Table of contents, International Statistical Review, vol. 71, no. 3 (2003)

Introduction authors: David J. Balding , Joseph L. Gastwirth

Interpreting DNA Evidence: A Review authors: L.A. Foreman , C. Champod , I.W. Evett , J.A. Lambert , S. Pope

The Statistical Interpretation of Forensic Glass Evidence authors: James M. Curran

Questioning a Courtroom Proof of the Uniqueness of Fingerprints authors: David H. Kaye

Sentencing Illicit Drug Traffickers: How do the Courts Handle Random Sampling Issues? authors: Alan J. Izenman

Copyright Damages and Statistics authors: Robert L. Basmann , Daniel J. Slottje

Statistical Issues Arising in Disparate Impact Cases and the Use of the Expectancy Curve in Assessing the Validity of Pre-Employment Tests authors: Joseph L. Gastwirth , Weiwen Miao , Gang Zheng

Multiple Imputation: Theory and Method authors: Paul Zhang

Proper and Improper Multiple Imputation authors: Soren Feodor Nielsen

Dimension Reduction with Linear Discriminant Functions Based on an Odds Ratio Parameterization authors: Angelika van der Linde

The word "statistics" derives from the collection and use of data to assist in the administration of a state (nation). The justice system is one of the fundamental pillars of a state, and is central to the political life of most countries. Ideas from probability and statistics have been used to try to model and improve methods of legal decision-making since the earliest days of our discipline in the 16th and 17th centuries. More recently, the debate over the Bayesian statistical paradigm, rooted in probability theory, as a model for legal reasoning has not subsided since being ignited by Finkelstein & Farley (1970); see Balding (1998) for a comment on a UK court judgment dealing directly with this issue.

Despite this interest in legal reasoning, in many respects the use of statistical inference to interpret data in the legal context is a relatively recent development. Statistical information has assisted in the enforcement of the Civil Rights Act, enacted in 1964 in the United States. Indeed, formal hypothesis testing comparing the racial composition of juries with the minority fraction of eligible members of the population was accepted by the US Supreme Court in the 1997 Castenda case. The data from that case, and related employment discrimination cases, are analyzed and discussed in Gastwirth (1988, Chapters 4--6) and Finkelstein & Levin (2000, Chapter 4). The tragedy of excessive birth defects in babies whose mothers used the drug thalidomide, which occurred in Germany and other parts of Europe, demonstrated the need for official surveillance of the effects of new drugs. When this surveillance fails, statisticians are called on to help evaluate the health-related studies in product liability cases. The US Federal Judicial Center has published an excellent reference manual on the current role of scientific evidence in courts, with a major emphasis on the role of statistics, which is freely downloadable over the internet (go to $\mathtt{http://www.fjc.gov/}$, and click on the "evidence" tab on the publications catalogue).

The court of law provides different challenges to statisticians than are demanded in other spheres of applied statistical work. All admissible evidence---not just "scientific" evidence---can play a role in the court's decision-making, for example evidence relating to possible motivations of the parties, and the time sequence of events. The concept of "admissibility" can cause further difficulties. Professional judgments by the statistician are inevitably required, concerning for example the alternative hypotheses that a court may wish to consider and hence require statistical assessment. Difficulties can arise when standard statistical conventions are not acceptable to the court (see the paper of Izenman). The papers in this issue concern both criminal and civil cases. In most legal systems the weight of evidence required to find an accused person guilty of a crime is greater than that needed to prevail in a civil case. The term "beyond a reasonable doubt" is often used for the criminal standard, and "preponderance of the evidence", or "balance of probabilities", for the civil standard. A statistical study that contributes to meeting the civil law standard might be insufficient in a criminal case.

The six papers in this issue deal with a range of legal settings and challenges to statistics. Foreman and co-authors from the UK Forensic Science Service review the interpretation of DNA evidence, an area which requires many judgments, for example about population genetic effects and about the appropriate hypotheses to consider. Complex genetic relationships among possible offenders provides a challenge which Bayesian networks are well suited to overcome, and this is an area of recent research reviewed by these authors.

Although glass evidence has been used for many years, issues concerning the quantification of this evidence have continued to evolve, in part because of the changing technology of glass manufacture and analysis techniques, but also in response to the increasing sophistication of courts in their handling of scientific evidence, reflected in an increasing role for likelihood ratios. Curran, a New Zealand statistician who has developed statistical techniques and software for glass and other forms of forensic evidence, discusses these developments.

For many years it has been assumed that fingerprints are unique to an individual, even an identical twin. However, fingerprint evidence has been used for over a century, and whether or not the basis for this claim meets modern scientific standards has been questioned. Specifically, in the mid-1990s the US Supreme Court changed the standards for admitting scientific evidence. Kaye, a leading legal scholar on the use of statistical and scientific evidence, describes and questions a study commissioned by the FBI to provide scientific support for the uniqueness claim. He highlights many apparent weaknesses of the study, in particular its failure to address the question of the quality of crime-scene fingerprints relative to those taken under ideal conditions. The lack of openness surrounding the study, and the fact that the FBI consulted a private contractor rather than researchers from the academic or voluntary sectors, are also causes for concern.

Izenman's contribution to the issue highlights a clash between scientific and legal cultures. Once a defendant has been convicted of drug possession, the issue arises as to whether the severity of the sentence should depend on the amount of the drugs seized and, if so, how accurately this needs to be measured, given that exhaustive testing is infeasible. If the seizure includes many containers, it would seem natural to statisticians to estimate drug quantity via a random sampling of the containers followed by testing of a sample of the contents of the chosen containers. However, this practice may not suffice to establish the amount of drugs to the criminal evidence standard that, a recent US Supreme Court judgment seems to imply, is required in such cases.

The right of an individual or corporation to profit from their creative products underlies intellectual property law. As world trade increases, the different systems protecting the value of such products adopted by the nations of the world are becoming harmonized. Basmann and Slottje describe a case in which another company without authorization used the insignia of a sports team. The sales of the team's official merchandise were diminished and the authors developed a regression model to estimate the lost profit (damages). Their discussion of the trial testimony illustrates that statisticians testifying in court need to be very careful to accurately describe the underlying statistical concept. The misinterpretation of the $p$-value of a test as a "posterior" probability of the hypothesis being true is all too common in legal opinions.

Gastwirth, Miao and Zheng discuss the types of statistical data and techniques used in employment discrimination cases concerned with the potential use of a non-job related requirement to reduce the employment or promotion prospects of minority groups. While the basic data often is a simple two by two table when several years of data are involved, the issue of when it is proper to combine the yearly tables arises. In order to demonstrate that a job-requirement is appropriate, typically employers show it is correlated to an important aspect of the job, e.g., yearly performance evaluations. Psychologists have used the expectancy concept, that is the probability that an applicant at a certain percentile on a pre-employment exam will be an "above average" employee, to illustrate the productivity gain of requiring prospective employees to pass the exam. The authors introduce a notion of productivity gain, derived from the expectancy curve, and illustrate its use on data from an actual case.

While the papers in this issue illustrate the uses of statistical reasoning in important areas of the law, they are far from exhaustive. In particular, the proper design of quality control sampling for checking compliance with environmental regulations has not received adequate attention. For instance, due to concern that chemicals used in de-icing roadways increased leaching of nitrogen and heavy metals into the country's water supply, in 2002 Finland introduced a system of using 200 monitoring points to check the ground water quality of 100 aquifers. Statistical input will be needed in order that the samples are taken often enough to detect an increase in harmful chemicals to ensure the safety of drinking water. Similarly, a French institute has raised concerns about residues of medicines in surface and groundwater both on the aquatic environment and humans (see $\mathtt{http://www.cemagref.fr}$). An environmental group in Italy found microbiological contamination in 78.5\% of samples from 18 rivers, including the Arno, Po and Tiber (see $\mathtt{http://www.legambiente.it}$).

Environmental protection often interacts with free trade and other economic issues as regulations are subject to cost-benefit analyses. Several Eastern European and Central Asian countries recently agreed to protect various ecosytems. The framework adopted can be found at $\mathtt{http://www.kyiv-}$ $\mathtt{2003.info/main/index.php?mode=5}$. The Mediterranean nations agreed to reduce industrial pollution substantially by 2010. Recommendations and agreements are available at $\mathtt{http://www.}$ $\mathtt{unepmap.gr}$.

Environmental issues seem likely to continue to raise interesting statistical questions, and require the input of statistical experts. Other regulatory issues, especially those related to international trade, will also provide new problems to challenge statisticians, as well as experts from other disciplines.

Balding, D.J. (1998). Court condemns Bayes. Royal Statistical Society News, {\bf 25}(8), 1--2.

Finkelstein, M.O. & Fairley, W.B. (1970). A Bayesian approach to identification and evidence. Harvard Law Review, {\bf 83}, 489--517.

Finkelstein, M.O. & Levin, B. (2000). Statistics for Lawyers. New York: Springer.

Gastwirth J.L. (1988). Statistical Reasoning in Law and Public Policy. San Diego: Academic Press.

The paper provides a review of current issues relating to the use of DNA profiling in forensic science. A short historical section gives the main statistical milestones that occurred during a rapid development of DNA technology and operational uses. Greater detail is then provided for interpretation issues involving STR DNA profiles, including:

methods that take account of population substructure in DNA calculations;

parallel work carried out by the US National Research Council;

the move away from multiple independence testing in favour of experiments that demonstrate the robustness of casework procedures;

the questionable practice of source attribution `with reasonable scientific certainty';

the effect on the interpretation of profiles obtained under increasingly sensitive techniques, the LCN technique in particular;

the use of DNA profiles as an intelligence tool;

the interpretation of DNA mixtures.

Experience of presenting DNA evidence within UK courts is also discussed. The paper then summarises a generic interpretation framework based on the concept of likelihood ratio within a hierarchy of propositions. Finally the use of Bayesian networks to interpret DNA evidence is reviewed.

When examining a sample of glass fragments recovered from a suspect in a forensic case, many questions arise: "Did this man break that window?", "Are these fragments from the crime scene source?", "Do the fragments recovered from the suspect come from more than one source?", "How common is it to find glass on someone unrelated with crime?" etc. Such questions are usually answered with the help of statistical methods. This paper reviews some of the statistical solutions and problems encountered in the interpretation and evaluation of forensic glass evidence.

Forensic scientists or analysts concerned with "individualization" often presume that features such as fingerprint minutia are unique to each individual. In the United States, defendants in criminal cases have been demanding proof of such assumptions. In at least two cases, the government of the United States has successfully relied on an unpublished statistical study prepared specifically for litigation to demonstrate the uniqueness of fingerprints. This article suggests that the study is neither designed nor executed in a way that can show whether an individual's fingerprint impressions are unique.

While many European justice systems distinguish between possession of and trafficking in illicit drugs, sentencing in drug cases in those countries tends not to depend (at least formally) upon the quantity of drugs seized from a defendant, but rather on the circumstances in which the defendant was found with drugs. Courts in the United States, on the other hand, penalize those convicted of drug crimes through an elaborate system of sentencing rules and guidelines. These sentences depend only upon the amount of drugs (possibly adjusted for circumstances) and the defendant's criminal history. Because of the enormous amount of work needed to determine drug type and quantity in each case, sampling the evidence and estimates of drug quantity have been accepted in most courts in the United States. During the last two years, several major developments in U.S. federal sentencing have taken place. It all started in June 2000 with the Supreme Court case Apprendi v. New Jersey, in which Justice O'Connor, in dissent, termed a "watershed in constitutional law". Prior to Apprendi, a judge would sentence a convicted drug trafficker to imprisonment using the preponderance standard of proof on the quantities of drugs seized. The affect of Apprendi is to make more juries decide the quantity of drugs by the reasonable doubt standard. Although Apprendi had nothing to do with illicit drugs, the implication now is that all federal sentencing protocols are under revision. Since Apprendi was decided, huge numbers of cases involving convicted drug traffickers have relied upon its reasoning on appeal, with the result that a ruling from the Supreme Court on further clarification of this issue is expected. This article discusses the impact so far of Apprendi on statistical issues involved in estimating total drug quantity and considers possible future directions under the changing protocols.

This paper discusses the statistical issues that arise in conducting an economic damages analysis in the context of a litigation matter involving copyrights. Calculating damages in copyright cases turns out to be a natural application for econometric modelling methods. Surprisingly, elementary statistical issues can be a source of significant debate between the experts in such matters. In this paper, we present a case study and illustrate how issues such as interpretation of $p$-values and what "rejection of the null hypothesis" really "means" in such matters.

Disparate impact cases concern the potential adverse effect seemingly neutral employment practices, such as passing a pre-employment test or possessing a fixed level of education, have on minority applicants. Their purpose is to eliminate discrimination by subterfuge, i.e., imposing a requirement that eliminates many minority individuals who could do the job but who do not meet the requirement. When a significantly higher fraction of applicants from minority groups fail the requirement compared to majority applicants, the requirement needs to be shown to be job-related. Statistical techniques used at the various stages of a disparate impact claim are described. Properties of the expectancy curve, which describes the utility of a pre-employment test and helps in defining a band of scores defining "equivalently skilled" applicants are discussed.

In this review paper, we discuss the theoretical background of multiple imputation, describe how to build an imputation model and how to create proper imputations. We also present the rules for making repeated imputation inferences. Three widely used multiple imputation methods, the propensity score method, the predictive model method and the Markov chain Monte Carlo (MCMC) method, are presented and discussed.

Multiple imputation has become viewed as a general solution to missing data problems in statistics. However, in order to lead to consistent asymptotically normal estimators, correct variance estimators and valid tests, the imputations must be proper. So far it seems that only Bayesian multiple imputation, i.e.\ using a Bayesian predictive distribution to generate the imputations, or approximately Bayesian multiple imputations has been shown to lead to proper imputations in some settings. In this paper, we shall see that Bayesian multiple imputation does not generally lead to proper multiple imputations. Furthermore, it will be argued that for general statistical use, Bayesian multiple imputation is inefficient even when it is proper.

The association of two random elements with positive joint probability density function is given by an odds ratio function. The covariance is an adequate description only in the case of two jointly Gaussian variables. The impact of the association structure on the set-up and solution of problems of linear discrimination is investigated, and the results are related to standard techniques of multivariate analysis, particularly to canonical correlation analysis, analysis of contingency tables, discriminant analysis and multidimensional scaling.

Annual index ISR vol. 71, no. 3

Acknowledgement of refereeing work