Jurimetrics is the application of quantitative methods, and often especially
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
and
statistics
Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, to
law
Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...
.
In the United States, the journal ''
Jurimetrics'' is published by the
American Bar Association
The American Bar Association (ABA) is a voluntary bar association of lawyers and law students, which is not specific to any jurisdiction in the United States. Founded in 1878, the ABA's most important stated activities are the setting of aca ...
and
Arizona State University
Arizona State University (Arizona State or ASU) is a public research university in the Phoenix metropolitan area. Founded in 1885 by the 13th Arizona Territorial Legislature, ASU is one of the largest public universities by enrollment in the ...
. 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
Lee Loevinger (April 24, 1913 – April 26, 2004) was an American jurist and lawyer.
Born in Saint Paul, Minnesota, Loevinger received his bachelor's degree from University of Minnesota in 1933 and his law degree from University of Minnesota ...
in his article "Jurimetrics: The Next Step Forward".
Showing the influence of
Oliver Wendell Holmes, Jr., Loevinger quoted Holmes' celebrated phrase that:
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
Bayesian inference is a method of statistical inference in which Bayes' theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and ...
*
Causal inference
Causal inference is the process of determining the independent, actual effect of a particular phenomenon that is a component of a larger system. The main difference between causal inference and inference of association is that causal inference ana ...
**
Instrumental variables
In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered ...
*
Design of experiments
The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
** Vital for
epidemiological studies
Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and determinants of health and disease conditions in a defined population.
It is a cornerstone of public health, and shapes policy decisions and evid ...
*
Generalized linear model
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and by ...
s
**
Ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the ...
,
logistic regression
In statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression an ...
,
Poisson regression
*
Meta-analysis
A meta-analysis is a statistical analysis that combines the results of multiple scientific studies. Meta-analyses can be performed when there are multiple scientific studies addressing the same question, with each individual study reporting m ...
*
Probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
s
**
Binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no ques ...
,
hypergeometric distribution,
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
*
Survival analysis
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysi ...
**
Kaplan-Meier estimator,
proportional hazards model
Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes, before some event occurs, to one or more covariates that may be associated with that quantity of time. In a proportional haza ...
,
Weibull distribution
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Re ...
Applications
*
Accounting fraud detection (
Benford's law)
*
Airline deregulation
Airline deregulation is the process of removing government-imposed entry and price restrictions on airlines affecting, in particular, the carriers permitted to serve specific routes. In the United States, the term usually applies to the Airline Der ...
*Analysis of
police stops (
Negative binomial regression
In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable ''Y'' has a Poisson distribution, and assumes the log ...
)
*
Ban the Box legislation and subsequent impact on job applications
**
Statistical discrimination (economics)
Statistical discrimination is a theorized behavior in which racial or gender inequality results when economic agents (consumers, workers, employers, etc.) have imperfect information about individuals they interact with. According to this theory, ...
*
Calorie labeling mandates and food consumption
**
Risk compensation
Risk compensation is a theory which suggests that people typically adjust their behavior in response to perceived levels of risk, becoming more careful where they sense greater risk and less careful if they feel more protected. Although usually ...
*Challenging election results (
Hypergeometric distribution)
*
Condorcet's jury theorem
Condorcet's jury theorem is a political science theorem about the relative probability of a given group of individuals arriving at a correct decision. The theorem was first expressed by the Marquis de Condorcet in his 1785 work ''Essay on the App ...
*
Cost-benefit analysis of
renewable portfolio standards for greenhouse gas abatement
*Effect of
compulsory schooling
Compulsory education refers to a period of education that is required of all people and is imposed by the government. This education may take place at a registered school or at other places.
Compulsory school attendance or compulsory schooling ...
on future earnings
*Effect of
corporate board size on firm performance
*Effect of
damage caps on
medical malpractice
Medical malpractice is a legal cause of action that occurs when a medical or health care professional, through a negligent act or omission, deviates from standards in their profession, thereby causing injury or death to a patient. The neglige ...
claims
*Effect of a
fiduciary
A fiduciary is a person who holds a legal or ethical relationship of trust with one or more other parties (person or group of persons). Typically, a fiduciary prudently takes care of money or other assets for another person. One party, for exa ...
standard on
financial advice
A financial adviser or financial advisor is a professional who provides financial services to clients based on their financial situation. In many countries, financial advisors must complete specific training and be registered with a regulatory ...
*
False conviction rate of inmates
sentenced to death
Capital punishment, also known as the death penalty, is the state-sanctioned practice of deliberately killing a person as a punishment for an actual or supposed crime, usually following an authorized, rule-governed process to conclude that t ...
*
Legal evidence
The law of evidence, also known as the rules of evidence, encompasses the rules and legal principles that govern the proof of facts in a legal proceeding. These rules determine what evidence must or must not be considered by the trier of f ...
(
Bayesian network
A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Ba ...
)
*Impact of
"pattern-or-practice" investigations on crime
*
Legal informatics
*
Ogden tables
The Ogden tables are a set of statistical tables and other information for use in court cases in the UK. Their purpose is to make it easier to calculate future losses in personal injury and fatal accident cases.
The tables take into account li ...
*
Optimal stopping of
clinical trial
Clinical trials are prospective biomedical or behavioral research studies on human participants designed to answer specific questions about biomedical or behavioral interventions, including new treatments (such as novel vaccines, drugs, diet ...
s
*
Peremptory challenges in jury selection
*
Personality
Personality is the characteristic sets of behaviors, cognitions, and emotional patterns that are formed from biological and environmental factors, and which change over time. While there is no generally agreed-upon definition of personality, ...
predictors of
antisocial behavior
Antisocial may refer to:
Sociology, psychiatry and psychology
* Anti-social behaviour
*Antisocial personality disorder
*Psychopathy
*Conduct disorder
Law
*Anti-social Behaviour Act 2003
*Anti-Social Behaviour Order
*Crime and Disorder Act 1998
* ...
*
Predictive policing
Predictive policing is the usage of mathematics, predictive analytics, and other analytical techniques in law enforcement to identify potential criminal activity. A report published by the RAND Corporation identified four general categories pred ...
*Predictors of
criminal recidivism
*Prevalence of
Caesarean delivery
Caesarean section, also known as C-section or caesarean delivery, is the surgical procedure by which one or more babies are delivered through an incision in the mother's abdomen, often performed because vaginal delivery would put the baby or mo ...
and malpractice claims risk
*
Prosecutor's fallacy (
People v. Collins)
*
Reference class problem
In statistics, the reference class problem is the problem of deciding what class to use when calculating the probability applicable to a particular case.
For example, to estimate the probability of an aircraft crashing, we could refer to the fr ...
Gender quotas on corporate boards
In 2018, California's
legislature
A legislature is an assembly with the authority to make laws for a political entity such as a country or city. They are often contrasted with the executive and judicial powers of government.
Laws enacted by legislatures are usually known ...
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
A board of directors (commonly referred simply as the board) is an executive committee that jointly supervises the activities of an organization, which can be either a for-profit or a nonprofit organization such as a business, nonprofit orga ...
.
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
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no ques ...
, 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
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
for the binomial distribution is:
where
is the probability of getting
successes in
trials, and
is the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. For this computation,
is the probability that a person qualified for board service is female,
is the number of female board members, and
is the number of board seats. We will assume that
.
Depending on the number of board members, we are trying compute the
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Eve ...
:
With these formulas, we are able to compute the probability of violating Senate Bill 826 by chance:
As
Ilya Somin
Ilya Somin (born 1973) is a law professor at George Mason University, an adjunct scholar at the Cato Institute, a blogger for the Volokh Conspiracy, and a former co-editor of the '' Supreme Court Economic Review'' (2006–2013). His research focu ...
points out,
a significant percentage of firms - without any history of
sex discrimination
Sexism is prejudice or discrimination based on one's sex or gender. Sexism can affect anyone, but it primarily affects women and girls.There is a clear and broad consensus among academic scholars in multiple fields that sexism refers primari ...
- could be in violation of the law.
In more male-dominated industries, such as
technology
Technology is the application of knowledge to reach practical goals in a specifiable and reproducible way. The word ''technology'' may also mean the product of such an endeavor. The use of technology is widely prevalent in medicine, scien ...
, 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:
Bayesian analysis of evidence
Bayes' theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
states that, for events
and
, the
conditional probability
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occu ...
of
occurring, given that
has occurred, is:
Using the
law of total probability, we may expand the denominator as: