385–404. doi:10.1080/13518470802466121. ISSN 1351-847X.
16. Donohue III, John J.; Ho, Daniel E. (2007). "The Impact of Damage Caps on Malpractice Claims: Randomization Inference with Difference-in-Differences". Journal of Empirical Legal Studies 4 (1): 69–102. doi:10.1111/j.1740-1461.2007.00082.x.
17. Linnainmaa, Juhani T.; Melzer, Brian; Previtero, Alessandro (2018). "The Misguided Beliefs of Financial Advisors". SSRN.
18. Van Doren, Peter (2018-06-25). "The Fiduciary Rule and Conflict of Interest" (in en). Cato Institute.
19. Kennedy, Edward H.; Hu, Chen; O’Brien, Barbara; Gross, Samuel R. (2014-05-20). "Rate of false conviction of criminal defendants who are sentenced to death" (in en). Proceedings of the National Academy of Sciences 111 (20): 7230–7235. doi:10.1073/pnas.1306417111. ISSN 0027-8424. PMID 24778209. Bibcode2014PNAS..111.7230G.
20. Fenton, Norman; Neil, Martin; Lagnado, David A. (2013). "A General Structure for Legal Arguments About Evidence Using Bayesian Networks" (in en). Cognitive Science 37 (1): 61–102. doi:10.1111/cogs.12004. ISSN 1551-6709. PMID 23110576.
21. Vlek, Charlotte S.; Prakken, Henry; Renooij, Silja; Verheij, Bart (2014-12-01). "Building Bayesian networks for legal evidence with narratives: a case study evaluation" (in en). Artificial Intelligence and Law 22 (4): 375–421. doi:10.1007/s10506-014-9161-7. ISSN 1572-8382.
22. Kwan, Michael; Chow, Kam-Pui; Law, Frank; Lai, Pierre (2008). "Reasoning About Evidence Using Bayesian Networks". in Ray, Indrajit; Shenoi, Sujeet (in en). Advances in Digital Forensics IV. IFIP — The International Federation for Information Processing. 285. Springer US. pp. 275–289. doi:10.1007/978-0-387-84927-0_22. ISBN 9780387849270.
23. Devi, Tanaya; Fryer, Roland G. Jr. (2020). "Policing the Police: The Impact of "Pattern-or-Practice" Investigations on Crime". NBER Working Paper Series No. 27324.
24. Lai, T. L.; Levin, Bruce; Robbins, Herbert; Siegmund, David (1980-06-01). "Sequential medical trials" (in en). Proceedings of the National Academy of Sciences 77 (6): 3135–3138. doi:10.1073/pnas.77.6.3135. ISSN 0027-8424. PMID 16592839. Bibcode1980PNAS...77.3135L.
25. Levin, Bruce (2015). "The futility study—progress over the last decade". Contemporary Clinical Trials 45 (Pt A): 69–75. doi:10.1016/j.cct.2015.06.013. ISSN 1551-7144. PMID 26123873.
26. Deichmann, Richard E.; Krousel-Wood, Marie; Breault, Joseph (2016). "Bioethics in Practice: Considerations for Stopping a Clinical Trial Early". The Ochsner Journal 16 (3): 197–198. ISSN 1524-5012. PMID 27660563.
27. "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.
28. Finkelstein, Michael O.; Levin, Bruce (1997). "Clear Choices and Guesswork in Peremptory Challenges in Federal Criminal Trials". Journal of the Royal Statistical Society. Series A (Statistics in Society) 160 (2): 275–288. doi:10.1111/1467-985X.00062. ISSN 0964-1998.
29. Jones, Shayne E.; Miller, Joshua D.; Lynam, Donald R. (2011-07-01). "Personality, antisocial behavior, and aggression: A meta-analytic review". Journal of Criminal Justice 39 (4): 329–337. doi:10.1016/j.jcrimjus.2011.03.004. ISSN 0047-2352.
30. Perry, Walter L.; McInnis, Brian; Price, Carter C.; Smith, Susan; Hollywood, John S. (2013) (in en). Predictive Policing: The Role of Crime Forecasting in Law Enforcement Operations. Retrieved 2019-08-16.
31. Spivak, Andrew L.; Damphousse, Kelly R. (2006). "Who Returns to Prison? A Survival Analysis of Recidivism among Adult Offenders Released in Oklahoma, 1985 – 2004" (in en-US). Justice Research and Policy 8 (2): 57–88. doi:10.3818/jrp.8.2.2006.57. ISSN 1525-1071.
32. Localio, A. Russell; Lawthers, Ann G.; Bengtson, Joan M.; Hebert, Liesi E.; Weaver, Susan L.; Brennan, Troyen A.; Landis, J. Richard (1993). "Relationship Between Malpractice Claims and Cesarean Delivery". JAMA 269 (3): 366–373. doi:10.1001/jama.1993.03500030064034. PMID 8418343.
33. Somin, Ilya (2018-10-04). "California's Unconstitutional Gender Quotas for Corporate Boards" (in en-US). The Volokh Conspiracy.
34. Gillespie, Nick (2018-02-14). "Yes, This Is a Good Time To Talk About Gun Violence and How To Reduce It" (in en-US).
35. Buscaglia, Edgardo (2001). "The Economic Factors Behind Legal Integration: A Jurimetric Analysis of the Latin American Experience". German Papers in Law and Economics 1: 1.
36. Buscaglia, Edgardo (2001). "A Governance-Based Jurimetric Analysis of Judicial Corruption: Subjective versus Objective Indicators". International Review of Law and Economics 21: 231. doi:10.1016/S0144-8188(01)00058-8.