Showing posts with label Bayes. Show all posts
Showing posts with label Bayes. Show all posts

Sunday, September 22, 2024

Who Published Bayes' Theorem?

For the legal profession, "American Law Reports or ALR is a longstanding, highly trusted series of in-depth articles, called annotations, on specific legal issues."\1/ But the report on "Use of Bayes' Theorem in Criminal and Civil Cases"\2/ may be less worthy of trust than the average annotation. Historians and statisticians, at least, would be surprised at its opening sentences:

Bayes' Theorem, named for the English clergyman and scientist who published it in 1763, is a scientific principle of the likelihood ratio used to calculate conditional probabilities leading to a variety of statistical "Bayesian" methodologies. It speculates on the probability that a particular fact is true or that a particular event will occur, given our knowledge of one or more related facts or events.

There is no doubt (asymptotically speaking) that in 1763, Thomas Bayes did not publish the theorem that bears his name. He was dead. A close friend, the Reverend Richard Price, read the paper to the Royal Society of London after retrieving it from Bayes's papers and adding to it.\3/ This expanded and modified version appeared in the Society's Philosophical Transactions.

The ALR's rendering of 18th Century history won't cause any problems for lawyers employing or confronting Bayesian analyses in court. Posthumous or otherwise, discovered first by Bayes or by someone else,\4/ the theorem is what it is. But what will lawyers make of the idea that the theorem is "a scientific principle of the likelihood ratio" that "speculates" about conditional probability? Is the theorem mere speculation? Is it only about a likelihood ratio? No, and no.

The usual understanding is that a likelihood ratio is the data-based, objective part of the theorem and that speculation becomes an issue if and when the prior probability is not objective as well. As Judea Pearl and Dana MacKenzie explain,

Bayesian statistics gives us an objective way of combining the observed evidence with our prior knowledge (or subjective belief) to obtain a revised belief ... . Still, what frequentists could not abide was that Bayesians were allowing opinion, in the form of subjective probabilities, to intrude into the pristine kingdom of statistics. Mainstream statisticians were won over only grudgingly, when Bayesian analysis proved a superior tool for a variety of applications ... .\5/

Fitting expert prior probabilities and likelihoods into the legal process poses some special problems for mainstream statisticians. That accommodation is the subject of continuing, if often repetitive, multidisciplinary dialog that does not fit neatly and quickly into a blog.

Notes

  1. University of South Carolina School of Law, Legal Research, Analysis & Writing, https://guides.law.sc.edu/LRAWSpring/LRAW/alr.
  2. Deborah F. Buckman, Annot., Use of Bayes' Theorem in Criminal and Civil Cases, 47 A.L.R.7th Art. 4 (2019).
  3. Stephen M. Stigler, Richard Price, The First Bayesian, 33 Statistical Science 117 (2018), https://doi.org/10.1214/17-STS635; Martyn Hooper, Richard Price, Bayes’ Theorem, and God, Significance 36-39 (Feb. 2013), https://www.york.ac.uk/depts/maths/histstat/price.pdf.
  4. Stephen M. Stigler, Who Discovered Bayes’s Theorem?, 37 Am. Stat. 290-296 (1983).
  5. Judea Pearl & Dana MacKenzie, The Book of Why: The New Science of Cause and Effect 90 (2018).

Monday, September 18, 2023

Use with Caution: NIJ's Training Course in Population Genetics and Statistics for Forensic Analysts

The National Institute of Justice (NIJ) "is the research, development and evaluation agency of the U.S. Department of Justice . . . dedicated to improving knowledge and understanding of crime and justice issues through science." It offers a series of webpages and video recordings (a "training course") on Population Genetics and Statistics for Forensic Analysts. The course should be approached with caution. I have not worked through all the pages and videos, but here are a few things that rang alarm bells:


NIJ's Training Comment

Many statisticians have employed what is known as Bayesian probability ... which is based on probability as a measure of one's degree of belief. This type of probability is conditional in that the outcome is based on knowing information about other circumstances and is derived from Bayes Theorem. Bayes' rule applies to both objective and subjective probabilities. Both types of probability include conditional probabilities. The "type of probability" is not derived from Bayes' Theorem.

Conditional probability, by definition, is the probability P of an event A given that an event B has occurred. ... Take the example of a die with six sides. If one was to throw the die, the probability of it landing on any one side would be 1/6. This probability, however, assumes that the die is not weighted or rigged in any way, and that all of the sides contain a different number. If this were not true, then the probability would be conditional and dependent on these other factors. The "other factors" are nothing more than part of the description of the experiment whose outcomes are the events that are observed. They are not conditioning events in a sample space.

The following equation can be used to determine the probability of the evidence given that a presumed individual is the contributor rather than a random individual in the population: LR = P(E/H1) / P(E/H0) ... . In the case of a single source sample, the hypothesis for the numerator (the suspect is the source of the DNA) is a given, and thus reduces to 1. This reduces to: LR = 1/ P(E/H0) which is simply 1/P, where P is the genotype frequency. The hypothesis for the numerator of a likelihood ratio is always "a given"--that is, it goes on the right-hand-side of the expression for a conditional probability. So is the hypothesis in the denominator. Neither probability "reduces to 1" for that reason. Only if the "evidence" is the true genotype in both the recovered sample and the sample from the defendant can it be said that P(E|H1) = 1. In other words, to say that the probability of a reported match is 1 if the defendant is the source treats the probability of laboratory error as zero. That may be acceptable as a simplifying assumption, but the assumption should be made visible in a training course.

Although likelihood ratios can be used for determining the significance of single source crime stains, they are more commonly used in mixture interpretation. ... The use of any formula for mixture interpretation should only be applied to cases in which the analyst can reasonably assume "that all contributors to the mixed profile are unrelated to each other, and that allelic dropout has no practical impact." This limitation does not apply to modern probabilistic genotyping software!

Tuesday, May 24, 2022

The New York Court of Appeals Returns to Probabilistic Genotyping Software (Part II—General Acceptance)

The New York Court of Appeals returned to the contentious issue of “probabilistic genotyping software” (PGS) in People v. Wakefield, 2022 N.Y. Slip Op. 02771, 2022 WL 1217463 (N.Y. Apr. 26, 2022). As previously discussed, in People v. Williams, 147 N.E.3d 1131 (N.Y. 2020), a slim majority of the court held that the output of a computer program should not have been admitted without a full evidentiary hearing on its general acceptance within the scientific community. The majority opinion described a confluence of considerations:
  1. The program had only been tested in the laboratory that developed it (“an invitation to bias,” id. at 1141);
  2. The only evidentiary hearing ever conducted on the program had only shown “internal validation” and formal approval by a subcommittee of a state forensic science commission that was a “narrow class of reviewers, some of whom were employed by the very agency that developed the technology,” id. at 1142;
  3. Given “the ‘black box’ nature of that program,” the developer's “secretive approach ... was inconsistent with quality assurance standards” id.; and
  4. Submissions for hearings in other cases “suggested that the accuracy calculations of that program may be flawed,” id.

But which of these four factors were dispositive? Was it the combination of all four, or something in between, that rendered the evidence inadmissible? If the developer were to change its “secretive approach” so as to allow defense experts to study the program’s source code, would that, plus the “internal validation,” be enough to establish general scientific acceptance? Would it be sufficient for the state to refute the suggestions of flawed “accuracy calculations of the program” through testimony from its experts? Just what did the court mean when it summarized its analysis with the statement that “[i]n short, the [PGS] should be supported by those with no professional interest in its acceptance. Frye demands an objective, unbiased review”?

The opinion did not reveal how the majority might answer these questions. Of course, in holding that a hearing was necessary, the Williams majority implied that some information outside of the normal scientific literature could fill the gap created by the absence of replicated developmental validation studies from external (“objective, unbiased”) researchers. But what might that information be?

The court’s encounter with PGS last month did not answer this open question, for the court in Wakefield found that there were replicated studies from the developer of a more sophisticated computer program and other researchers. In addition, it pointed to other evaluations or uses of the program. The totality of the evidence, it reasoned, was stronger than the developer-only record in Williams and demonstrated the requisite general acceptance. But the opinion provoked one member of the court to complain of a "jarring turnabout" from "the same view unsuccessfully advocated by a minority in Williams two years ago."

This posting describes the case, the DNA evidence, and aspects of the discussions of general acceptance that struck me as interesting or puzzling.

The Crime, the Samples, and Some Misunderstood Probabilities of Exclusion

In 2010, John Wakefield strangled the occupant of an apartment with a guitar amplifier cord and made off with various items. The New York State Police laboratory analyzed samples from four areas: the front part of the collar of the victim's shirt, the rear part of the collar; the victim's forearm; and the amplifier cord. The laboratory concluded that the DNA on the collar was “consistent with at least two donors, one of which was the victim, and defendant could not be excluded as the other contributor”; that the DNA from the forearm “was consistent with DNA from the victim, as the major contributor, mixed with at least two additional donors” and the DNA on the cord was “a mixture of at least two donors, from which the victim could not be excluded as a possible contributor.” 2022 WL 1217463, at *1.

At this point, the court’s description of the State Police laboratory’s work becomes had to follow. The court wrote that:

[T]he analyst did not call any alleles based on peaks on the electropherogram below [the pre-established stochastic] threshold. As a result, there was insufficient data to allow the Lab to calculate probabilities for the unknown contributors to the DNA mixtures found on the amplifier cord and the front of the shirt collar.

No alleles at all? It takes only one allele to compute a probability of exclusion, although with such a limited profile, the exclusion probability might be close to zero, meaning that the data are uninformative. In any event, for the other two samples, “[t]he Lab was able to call ... 4 ... STR loci” that enabled “the analyst, using the combined probability of inclusion method, [to opine that] the probability an unrelated individual contributed DNA to the outside rear shirt collar was 1 in 1,088" and “that the probability an unrelated individual contributed DNA ... was 1 in 422" for “the profile obtained from the victim's forearm.”

Or so the court said. As explained in Box 1, these numbers are not “the probability an unrelated individual contributed DNA.” They are estimates of the probability that a randomly selected, unrelated individual could not be excluded as a possible source. Given a large number of unrelated individuals in the region, there easily could be more than a hundred people with STR profiles compatible with the mixtures.

BOX 1: TRANSPOSITION

The probability of inclusion is not the probability that an included individual is the contributor. It is the probability of not excluding an individual as a possible contributor. That probability is not necessarily equal to the probability that an included individual actually contributed to the sample from which he or she could not be excluded. If C stands for contributor and I for included, the probability of inclusion for any randomly selected individual can be written P(I given C). The source probability for the individual is different. It is P(C given I). Equating the two is known as the transposition fallacy (or the “prosecutor’s fallacy,” though it could be called the “judges” fallacy as well).

We do not need any symbols to see that the two conditional probabilities are not necessarily equal. The population of Schnectady county, where the crime occurred, was about 155,000 in 2010. Let’s round down to 150,000. That ought to remove all of Wakefield’s relatives. Excluding all but 1 in 1,088 individuals would leave 138 people as possible perpetrators. Of course, some would be far more plausible suspects than others, but based on the DNA evidence alone, how can the court claim that “the probability an unrelated individual contributed DNA to the outside rear shirt collar was 1 in 1,088”? That probability cannot be determined from the DNA evidence alone. It can be computed only if we are willing to assign a “prior probability” of being the murderer to each of the unrelated individuals in Schnectady (or anywhere else).

Suppose we assume that, ab initio, everyone in the county has an equal probability of being a source of the DNA on the collar. At that point, Wakefield’s probability is quite small. It is 1/150,000. Since the DNA testing would have excluded all but some 138 people, and because Wakefield is one of them, the probability attached to him is larger. Now the probability is 1/138. But that still leaves the vast bulk of the probability with the 137 unrelated individuals. Instead of transposing, we should say that “the probability an unrelated individual contributed DNA to the outside rear shirt collar was 137 out of in 138” rather than the court’s “1 in 1,088.” Of course, our assumption of equal probabilities for every unrelated individual is unrealistic, but that does not impeach the broader point that the mathematics does not make the probability of an unrelated individual the number that the court supplied.

Cybergenetics to the Rescue

To secure a better and more complete analysis, “the electronic data from the DNA testing of the four samples at issue was then sent to Cybergenetics [for] calculating a likelihood ratio—using all of the information generated on the electropherogram, including peaks that fall below a laboratory's stochastic threshold.” Cybergenetics is a private company whose “flagship TrueAllele® technology resolves complex forensic evidence, providing accurate and objective DNA match statistics.” TrueAllele's calculations of the likelihood ratios, using the hypothesis that the four samples contained DNA from an unrelated black individual as the alternative to the hypothesis that Wakefield’s DNA was present were 5.88 billion for the cord, 170 quintillion for the outside rear shirt collar, 303 billion for the outside front shirt collar, and 56.1 million for the forearm.

Wakefield moved to exclude these findings, The Schnectady County Supreme Court held a pretrial evidentiary hearing “over numerous days.” People v. Wakefield, 47 Misc.3d 850, 851, 9 N.Y.S.3d 540 (2015). (New York calls its trial courts supreme courts.) Finding “that Cybergenetics TrueAllele Casework is not novel but instead is ‘generally accepted’ under the Frye standard,” \1/ Justice Michael V. Coccoma (New York calls its trial judges justices) denied the motion. 47 Misc.3d at 859. A jury convicted Wakefield of first degree murder and robbery. The Appellate Division affirmed, and seven years after the trial, so did the Court of Appeals (New York calls its most supreme court the Court of Appeals).

Changes in New York’s Highest Court

Back in Williams, the Court of Appeals judges had split 4-3 on whether New York City's home-grown PGS had attained general acceptance. The three judges led by Chief Judge Janet M. DiFiore * objected to the majority’s negative comments about PGS and propounded a narrower rationale for requiring a Frye hearing. But even if one could have confidently applied the majority reasoning in Williams to the scientific status of TrueAllele in Wakefield, the exercise in legal logic might have been futile. In the two short years since Williams, the composition of the court had changed. One concurring judge died, and the majority-opinion bloc lost half its members, including the opinion’s author, to retirements. The reconstituted court gave Chief Judge DiFiore the opportunity to write a more laudatory opinion for a new and larger majority.

Only one judge stood apart from this new majority. Having been in the majority in Williams, Judge Jenny Rivera now found herself in the Chief Judge’s situation in Williams, composing a dissenting opinion with respect to the reasoning on general acceptance but concurring in the result. Drawing on Williams, Judge Rivera maintained that “the court erred in admitting the TrueAllele results but the error ... was harmless” in view of the other evidence of guilt.

The Court’s Understanding of TrueAllele

The opinions are vague about the inner workings of TrueAllele. The majority opinion suggests that what is distinctive about PGS is that it cranks out a likelihood ratio. \2/ But “likelihood ratio,” for present purposes, simply denotes the probability of data given one hypothesis divided by the probability of the same data given a (simple) alternative hypothesis. It has nothing to do with the probabilistic part of TrueAllelle. Indeed, TrueAllele only computes a likelihood ratio after the probability analysis is completed. It does this by dividing (i) the final posterior odds that favor one source hypothesis as compared to another by (ii) the initial prior odds. This division gives a “Bayes' factor” that states how much the data have changed the odds.

Let me try saying this another way. In effect, TrueAllele starts with prior odds based solely on the frequencies of various DNA alleles (and hence genotypes) in some population, performs successive approximations to converge on a better estimate of the odds, and divides the adjusted odds by the prior odds to yield what Cybergenetics calls “the match statistic.” If all goes well, this quotient (call it a likelihood ratio, a Bayes' factor, a match statistic, or whatever you want) reveals how powerful the DNA evidence is (which is not necessarily the same as the odds that any hypothesis is true). At least, that is what I think goes on. The court contents itself with warm and fuzzy statements such as “a probability model to assess the values of a genotype objectively” “based on mathematical computations from all the data in the electropherograms.” and “separates the genotypes using the mathematical probability principle of the Markov chain Monte Carlo (MCMC) search to calculate the probability for what the different genotypes could be.” (This last clause may not be so warm and fuzzy; it begins to unpack what I simplistically called successive approximations.)

The Timing for General Acceptance

Wakefield is a backwards-looking case. The main question before the Court of Appeals was whether, in 2015, TrueAllele reasonably could have been deemed to have been generally accepted in the scientific community. That is what New York law requires. \3/ The Chief Judge’s analysis of the general acceptance of TrueAllele starts with the observation that “[t]he well-known Frye test applied to the admissibility of novel scientific evidence (Frye v. United States, 293 F. 1013 [D.C. Cir.1923]) is 'whether the accepted techniques, when properly performed, generate results accepted as reliable within the scientific community generally' (People v. Wesley, 83 N.Y.2d 417, 422, 611 N.Y.S.2d 97, 633 N.E.2d 451 [1994]).”

Wesley is an interesting case to cite here. One would not know from the citation or the analysis in Wakefield that in Wesley there was no opinion for a majority of the seven judges on the court. There was one opinion for three judges and another opinion for two judges concurring only in the result. The remaining two judges did not participate. The concurring opinion was written by the late Chief Judge Judith S. Kaye, the longest-serving chief judge in New York history.

Chief Judge Kaye’s concurrence is memorable for its skepticism about finding general acceptance on the basis of studies from the developer of a method. Current Chief Judge Janet DiFiore briefly summarized that discussion (as did the majority in Williams). A more complete exposition is in Box 2. Chief Judge DiFiore then suggests that the Wesley concurrence was satisfied because “[n]otwithstanding these concerns, Chief Judge Kaye ultimately agreed that, at the time the appeal was decided, "RFLP-based forensic analysis [was] generally accepted as reliable" and those testing procedures were accepted as the standard methodology used in the scientific community until the advent of the PCR STR method used today.”

This presentation places an odd spin on the Wesley concurrence. The sole basis for the concurrence was that “it can fairly be said that use of DNA evidence was harmless beyond a reasonable doubt” because the DNA evidence “added nothing to the People's case.” 83 N.Y.2d at 444–45. The observations that five years after the hearing in Wesley, it had become clear that “in principle” RFLP-VNTR testing was “fundamentally sound” and was generally accepted were clearly dicta. Chief Judge Kaye was not suggesting that because a method had become generally accepted later, its earlier admission was vindicated. The dicta on later general acceptance was intended to inform trial courts that while they were at liberty to admit RFLP-VNTR evidence without pretrial hearings on general acceptance, they still needed to probe “the adequacy of the methods used to acquire and analyze samples ... case by case.” Id. at 445.

In contrast to Wesley, which emphasized the state of the science “at the time of the Frye hearing in 1988,” 83 N.Y.2d at 425 (plurality opinion), and whether “in 1988, ... there was consensus,” id. at 439 (concurring opinion), Chief Judge DiFiore’s opinion is less precise on when general acceptance came into existence:

BOX 2. PEOPLE v. WESLEY
83 N.Y.2d 417, 439–41, 611 N.Y.S.2d 97, 633 N.E.2d 451 (N.Y. 1994) (Chief Judge Kaye, concurring) (citations and footnote omitted)

The inquiry into forensic analysis of DNA in this case also demonstrates the "pitfalls of self-validation by a small group" Before bringing novel evidence to court, proponents of new techniques must subject their methods to the scrutiny of fellow scientists, unimpeded by commercial concerns.

A Frye court should be particularly cautious when — as here — "the supporting research is conducted by someone with a professional or commercial interest in the technique" DNA forensic analysis was developed in commercial laboratories under conditions of secrecy, preventing emergence of independent views. No independent academic or governmental laboratories were publishing studies concerning forensic use of DNA profiling. The Federal Bureau of Investigation did not consider use of the technique until 1989. Because no other facilities were apparently conducting research in the field, the commercial laboratory's unchallenged endorsement of the reliability of its own techniques was accepted by the hearing court as sufficient to represent acceptance of the technique by scientists generally. The sole forensic witness at the hearing in this case was Dr. Michael Baird, Director of Forensics at Lifecodes laboratory, where the samples were to be analyzed. While he assured the court of the reliability of the forensic application of DNA, virtually the sole publications on forensic use of DNA were his own or those of Dr. Jeffreys, the founder of Cellmark, one of Lifecodes' competitors. Nor had the forensic procedure been subjected to thorough peer review. ***

The opinions of two scientists, both with commercial interests in the work under consideration and both the primary developers and proponents of the technique, were insufficient to establish "general acceptance" in the scientific field. The People's effort to gain a consensus by having their own witnesses "peer review" the relevant studies in time to return to court with supporting testimony was hardly an appropriate substitute for the thoughtful exchange of ideas in an unbiased scientific community envisioned by Frye. Our colleagues' characterization of a dearth of publications on this novel technique as the equivalent of unanimous endorsement of its reliability ignores the plain reality that this technique was not yet being discussed and tested in the scientific community.

"Although the continuous probabilistic approach was not used in the majority of forensic crime laboratories at the time of the hearing, the methodology has been generally accepted in the relevant scientific community based on the empirical evidence of its validity, as demonstrated by multiple validation studies, including collaborative studies, peer-reviewed publications in scientific journals and its use in other jurisdictions. The empirical studies demonstrated TrueAllele's reliability, by deriving reproducible and accurate results from the interpretation of known DNA samples."

Presumably, and notwithstanding citations to materials appearing after 2015, \4/ she meant to write that the methodology had been generally accepted in 2015 because the indications listed were present then. (Whether the decisive time for general acceptance should be that of the trial rather than the appeal is not completely obvious. If a technique becomes generally accepted later, why should the defendant be entitled to a new trial in which the evidence that should have been excluded has become admissible anyway? The defendant's interest in the time-of-trial rule is the interest in not being convicted with the help of scientifically sound evidence (as per the general-acceptance standard based on the best current knowledge). A counter-argument is that a large pool of potential defense experts to question the application of the general accepted method in the particular case did not exist at the time of trial because the evidence was too novel.)

Quantifying the Accuracy of PGS

Turning to the question of the state of acceptance as of 2015, the majority opinion maintains that

]T]he methodology has been generally accepted in the relevant scientific community based on the empirical evidence of its validity, as demonstrated by multiple validation studies, including collaborative studies, peer-reviewed publications in scientific journals and its use in other jurisdictions. The empirical studies demonstrated TrueAllele's reliability, by deriving reproducible and accurate results from the interpretation of known DNA samples.

Both the fact that the software was written to implement uncontroversial mathematical ideas and the published empirical evidence are important. If the software were designed to implement a mathematically invalid procedure, the game would be over before it began. But techniques such as Bayes’ rule and sampling methods for getting a representative picture of the posterior distribution only work when they are developed appropriately for a particular application. Acknowledging that these tools have been used to solve problems in many fields of science is a bit like saying that the mathematics of probability theory is undisputed. The validity of the mathematical ideas are a necessary but hardly a sufficient condition for a finding that software intended to apply the ideas functions as intended. Using a particular mathematical formula or method to describe or predict real-world phenomena is an endeavor that is subject to and in need of empirical confirmation. Because PGS models the variability in the empirical data that emerge from chemical reactions and electronic detectors, “empirical evidence ... of its accuracy” is indispensable to establishing its accuracy.

Unfortunately, Wakefield is short on details from the “multiple validation studies” and “peer-reviewed publications.” What do the studies and publications reveal about the accuracy of output such as “5.88 billion times more probable” and “170 quintillion times more probable”? The Supreme Court opinion is devoid of any quantitative statement of how well the deconvoluted individual profiles and their Bayes’ factors reported by TrueAllele correspond to the presence or absence of those profiles in samples constructed with or otherwise known to contain DNA from given individuals. So is the Appellate Division opinion. So too with the Court of Appeals’ opinions. The court is persuaded that “[t]he empirical studies demonstrated TrueAllele's reliability, by deriving reproducible and accurate results from the interpretation of known DNA samples.” But how well did True Allele perform in the “many published and peer reviewed” validity studies?

A separate posting summarizes parts of the six studies circa 2015 that are both published and peer reviewed. The numbers in these studies suggest that within certain ranges (with regard to the quantity of DNA, the number of contributors, and the fractions from the multiple contributors), TrueAllele’s likelihood ratios discriminate quite well between samples paired with true contributors and the same samples paired with noncontributors. For example, in one experiment, LR was never greater than 1 for 600,000 simulations of false contributors to 10 two-person mixtures containing 1 nanogram of DNA—no observed false positives! Conversely, LR was never less than 1 for every true contributor to the same ten mixtures—no observed false negatives in 20 comparisons. Moreover, the program’s output behaves qualitatively as it should, generally producing smaller likelihood ratios for electrophoretic data that are more complex or more bedeviled by stochastic effects on peak heights and locations.

Such results suggest that TrueAllele’s LRs are in the ballpark. Yet, it is hard to gauge the size of the ballpark. Is a computed LR of 5.88 billion truly a probability ratio of 5.88 billion? Could the ratio be a lot less or a lot more? The validity studies do not give quantitative answers to these questions about “accuracy.” \5/

The Developer’s Involvement

On appeal, Wakefield had to convince the court that the unchallenged studies and other indicia of general acceptance were too weak to permit a finding of general acceptance. To do so, he pointed to “the dearth of independent validation as a result of Dr. Perlin's involvement in the large majority of studies produced at the hearing.” (Indeed, Dr. Perlin is the lead author of every one of the five published validity studies and a co-author of a sixth published study that also helps show validity.)

The majority acknowledged “legitimate concern” but decided that it was overcome “by the import of the empirical evidence of reliability demonstrated here and the acceptance of the methodology by the relevant scientific community.” However, the discussion of “the import of the empirical evidence” seems somewhat garbled.

1

First, the court notes that “the FBI Quality Assurance Standards requires ‘a developmental validation for a particular technology’ be published.” That the FBI might be satisfied with a single publication from the developer of a method does not speak to what the broader scientific community regards as essential to the validation. Along with the QAS, the court cites "NIST, DNA Mixture Interpretation: A NIST Scientific Foundation Review, at 64 (June 2021 Draft report)." The page merely reports that the NIST staff were able to examine “[p]ublicly available data on DNA mixture interpretation performance ... from five sources [including] published PGS studies” and that “conducting mixture studies may be viewed as a necessity to meet published guidelines or QAS requirements ... .” That scientists and other NIST personnel who choose to review a technology will read the scientific reports of the developers of the technology does not tell us much about defendant’s claim that Cybergenetics’ involvement in the published validation studies gravely diminishes “the import of the empirical evidence.”

2

Second, the Court of Appeals maintained that “the interest of the developer was addressed at the Frye hearing in this case.” As the court described the hearing, the response to this concern was that “[a]lthough Dr. Perlin was involved in and coauthored most of the validation studies, his interest in TrueAllele was disclosed as required by the journals who published the studies and the empirical evidence of the reliability of TrueAllele was not disputed.”

These responses seem rather flaccid. Some of the articles contain conflict-of-interest statements; most do not. \7/ But the presence or absence of obvious disclaimers does not come to grips with the complaint. Defendant’s argument is not that there are hidden funding sources or financial relationships. It is that interests in the outcomes of the studies somehow may affect the results. The claim is not that validation data were fabricated or that the data analysis was faulty. As with the movement for replication and “open science,” it is a response to more subtle threats.

3

Third, the opinion asserts that “the scientific method” is “entirely consistent with” proof of validity coming from the inventors, discoverers, or commercializers (citing President's Council of Advisors on Sci. and Tech., Exec. Office of the President, Forensic Science in Criminal Courts: Ensuring Scientific Validity of Feature-Comparison Methods, at 46 (2016)). Again, however, the argument is not that only disinterested parties do and should participate in scientific dialog. It is that "[w]hile it is completely appropriate for method developers to evaluate their own methods, establishing scientific validity also requires scientific evaluation by other scientific groups that did not develop the method.” Id. at 80 (https://obamawhitehouse.archives.gov/sites/default/files/microsites/ostp/PCAST/pcast_forensic_science_report_final.pdf [https://perma.cc/R76Y-7VU]).

4

That precept leads to the court’s last and most telling response to the “legitimate concern” over “the dearth of independent validation.” The Chief Judge finally wrote that “there were [not only] developer [but also] independent validation studies and laboratory internal validation studies, many published and peer-reviewed.”

But is this a fair characterization of the scientific literature as of 2015? From what I can tell, no more than five or six studies appear in peer-reviewed journals, and none are completely “independent validation studies.” The NIST report cited in Wakefield lists but a single “internal validation” study, from Virginia in 2013, apparently released in response to a Freedom of Information Act request, Although the NIST reviewers limited themselves to laboratory studies or data posted to the Internet, they concluded that “[c]urrently, there is not enough publicly available data to enable an external and independent assessment of the degree of reliability of DNA mixture interpretation practices, including the use of probabilistic genotyping software (PGS) systems.” 

Of course, this “Key Takeaway #4.3” is merely part of a draft report and is not a judgment as to what conclusions on validity should be reached on the basis of the published studies and the internal ones. Nevertheless, the court overlooks this prominent “takeaway” (and others). Instead, the Chief Judge asserts that “[t]he technology was approved for use by NIST”—even though NIST is not a regulatory agency that approves technologies—and that “NIST's use of the TrueAllele system for its standard reference materials likewise demonstrates confidence within the relevant community that the system generates accurate results.”

~~~

This is not to say that the scientific literature was patently insufficient to support the court’s assessment of the general scientific acceptance of TrueAllele for interpreting the DNA data in the case. But it does raise the question of whether the court’s assertions about the large number of “independent validity studies” and internal ones that have been “published and peer-reviewed” are exaggerated.

Source Code and General Acceptance

The defense also contended that the state’s testimony and exhibits from “the Frye hearing [were] insufficient because, absent disclosure of the TrueAllele source code for examination by the scientific community, its ‘proprietary black box technology’ cannot be generally accepted as a matter of law.” This argument bears two possible interpretations. On the one hand, it could be a claim that scientists demand open-source programs—those with every line of code deposited somewhere for everyone to see—before they will consider a program suitable for data analysis or other purposes. We can call this position the open-source theory.

On the other hand, the claim might be “that disclosure of the TrueAllele source code [to the defense, perhaps with an order to protect against more widespread dissemination of trade secrets] was required to properly conduct the Frye hearing” and that without at least that much discovery of the code, scientists would not regard TrueAllele as valid. We can call this position the discovery-based theory. It implies that, in establishing general scientific acceptance in a Frye hearing, pretrial discovery of secret code is an adequate substitute for exposing the code to the possible scrutiny of the entire scientific community. \8/

The Wakefield opinions are not entirely clear on about which theory they embrace or reject. Judge Rivera’s concurrence may have endorsed both theories. In addition to accentuating “the need to provide defendant with access to the source code,” she decried the absence of “objective, expert third-party access,” writing that

The court's decision was an abuse of discretion as a matter of law because it relied on validation studies by interested parties and evaluations founded on incomplete information about TrueAllele's computer-based methodology. Without defense counsel and objective, expert third-party access to and evaluation of the underlying algorithms and source code, the court could not conclude that TrueAllele's brand of probabilistic genotyping was generally accepted within the forensic science community.

The “evaluations founded on incomplete information” were from a standards developing organization, a state forensic science commission, and NIST. They were incomplete because, according to Judge Rivera, “without the source code, the agencies could not adequately evaluate the use of TrueAllele for this type of DNA mixture analysis ... .”

Focusing on the discovery-based theory, the rest of the court determined that “[d]isclosure ... was not needed in order to establish at the Frye hearing the acceptance of the methodology by the relevant scientific community. The Chief Judge gave two, somewhat confusingly stated, reasons. The first was that Wakefield sought the source code under a rule for discovery that did not apply and then “made no further attempt to demonstrate a particularized need for the source code by motion to the court.” But it is not clear how the failure “to demonstrate a particularized need” overcomes (or even responds to) the argument that the scientific community will not accept software as validly implementing algorithms unless the source code is either open source or given only to the defendant.

The Chief Judge continued:

Moreover, defendant's arguments as to why the source code had to be disclosed pay no heed to the empirical evidence in the validation studies of the reliability of the instrument or to the general acceptance of the methodology in the scientific community—the issue for the Frye hearing—and are directed more toward the foundational concern of whether the source code performed accurately and as intended (see Wesley, 83 N.Y.2d at 429, 611 N.Y.S.2d 97, 633 N.E.2d 451).

The meaning of the sentence may not be immediately apparent. The defense argument is that giving a defendant (or perhaps the scientific community generally) access to source code is a prerequisite to general acceptance of the proposition that the software correctly implements theoretically sound algorithms. If this broad proposition is false dogma, the court should simply say so. It should announce that source code need not be disclosed because there is an alternative, reasonably effective means for establishing that the software performs as it should. The first part of the first sentence starts out that way, but the sentence then states that “whether the source code performed accurately and as intended” is not a matter of general acceptance at all. It is only “foundational” in the sense identified by Chief Judge Kaye in Wesley, who, as we saw (Box 2), wrote that even though RFLP-VNTR testing was generally accepted, the complete “foundation” for admitting DNA evidence entails proof that the generally accepted procedure was performed properly in the case at bar.

But regarding the argument about source code as falling outside of the Frye inquiry misapprehends the defense argument. Neither the open-source nor the discovery-based theories pertain to the execution of valid software. They question the premise that validity can be generally accepted without disclosure of the program’s source code. Yet, the majority elaborates on its non-Frye "foundational" classification for the source-code argument as follows:

To the extent the testimony at the hearing reflected that the TrueAllele Casework System may generate less reliable results when analyzing more complex mixtures (see also President's Council of Advisors on Sci. and Tech., Exec. Office of the President, Forensic Science in Criminal Courts: Ensuring Scientific Validity of Feature-Comparison Methods, at 80 [2016] https://obamawhitehouse.archives.gov/sites/default/files/microsites/ostp/PCAST/pcast_forensic_science_report_final.pdf [published after the Frye hearing was held]), defendant did not refine his challenge to address the general acceptance of TrueAllele on such complex mixtures or how that hypothesis would have been applicable to the particular facts of this case. As a result, it is unclear that any such objection would have been relevant to defendant's case, where the samples consisted largely of simple (two-contributor) mixtures with the victim as a known contributor (see also NIST, DNA Mixture Interpretation: A NIST Scientific Foundation Review, at 3 [June 2021 Draft report] https://nvlpubs.nist.gov/nistpubs/ir/2021/NIST.IR.8351-draft.pdf).

These citations to the PCAST and NIST reports actually undercut any suggestion that source-code secrecy does not implicate Frye. The NIST draft repeatedly states that

Forensic scientists interpret DNA mixtures with the assistance of statistical models and expert judgment. Interpretation becomes more complicated when contributors to the mixture alleles. Complications can also arise when random variations, also known as stochastic effects, make it more difficult to confidently interpret the resulting DNA profile.

Not all DNA mixtures present these types of challenges. We agree with the President’s Council of Advisors on Science and Technology (PCAST) that “DNA analysis of single-source samples or simple mixtures of two individuals, such as from many rape kits, is an objective method that has been established to be foundationally valid” (PCAST 2016).

NIST, DNA Mixture Interpretation: A NIST Scientific Foundation Review, at 2-3 & 11-12 (June 2021 draft) (citations omitted). To demand that “defendant ... refine his challenge to address the general acceptance of TrueAllele on ... complex mixtures or ... the particular facts of this case” is to hold that TrueAllele is generally accepted for use with “single-source samples or simple mixtures of two individuals”—even though the source code is hidden. But if science does not demand the disclosure of source code for general acceptance inside the "single" or simple "zone," then why would it demand disclosure for general acceptance outside that zone?

The court's remarks make more sense as a response to Wakefield’s different discovery argument about the need for source code for trial purposes. This argument does not claim that disclosure of source is essential to general acceptance to exist. It looks to the trial rather than the pretrial Frye hearing. The thought may be that if the accuracy of the program for the “simple” cases is assured, then the need for discovery of the code to prepare for trial testimony is less compelling. The court appears to be responding that because “the samples consisted largely of simple (two-contributor) mixtures with the victim as a known contributor,” there was little need for discovery of the code in this case.

Although this rejoinder departs from the topic of what Wakefield teaches us about general acceptance, I would note that it is difficult to reconcile this characterization of the case with Chief Judge DeFiore’s own description of the samples. The court mentioned four samples. Its initial description of them indicates that the New York laboratory deemed the sample on the amplifying cord to be “at least” a three-person mixture and stated that “because of the complexity of the mixture,” the laboratory could not even compare “results generated from the amplifier cord ... to defendant's DNA profile.” 2022 WL 1217463, at *1. Because of the “stochastic threshold,” the laboratory could discern peaks at only 4 out of 15 loci for “the outside rear shirt collar” and “for the profile obtained from the victim's forearm.” Id. Presumably, the “insufficient data” on “the unknown contributors to the DNA mixtures found on the amplifier cord and the front of the shirt collar” is what led the state to call Cybergenetics for help. These samples are not instances of what PCAST called “DNA analysis of single-source samples or simple mixtures of two individuals, such as from many rape kits” or what the NIST group called “two-person mixtures involving significant quantities of DNA.” They are “more complicated” situations that arise “when contributors to the mixture share common alleles [and] when random variations, also known as stochastic effects” are present.

In sum, the deeper one looks into the Wakefield opinions, the more there is to wonder about. But whatever quirks and quiddities reside in the writing, the nearly unanimous opinion of the Court of Appeals signals that a trial court can choose to dispense with the general-acceptance inquiry for at least one PGS program—TrueAllele—for nonchallenging single samples or two-person mixtures and for samples of somewhat greater complexity as well.

NOTES

* UPDATE: On July 12, 2022, Chief Judge DiFiore announced that she will resign on August 31. See, e.g., Jimmy Vielkind & Corinne Ramey, New York’s Top Judge Resigns Amid Misconduct Proceeding: Attorney for Court of Appeals Judge Janet DiFiore Said Her Resignation Wasn’t Related to a Claim that She Improperly Attempted to Influence a Disciplinary Hearing, Wall St. J., July 12, 2022 8:31 am ET, https://www.wsj.com/articles/new-yorks-top-judge-resigns-amid-misconduct-proceeding-11657629111.
  1. This formulation conflates the issue of novelty with the issue of general acceptance, which can change over time. See Williams, 35 N.Y.3d at 43, 147 N.E.3d at 1143.
  2. The description begins with the remark that “The likelihood ratio in its modern form was developed by Alan Turing during World War II as a code-breaking method.” That is a possibly defective bit of intellectual history, inasmuch as Turing did not develop the likelihood ratio. To decipher messages, Turing relied on a logarithmic scale for the Bayes’ factor in two ways—as indicating the strength of evidence, and as a tool for sequential analysis. Sir Harold Jeffreys had done the former in his 1939 Theory of Probability book. The sequential analysis problem is not clearly connected to PGS. It arises when the sample size is not fixed in advance and the data are evaluated continuously as they are collected. PGS processes all the data at once.
  3. As the court wrote in People v. Williams, 35 N.Y.3d 24, 147 N.E.3d 1131, 1139–40, 124 N.Y.S.3d 593 (N.Y. 2020), “[r]eview of a Frye determination must be based on the state of scientific knowledge and opinion at the time of the ruling (see Cornell, 22 N.Y.3d at 784-785, 986 N.Y.S.2d 389, 9 N.E.3d 884 (‘a Frye ruling on lack of general causation hinges on the scientific literature in the record before the trial court in the particular case’”).
  4. E.g., 2022 WL 1217463 at *7 n.10 (“TrueAllele is not an outlier in the use of the continuous probabilistic genotyping method. Other types of probabilistic genotyping software, such as STRMix, have likewise been found to be generally accepted (see e.g. United States v. Gissantaner, 990 F.3d 457, 467 (6th Cir.2021)).”
  5. Cf. David H. Kaye, Theona M. Vyvial & Dennis L. Young, Validating the Probability of Paternity, 31 Transfusion 823 (1991) (comparing the empirical LR distribution for parentage using presumably true and false mother-child-father trios derived from a set of civil paternity cases to the “paternity index” (PI), a likelihood ratio computed with software applying simple genetic principles to the inheritance of HLA types, and reporting that the theoretical PI diverged from the empirical LR for PI > 80 or so).
  6. At trial, “Gary Skuse, Ph.D., a professor of biological sciences at the Rochester Institute of Technology, testified at trial as a defense witness [and] agreed ... that defendant's DNA was present in the mixtures found on the shirt collar and amplifier cord and that it was ‘most likely’ present on the victim's forearm.”
  7. The articles in the Journal of Forensic Sciences and Science and Justice have no such statements. The “Competing Interests” paragraph in a PloS One article advises that “I have read the journal’s policy and have the following conflicts. Mark Perlin is a shareholder, officer and employee of Cybergenetics in Pittsburgh, PA, a company that develops genetic technology for computer interpretation of DNA evidence. Cybergenetics manufactures the patented TrueAllele Casework system, and provides expert testimony about DNA case results. Kiersten Dormer and Jennifer Hornyak are current or former employees of Cybergenetics. Lisa Schiermeier-Wood and Dr. Susan Greenspoon are current employees of the Virginia Department of Forensic Science, a government laboratory that provides expert DNA testimony in criminal cases and is adopting the TrueAllele Casework system. This does not alter our adherence to all the PLOS ONE policies on sharing data and materials.”
  8. The defense advanced another different discovery theory in arguing that it could not adequately cross-examine and confront Dr. Perlin at trial unless it could access the source code. The court rejected this theory too.

Friday, November 27, 2020

Mysteries of the Department of Justice's ULTR for Firearm-toolmark Pattern Examinations

The Department of Justice's Uniform Language for Testimony and Reports (ULTR) for the Forensic Firearms/toolmarks Discipline – Pattern Examination" offers a ready response to motions to limit overclaiming or, to use the pedantic term, ultracrepidarianism, in expert testimony. Citing the DoJ policy, several federal district courts have indicated that they expect the government's expert witnesses to follow this directive (or something like it). \1/

Parts of the current version (with changes from the original) are reproduced in Box 1. \2/ This posting poses three questions about this guidance. Although the ULTR is a step in the right direction, it has a ways to go in articulating a clear and optimal policy.

Box 1. The ULTR

DEPARTMENT OF JUSTICE
UNIFORM LANGUAGE FOR TESTIMONY AND REPORTS
FOR THE FORENSIC FIREARMS/TOOLMARKS DISCIPLINE –
PATTERN MATCH EXAMINATION
...
III. Conclusions Regarding Forensic Pattern Examination of Firearms/Toolmarks Evidence for a Pattern Match

The An examiner may offer provide any of the following conclusions:
1.Source identification (i.e., identified)
2.Source exclusion (i.e., excluded)
3.Inconclusive
Source identification
‘Source identification’ is an examiner’s conclusion that two toolmarks originated from the same source. This conclusion is an examiner’s decision opinion that all observed class characteristics are in agreement and the quality and quantity of corresponding individual characteristics is such that the examiner would not expect to find that same combination of individual characteristics repeated in another source and has found insufficient disagreement of individual characteristics to conclude they originated from different sources.

The basis for a ‘source identification’ conclusion is an examiner’s decision opinion that the observed class characteristics and corresponding individual characteristics provide extremely strong support for the proposition that the two toolmarks came originated from the same source and extremely weak support for the proposition that the two toolmarks came originated from different sources.

A ‘source identification’ is the statement of an examiner’s opinion (an inductive inference2) that the probability that the two toolmarks were made by different sources is so small that it is negligible. A ‘source identification’ is not based upon a statistically-derived or verified measurement or an actual comparison to all firearms or toolmarks in the world.

Source exclusion
‘Source exclusion’ is an examiner’s conclusion that two toolmarks did not originate from the same source.

The basis for a ‘source exclusion’ conclusion is an examiner’s decision opinion that the observed class and/or individual characteristics provide extremely strong support for the proposition that the two toolmarks came from different sources and extremely weak or no support for the proposition that the two toolmarks came from the same source two toolmarks can be differentiated by their class characteristics and/or individual characteristics.

Inconclusive
‘Inconclusive’ is an examiner’s conclusion that all observed class characteristics are in agreement but there is insufficient quality and/or quantity of corresponding individual characteristics such that the examiner is unable to identify or exclude the two toolmarks as having originated from the same source.

The basis for an ‘inconclusive’ conclusion is an examiner’s decision opinion that there is an insufficient quality and/or quantity of individual characteristics to identify or exclude. Reasons for an ‘inconclusive’ conclusion include the presence of microscopic similarity that is insufficient to form the conclusion of ‘source identification;’ a lack of any observed microscopic similarity; or microscopic dissimilarity that is insufficient to form the conclusion of ‘source exclusion.’

IV. Qualifications and Limitations of Forensic Firearms/Toolmarks Discipline Examinations
• A conclusion provided during testimony or in a report is ultimately an examiner’s decision and is not based on a statistically-derived or verified measurement or comparison to all other firearms or toolmarks. Therefore, an An examiner shall not assert that two toolmarks originated from the same source to the exclusion of all other sources. This may wrongly imply that a ‘source identification’ conclusion is based upon a statistically-derived or verified measurement or an actual comparison to all other toolmarks in the world, rather than an examiner’s expert opinion.
○ assert that a ‘source identification’ or a ‘source exclusion’ conclusion is based on the ‘uniqueness’3 of an item of evidence.

○ use the terms ‘individualize’ or ‘individualization’ when describing a source conclusion.

○ assert that two toolmarks originated from the same source to the exclusion of all other sources.
• An examiner shall not assert that examinations conducted in the forensic firearms/toolmarks discipline are infallible or have a zero error rate.

• An examiner shall not provide a conclusion that includes a statistic or numerical degree of probability except when based on relevant and appropriate data.

• An examiner shall not cite the number of examinations conducted in the forensic firearms/toolmarks discipline performed in his or her career as a direct measure for the accuracy of a proffered conclusion provided. An examiner may cite the number of examinations conducted in the forensic firearms/toolmarks discipline performed in his or her career for the purpose of establishing, defending, or describing his or her qualifications or experience.

• An examiner shall not assert that two toolmarks originated from the same source with absolute or 100% certainty, or use the expressions ‘reasonable degree of scientific certainty,’ ‘reasonable scientific certainty,’ or similar assertions of reasonable certainty in either reports or testimony unless required to do so by a judge or applicable law.34


2 Inductive reasoning (inferential reasoning):
A mode or process of thinking that is part of the scientific method and complements deductive reasoning and logic. Inductive reasoning starts with a large body of evidence or data obtained by experiment or observation and extrapolates it to new situations. By the process of induction or inference, predictions about new situations are inferred or induced from the existing body of knowledge. In other words, an inference is a generalization, but one that is made in a logical and scientifically defensible manner. Oxford Dictionary of Forensic Science 130 (Oxford Univ. Press 2012).
3 As used in this document, the term ‘uniqueness’ means having the quality of being the only one of its kind.’ Oxford English Dictionary 804 (Oxford Univ. Press 2012).
34 See Memorandum from the Attorney General to Heads of Department Components (Sept. 9. 2016), https://www.justice.gov/opa/file/891366/download.

1

Are the two or three conclusions -- identification, exclusion, and inconclusive -- the only way the examiners are allowed to report their results?

In much of the world, examiners are discouraged from reaching only two conclusions--included vs. excluded (with the additional option of denominating the data as too limited to permit such a classification). They are urged to articulate how strongly the data support one classification over the other. Instead of pigeonholing, they might say, for example, that the data strongly support the same-source classification (because those data are far more probable for ammunition fired from the same gun than for ammunition discharged from different guns). 

The ULTR studiously avoids mentioning this mode of reporting. It states that examiners "may provide any of the following ... ." It does not state whether they also may choose not to -- and instead report only the degree of support for the same-source (or the different-source) hypothesis. Does the maxim of expressio unius est exclusio alterius apply? Department of Justice personnel are well aware of this widely favored alternative. They have attended meetings of statisticians at which straw polls overwhelmingly endorsed it over the Department's permitted conclusions. Yet, the ULTR does not list statements of support (essentially, likelihood ratios) as permissible. But neither are they found in the list of thou-shalt-nots in Part IV. \3/ Is the idea that if the examiners have a conclusion to offer, they must state it as one of the two or three categorical ones -- and that they may give a qualitative likelihood ratio if they want to?

2

Is the stated logic of a "source identification" internally coherent and intellectually defensible?

The ULTR explains that "[t]he basis for a 'source identification' is

an examiner’s opinion that the observed class characteristics and corresponding individual characteristics provide extremely strong support for the proposition that the two toolmarks originated from the same source and extremely weak support for the proposition that the two toolmarks originated from different sources.

Translated into likelihood language, the DoJ's "basis for a source identification" is the belief that the likelihood ratio is very large -- the numerator of L is close to one, and the denominator is close to zero (see Box 2).

On this understanding, a "source identification" is a statement about the strength of the evidence rather than a conclusion (in the sense of a decision about the source hypothesis). However, the next paragraph of the ULTR states that "[a] ‘source identification’ is the statement of an examiner’s opinion (an inductive inference2) that the probability that the two toolmarks were made by different sources is so small that it is negligible."

Box 2. A Technical Definition of Support
The questioned toolmarks and the known ones have some degree of observed similarity X with respect to relevant characteristics. Let Lik(S) be the examiner's judgment of the likelihood of the same-source hypothesis S. This likelihood is proportional to Prob(X | S), the probability of the observed degree of similarity X given the hypothesis (S). For simplicity, we may as well let the constant be 1. Let Lik(D) be the examiner's judgment of the likelihood of the different-source hypothesis (D). This likelihood is Prob(X | D). The support for S is the logarithm of the likelihood ratio L = Lik(S) / Lik(D) = Prob(X | S) / Prob(X | D). \4/

In this way, the ULTR jumps from a likelihood to a posterior probability. To assert that "the probability that the two toolmarks were made by different sources ... is negligible" is to say that Prob(D|X) is close to 0, and hence that Prob(S|X) is nearly 1. However, the likelihood ratio L = Lik(S) / Lik(D) is only one factor that affects Prob(D|X). Bayes' theorem establishes that

Odds(D|X) = Odds(D) / L.

Consequently, a very large L (great support for S) shrinks the odds in favor of S, but whether we end up with a "negligible" probability for D depends on the odds on D without considering the strength of the toolmark evidence. Because the expertise of toolmark analysts only extends to evaluating the toolmark evidence, it seems that the ULTR is asking them to step outside their legitimate  sphere of expertise by assessing, either explicitly or implicitly, the strength of the particular non-scientific evidence in the case.

There is a way to circumvent this objection. To defend a "source identification" as a judgment that Prob(D|X) is negligible, the examiner could contend that the likelihood ratio L is not just very large, as the ULTR's first definition required, but that it is so large that it swamps every probability that a judge or juror reasonably might entertain in any possible case before learning about the toolmarks. A nearly infinite L would permit an analyst to dismiss the posterior odds on D as negligible without attempting to estimate the odds on the basis of other evidence in the particular case (see Box 3).

Box 3. How large must L be to swamp all plausible prior odds?

Suppose that the smallest prior same-source probability in any conceivable case were p = 1/1,000,000. The prior odds on the different-source hypothesis would be approximately 1/p = 1,000,000. According to Bayes' rule, the posterior odds on D then would be about (1/p)/L = 1,000,000/L.

How large would the support L for S have to be to make D a "negligible" possibility? If "negligible" means a probability below, say 1/100,000, then the threshold value of L (call it L*) would be such that 1,000,000 / L* < 1/100,000 (approximately). Hence L* > 10^11. Are examiners are able to reliably tell whether the toolmarks are such that L > 100 billion?

One can use different numbers, of course, but whether the swamping defense of the ULTR really works to justify actual testimony as to "source identification" defined according to the ULTR is none too clear.

The ULTR seems slightly embarrassed with the characterization of a "source identification" as an "opinion" on the small size of a probability. Parenthetically it calls the opinion "an inductive inference," which sounds more impressive. But the footnote that is supposed to explain the the more elegant phrase only muddies the waters. It reads as follows:

Inductive reasoning (inferential reasoning): A mode or process of thinking that is part of the scientific method and complements deductive reasoning and logic. Inductive reasoning starts with a large body of evidence or data obtained by experiment or observation and extrapolates it to new situations. By the process of induction or inference, predictions about new situations are inferred or induced from the existing body of knowledge. In other words, an inference is a generalization, but one that is made in a logical and scientifically defensible manner. Oxford Dictionary of Forensic Science 130 (Oxford Univ. Press 2012) [sic]. \4/

The flaws in this definition are many. First, "inferential reasoning" is not equivalent to "inductive reasoning." Inference is reaching a conclusion from stated premises. The argument from the premises to the conclusion can be deductive or inductive. Deductive arguments are valid when the conclusion is true given that the premises are true. Inductive arguments are sound when the conclusion is sufficiently probable given that the premises are true. Second, inductive reasoning can be based on a small body of evidence as well as on a large body of evidence. In other words, deduction produces logical certainty, whereas induction can yield no more than probable truth. Third, an induction -- that is, the conclusion of an inductive argument -- need not be particularly scientific or "scientifically defensible." Fourth, an inductive conclusion is not necessarily "a generalization." An inductive argument, no less than a deductive one, can go from the general to the specific -- as is the case for an inference that two toolmarks were made by the same source. Presenting an experience-based opinion as the product of "the scientific method" by the fiat of a flawed definition of "inductive reasoning" is puffery.

3

If the examiner has correctly discerned matching "individual characteristics" (as the ULTR calls them), why cannot the examiner "assert that a ‘source identification’ ... is based on ... ‘uniqueness’" or that there has been an "individualization"?

The ULTR states that a "source identification" is based on an examination of "class characteristics" and "individual characteristics." Presumably, "individual characteristics" are ones that differ in every source and thus permit "individualization." The dictionary on which the ULTR relies defines "individualization" as "assigning a unique source for a given piece of physical evidence" (which it distinguishes from "identification"). But the ULTR enjoins an examiner from using "the terms ‘individualize’ or ‘individualization’ when describing a source conclusion," from asserting "that a ‘source identification’ or a ‘source exclusion’ conclusion is based on the ‘uniqueness’ of an item of evidence," and from stating "that two toolmarks originated from the same source to the exclusion of all other sources."

The stated reason to avoid these terms is that a source attribution "is not based on a statistically-derived or verified measurement or comparison to all other firearms or toolmarks." But who would think that an examiner who "assert[s] that two '[t]oolmarks originated from the same source to the exclusion of all other sources'" is announcing "an actual comparison to all other toolmarks in the world"? The examiner apparently is allowed to report a plethora of matching "individual characteristics" and to opine (or "inductively infer") that there is virtually no chance that the marks came from a different source. Allowing such testimony cuts the heart out of the rules against asserting "uniqueness" and claiming "individualization."

NOTES

  1. E.g., United States v. Hunt, 464 F.Supp.3d 1252 (W.D. Okla. 2020) (discussed on this blog Aug. 10, 2020).
  2. The original version was adopted on 7/24/2018. It was revised on 6/8/2020.
  3. Are numerical versions of subjective likelihood ratios prohibited by the injunction in Part IV that "[a]n examiner shall not provide a conclusion that includes a statistic or numerical degree of probability except when based on relevant and appropriate data"? Technically, a likelihood ratio is not a "degree of probability" or (arguably) a statistic, but it seems doubtful that the drafters of the ULTR chose their terminology with the niceties of statistical terminology in mind.
  4. A.W.F. Edwards, Likelihood 31 (rev. ed. 1992) (citing H. Jeffreys, Further Significance Tests, 32 Proc. Cambridge Phil. Soc'y 416 (1936)).
  5. The correct name of the dictionary is A Dictionary of Forensic Science, and its author is Suzanne Bell. The quotation in the ULTR omits the following part of the definition of "inductive inference": "A forensic example is fingerprints. Every person's fingerprints are unique, but this is an inference based on existing knowledge since the only way to prove it would be to take and study the fingerprints of every human being ever born."

Monday, April 6, 2020

Using Serological Tests for COVID-19

The next wave of tests developed in response to the COVID-19 pandemic are designed to detect antibodies in the blood of infected individuals. Some of the antibodies (known as IgM) are short-term. Others (IgG) hang around longer. Just how long -- and just how much immunity the antibodies confer -- remains to be determined, \1/ but one hope is that these immunological tests will enable public health authorities to identify individuals with immunity who could be freed from distancing or isolation rules.

Will there be a database of the names of the immune among us? "If I were emperor and had unlimited resources, I’d do serological testing on every person in this country as fast as I could,” said one ER physician interviewed by the Los Angeles Times. \2/ The article reports that "Germany may issue 'immunity certificates' allowing release from quarantine for those who show they have already been exposed and fought off the coronavirus."

According to Times reporters Anita Chabria and Emily Baumgaertner, the FDA has been "lax" in allowing "the sale of more than 40 serological tests, not requiring them to undergo a formal emergency-use approval process," creating "potential problems because there is no centralized collection of results, no uniformity of methods and slim evidence that the tests work with acceptable accuracy."

Of course, what accuracy is acceptable depends on the use to which the tests are put, and a formal but abbreviated EUA process is required for tests to be marketed to diagnose the disease. The FDA granted its first EUA for a serological test on April 1, 2020, to the North Carolina phamaceutcial company Cellex. \3/ Other countries have been using other tests (that probably are as well validated) for some time. \4/

Cellex's fact sheet for physicians, \5/ like the ones that accompany molecular diagnostics tests for the virus itself, contains no estimates of the sensitivity and specificity of the test -- either for the particular antibodies it is intended to identify, for the virus (SAR-CoV-2), or for the disease (COVID-19). With respect to disease diagnosis, the fact sheet states that a "positive test result with the qSARS-CoV-2 IgG/IgM Rapid Test indicates that antibodies to SARS-CoV-2 were detected, and the patient has potentially been exposed to COVID-19." \6/ Well, yes, a positive test result for the antibodies is a positive test result for the antibodies. But what is known about the sensitivity for antibody detection--how probable is it to have a positive test result when the antibodies actually are circulating in the bloodstream? As for false positives (which arise from imperfect specificity), the fact sheet merely announces that the "qSARS-CoV-2 lgG/lgM Rapid Test has been designed to minimize the likelihood of false positive test results" -- whatever that means. \7/

Moreover, Cellex stays away from saying much about the implications of correctly detected antibodies. To be sure, "[w]hen IgM antibodies are present, they can indicate that a patient has an active or recent infection with SARS-CoV-2," but once again,  the fact sheet does not state how strong this possible indication is. Instead, it only warns that
A positive result for IgM or IgG may not mean that a patient’s current symptoms are due to COVID-19 infection. Laboratory test results should always be considered in the context of clinical observations and epidemiological data in making a final diagnosis and patient management decisions.
The fact sheet also steers clear of quantitative statements for negative test results. It gives the definition of a negative test result -- "A negative test result with this test means that SARS-CoV-2 specific antibodies were not present in the specimen above the limit of detection." Then it adds:
However, a negative result does not rule out COVID-19 and should not be used as the sole basis for treatment, patient management decisions, or to rule out active infection. Patients tested early after infection may not have detectable IgM antibody despite active infection; in addition, not all patients will develop a detectable IgM and/or IgG response to SARS-CoV-2 infection. The absolute sensitivity of the qSARS-CoV-2 IgG/IgM Rapid test is unknown.

When diagnostic testing is negative, the possibility of a false negative result should be considered in the context of a patient's recent exposures and the presence of clinical signs and symptoms consistent with COVID-19. ... Direct testing for virus (e.g., PCR testing) should always be performed in any patient suspected of COVID-19, regardless of the qSARS-CoV-2 IgG/IgM Rapid test.
Despite the absence of quantitative information in the fact sheet, Cellex did perform a validity study, which it summarized in the package insert. \8/ In one experiment, fifty normal whole blood samples were spiked with positive serum (diluted to 1:100), and fifty others were spiked with negative serum at the same dilution. The result was that "[a]ll spiked samples were correctly identified by the test except for one of the negative samples, which was tested positive with the test. Thus, there was a 99% concordance rate with expected results when venous whole blood specimens are used." The two-by-two table for these results is
No antibodies
in serum
Antibodies
in serum
Test –490
Test +150

The estimated sensitivity is therefore 50/50 = 1.00 (95% CI of 0.93 to 1), and the estimated specificity is 49/50 = 0.98 (95% CI of 0.89 to 1). \9/

A test of "clinical performance" compared (1) antibody-test outcomes on 98 serum or plasma samples from individuals who tested positive with a RT-PCR method for SARS-CoV-2 infection and who had mild or no symptoms with (2) antibody-test outcomes on 180 negative serum or plasma samples that had been collected before September 2019. The classifications were not as accurate.
No VirusVirus
Test –1747
Test +691
Sensitivity91/98 = 0.93 [0.86, 0.97]
Specificity174/180 = 0.97 [0.93, 0.99]

An additional 30 samples were collected from hospitalized individuals who were clinically confirmed positive for SARS-CoV-2 infection and exhibited severe symptoms. These samples, along with 70 negative serum or plasma samples collected prior to September 2019 generated the following results: \10/

NoVirusSeverely Ill
Test –651
Test +529
Sensitivity29/30 = 0.97 [0.83, 0.999]
Specificity65/70 = 0.93 [0.84, 0.98]

Although the population from which the samples came is not stated, let's assume that they are fully applicable to the U.S. population. How well would the serological test work if extended, as suggested earlier, to "every person in this country," or perhaps, to every person in the labor force? The Census Bureau estimates the July 2019 population at a bit under 300 million, and it estimates the labor force percentage from 2014-2018 at almost 64%, for a total of about 211 million workers. Let's round this off to 200 million. In widespread testing, the error probabilities would be higher than those obtained under laboratory conditions, but I'll use the ballpark sensitivity and specificity estimates of 0.95 from the tables above. \11/

Lets assume that 10% of the 200 million workers have been infected and are immune. This percentage is far higher than the percentage of infections based on the number of reported cases to date in the country (about 300,000), but the ratio of unreported to reported cases could well be on the order of 10 to 1. \12/ For this guess as to the prevalence of past infection, and under the optimistic assumption that every infection confers immunity, we have some 20 million immune workers and 180 million non-immune workers. Now we apply the sensitivity and specificity estimates of 0.95 = 19/20:
  • Of the 20 million immune workers, 19/20, will test positive (antibodies detected). That is 19 million true positives.
  • Of the 20 million immune workers, 1/20 will test negative. That is 1 million false negatives.
  • Of the 180 million non-immune workers, 19/20 will test negative. That is 171 million true negatives.
  • Of the 180 million non-immune workers, 1/20 will test positive. That is 9 million false positives.
The negatives (apparently non-immune workers) will continue to be excluded from the workplace. There will only be 1/(1 + 171) false exclusions -- about half a percent. That is pretty good.

The positives (apparently immune workers) will return to work. There will be 9/(9 + 19) = 9/28 -- about a third -- false inclusions. If these workers mistakenly believe that the blood test indicates a 95% probability of immunity (as they might if they were told that the false positive rate is only 5%), they will be sorely mistaken about how likely it is that they are immune. \13/

This arithmetic is a "mind-bending" instance of Bayes' rule in operation. \14/ Before testing, the probability of a randomly selected worker being immune was (assumed to be) 1/10, or 10%. A positive test result raises the probability of immunity to 19/28, or 68%. The odds have improved from 1:9 to about 2:1. More precisely, the posterior-to-prior-odds ratio is [19/(28 – 19)] / (1/9) = 19. This Bayes factor is just the positive likelihood ratio LR+ = (sensitivity) /  (1 – specificity) = 0.95 / 0.05 = 19. For any set of prior odds, the posterior odds are this same likelihood ratio times the stipulated prior odds, and any such test:

Posterior Odds = [(sensitivity) /  (1 – specificity)] × Prior Odds.

Proposals from economists and public policy analysts to relax employment limitations according to serological test results are emerging. \15/ To think through their merits, we would also need to consider the cost of the testing and the consequences of the false positives and negatives for individuals and the rest of society. However, in the Washington world of volatile decisionmaking, policies are likely to change before comprehensive and convincing analyses of their probable costs and benefits are available.

UPDATED: Note 14 added 6/20/20 9:15 AM ET

NOTES
  1. The period of immunity could be short. An article in Scientific American reported that
    [I]mmunity functions on a continuum. ... Immunity to seasonal coronaviruses (such as those that cause common colds), for example, starts declining a couple of weeks after infection. And within a year, some people are vulnerable to reinfection. ... But studies of SARS-CoV—the virus that causes severe acute respiratory syndrome, or SARS, which shares a considerable amount of its genetic material with SARS-CoV-2—are more promising. Antibody testing shows SARS-CoV immunity peaks at around four months and offers protection for roughly two to three years. ...

    Even if the antibodies stick around in the body, however, it is not yet certain that they will prevent future infection. What we want, [Dawn Bowdish, a professor of pathology and molecular medicine and Canada Research Chair in Aging and Immunity at McMaster University in Ontario] says, are neutralizing antibodies. These are the proteins that reduce and prevent infection by binding to the part of a virus that connects to and “unlocks” host cells. ... In contrast, nonneutralizing antibodies still recognize parts of the pathogen, but they do not bind effectively and so do not prevent it from invading cells. ...

    Nevertheless, a few small studies of cells in laboratory dishes suggest that SARS-CoV-2 infection triggers the production of neutralizing antibodies. And animal studies indicate such antibodies do prevent reinfection, at least for a couple of weeks. ...
    Stacey McKenna, What Immunity to COVID-19 Really Means, Sci. Am. Apr. 10, 2020, https://www.scientificamerican.com/article/what-immunity-to-covid-19-really-means/. See also Marc Lipsitch, Who Is Immune to the Coronavirus?, N.Y. Times, Apr. 13, 2020, https://www.nytimes.com/2020/04/13/opinion/coronavirus-immunity.html (sketching existing information and the major gaps in what is known).
  2. Anita Chabria & Emily Baumgaertner, A Coronavirus Immunity Test Is Essential for the U.S. But Will It Work?, L.A. Times, Apr. 2, 2020, 12:06 PM, https://www.latimes.com/science/story/2020-04-02/coronavirus-test-immunity-detection-accuracy
  3. Letter of Apr. 1, 2020, from RADM Denise M. Hinton, Chief Scientist, Food and Drug Administration, to James X. Li, Chief Executive Officer, Cellex Inc., available at https://www.fda.gov/media/136622/download.
  4. Apoorva Mandavilli, F.D.A. Approves First Coronavirus Antibody Test in U.S., N.Y. Times, Apr. 2, 2020, https://www.nytimes.com/2020/04/02/health/coronavirus-antibody-test.html.
  5. Cellex, Fact Sheet for Healthcare Providers, qSARS-CoV-2 lgG/lgM Rapid Test– Cellex Inc., Apr. 1, 2020, available at https://www.fda.gov/media/136623/download
  6. Presumably, "exposed to COVID-19" really means "exposed to the virus (SARS-CoV-2)."
  7. For an effort to unpack the meaning of minimize, see  https://for-sci-law.blogspot.com/2020/04/he-or-she-tested-negative-or-positive.html. 
  8. Cellex qSARS-CoV-2 IgG/IgM Rapid Test, Package Insert, available at https://www.fda.gov/media/136625/download
  9. In an unstated number of tests and specimens, no reactivity to a number of other virus was observed, and spiking specimens with a variety of substances that might be found in blood plasma did not generate false negatives or positives. Id.
    UPDATE (June 11, 2020): The FDA has posted a summary of sensitivity, specificity, and predictive values for serologic tests that it has authorized for emergency use. It reports a sensitivity of 93.8% (120/128) and a specificity of 96.0% (240/250). U.S. Food & Drug Administration, EUA Authorized Serology Test Performance, June 9, 2020, https://www.fda.gov/medical-devices/emergency-situations-medical-devices/eua-authorized-serology-test-performance
  10. The day of collection relative to the onset of illness was unknown. The package insert reports "a Positive Percent Agreement and Negative Percent Agreement of 93.75% (95% CI: 88.06-97.26%) and 96.40% (95% CI: 92.26-97.78%), respectively." Id.
  11. To the extent that many business and government activities are considered essential and continue to employ their workers, I am also overstating the number of workers who have been idled by precautionary measures, but we'll be looking at ratios that do not depend on the absolute numbers.
  12. See Kathleen M. Jagodnik, Forest Ray, Federico M. Giorgi & Alexander Lachmann, Correcting Under-reported COVID-19 Case Numbers: Estimating the True Scale of the Pandemic, .
  13. This overstates the impact of false negative test results because workers who already recovered from a clear case of COVID-19 will know that their probability of immunity is high notwithstanding the more general probability of a false inclusion.
  14. A Nature weekly briefing of 6/19/20 noted that "A mathematical pitfall plagues antibody tests that are less than 100% accurate: the lower the infection rate, the more likely it is that a positive result is wrong. Scientific American explains, with a very handy graphic, how this mind-bending fact arises. (Scientific American | 3 min read)."
  15. E.g., Aaron Edlin & Bryce Nesbitt, The ‘Certified Recovered’ from Covid-19 Could Lead the Economic Recovery, STAT, Apr. 6, 2020, https://www.statnews.com/2020/04/06/the-certified-recovered-from-covid-19-could-lead-the-economic-recovery/; David M. Studdert and Mark A. Hall, Disease Control, Civil Liberties, and Mass Testing — Calibrating Restrictions during the Covid-19 Pandemic, New Eng. J. Med., Apr. 9, 2020, DOI: 10.1056/NEJMp2007637.

Friday, April 19, 2019

The Latest Guidance on Expert Reports from England's Forensic Science Regulator

This week, the Forensic Science Regulator for England and Wales issued a third version of guidance on the content of expert reports for criminal cases. Forensic Science Regulator, Guidance on the Content of Reports Issued by Expert Witnesses in the Criminal Justice System in England and Wales, Apr. 17, 2019.

 These reports are required "when the witness will, either in the report or in testimony at court, provide evidence of opinion." Some items are rather mundane -- for instance, a report must be signed by its author. But several parts of the guidance are noteworthy:
  • The author must certify that the report is truthful and, if introduced into evidence, wilful falsehoods would be criminal. (If included in a report in the United States, such declarations could be significant on the question of whether the report is "testimonial" hearsay that would trigger Con frontation Clause protections.)
  • The report must state that "the witness will inform all parties and where appropriate the court in the event that his/her opinion changes on any material issues."
  • "The witness must state any limitations to their experience and whether any particular issue, which is discussed in the statement (or on which they had been requested to discuss), falls outside their expertise."
  • Consideration should be given to stating "[w]hether the witness tends to work for the prosecution, defence or both" amd "[w]hether the witness has been the subject of criticism and is aware of that criticism."
  • Qualifications must be impartially listed, and "[t]he membership of an organisation which is obtained solely by applying for membership (and perhaps, payment of a fee) is not a qualification," and presenting such memberships as proof of expertise "is, at the least, irrelevant and may be misleading."
  • "Where the witness is aware that reputable experts in the field would hold a range of opinions on the issue under consideration, the report must set out what that range is and justify why the witness’s opinion falls at a particular point within that range."
  • The report should delineate "[a]ny results obtained which would undermine the opinion expressed"; [a]ny work, or other information, the witness is aware of which would undermine the opinion"; "[a]ny limitations to the analytical approach adopted"; and "[t]he uncertainty of measurement associated with the methods employed and the manner in which this has been addressed."
The document also advises experts "to avoid the term 'consistent'" because it "has been criticised by the Courts. It can sound significant but, in reality, mean little." On the other hand, "'not consistent' is usually clear and does not give rise to the same issues."

The guidance on "statistics" asks the expert to keep "detailed consideration" of "complex statistical analysis" out of the report but to make it available some other way. Moreover, it seems to place all "Bayesian inference" into this category! In full, the guidance is that
The Courts have, on a number of occasions, expressed concern about the evidence of complex statistical analysis (e.g. Bayesian inference) being discussed before the jury.
It is therefore advisable to avoid incorporating detailed consideration of such analysis in reports. The statement should set out the model employed and the conclusions drawn.
The details of the analysis should be available for disclosure.
This is a matter that the Regulator should reconsider in the next revision of the guidance -- along with the suggestion that "reference to statistical significance" is an "appropriate term" for explaining "how safe or unsafe" is "an inference from any findings." The phrase "statistical significance" has not fared well in recent policy pronouncements from statisticians.