DNA Match Probability Calculation

A forensic STR profile consists of genotypes at multiple loci, typically 15 to 24 in current practice (the FBI CODIS system expanded from 13 to 20 core loci in 2017; the UK National DNA Database uses 16 loci under the ESS standard). At each locus, the analyst observes either a heterozygote (two different alleles) or a homozygote (two copies of the same allele). The genotype frequency at each locus is estimated from a population reference database.

For a heterozygote with alleles a and b at frequencies p and q in the database, the Hardy-Weinberg genotype frequency is 2pq. For a homozygote with allele a at frequency p, the frequency is p-squared. The product rule then multiplies these frequencies across all loci to give the multi-locus RMP. If a profile has 20 loci and the per-locus genotype frequencies average around 0.05, the RMP is roughly 0.05 to the power of 20, which is 10 to the power of minus 26. In practice, per-locus frequencies vary considerably, and the final RMP for a full profile routinely falls in the range of one in several trillion to one in a quadrillion or beyond.

The product rule is valid only if the loci are statistically independent. For STR loci on different chromosomes, independence holds by Mendel's law of independent assortment. For loci on the same chromosome, independence holds if they are far enough apart that recombination between them is effectively random. The loci in the CODIS and ESS panels were selected partly on this criterion, and published tests of independence across major population databases support the assumption for these specific loci, though departures are occasionally observed and should be noted.

National DNA databases are assembled from large, broadly defined population groups: European-American, African-American, Hispanic, South Asian, East Asian, and so on. Within each group, however, there are subpopulations, for example Gujarati Indians within the South Asian group, or Puerto Rican individuals within the Hispanic group, in which recent shared ancestry means that alleles are more correlated than the broad-group frequencies suggest. This is called population substructure, and it means the simple product rule underestimates how often two people from the same subgroup will share a profile.

The NRC-II report (1996) proposed the theta (FST) correction to handle this. Theta is a measure of genetic coancestry between members of the same subpopulation; typical values range from 0.01 for large, well-mixed populations to 0.03 for smaller, more isolated ones. For a heterozygote with alleles a and b at frequencies p and q, the theta-corrected frequency is:

2 x [(1-theta)p + theta] x [(1-theta)q + theta] / (1 + theta). For a homozygote with allele a at frequency p, the corrected formula is: [(2theta + (1-theta)p) x (3theta + (1-theta)p)] / [(1+theta)(1+2theta)]. These corrections make the genotype frequency larger (more conservative for the prosecution) than the uncorrected 2pq or p-squared estimate, reducing the risk that the product rule overstates the rarity of the profile.

Different jurisdictions have settled on different default theta values. US practice under FBI guidelines uses theta = 0.01 for major population groups and 0.03 for Native American populations. UK Forensic Science Regulator guidance recommends 0.02 as the default. Indian forensic laboratories applying DNA profiling typically draw on FST estimates derived from studies of Indian subpopulations, where values between 0.01 and 0.04 have been reported depending on the group. Courts in all these jurisdictions have accepted the theta correction as scientifically sound, though some defence challenges have argued for higher theta values specific to the defendant's subgroup.

The RMP is a probability of the evidence under the defence hypothesis that a random unrelated person is the source of the crime stain. On its own it answers only half the question. The full evaluative framework requires a second number: the probability of the evidence under the prosecution hypothesis that the suspect is the source. If the profile is complete and the suspect's profile was correctly typed, this probability is one: the evidence is certain if the suspect is the source. The likelihood ratio is then 1 divided by the RMP, which equals the reciprocal of the RMP.

In more complex cases, the prosecution probability is not simply one. In a mixture, for example, the probability that the observed mixed profile would be seen even if the suspect were a contributor depends on the proportions and the number of contributors. In partial profiles where some loci could not be typed, the probability under the prosecution hypothesis may be less than one if the analyst cannot confirm all loci match. The LR framework handles these cases consistently: it always takes the ratio of the two conditional probabilities.

The LR is preferred over stating the RMP alone because it separates scientific inference from legal fact-finding. The scientist calculates the LR and presents it to the court. The court, applying Bayes' theorem implicitly, multiplies the LR by its prior odds of guilt to reach posterior odds. The scientist does not express an opinion on the posterior probability of guilt. This separation is formally endorsed by the Association of Forensic Science Providers (UK), SWGDAM (US), and the European Network of Forensic Science Institutes, and is increasingly adopted by forensic science regulators worldwide, including in Australia.

The RMP depends on the allele frequencies used, which come from a population database. Forensic laboratories maintain databases for major population groups in their jurisdiction, and the choice of which database to apply to a given case can change the calculated RMP by an order of magnitude or more. Best practice, articulated in the SWGDAM guidelines (2016 revision) and UK Forensic Science Regulator guidance, is to calculate the RMP using the database whose population most closely matches the defendant's ancestry, and to report results from multiple databases.

This creates a practical challenge when the defendant belongs to a group not represented in available databases. In the United Kingdom, this issue arose in cases involving defendants of South Asian, East African, or mixed-ethnicity backgrounds. The Court of Appeal in R v Doheny and Adams [1997] set out principles for presenting DNA evidence that include the requirement to use appropriate population databases and to acknowledge their limitations. In India, where over 4,000 ethnic and caste subpopulations exist, the absence of large STR frequency databases for specific communities has been a persistent limitation. The DNA Technology (Use and Application) Regulation Bill, introduced in Parliament in 2019, addressed laboratory accreditation and database governance but did not resolve the underlying population genetics gap.

When no exact-match database exists, analysts use the closest available database and apply a conservative (higher) theta correction to account for the additional uncertainty. Some laboratories report the RMP across three or four databases and present the least favourable (lowest RMP, most favourable to the defendant) to the court. This practice is conservative toward the defence and is consistent with the general principle that forensic statistics should not overstate the strength of evidence.

Single-source, full profiles are the simplest case. A substantial proportion of forensic DNA samples are mixtures from two or more contributors, partial profiles where some loci could not be typed because of degradation or inhibition, or low-template samples where stochastic effects introduce dropout (alleles that should be present are absent) and drop-in (sporadic extra alleles). Each complication changes the probability calculation.

For mixtures, the standard approach through the early 2010s was to use conservative inclusion or exclusion criteria: if a suspect's alleles were all present somewhere in the mixture, they were included; the RMP was then calculated for the minor contributor using a method that summed all alleles present. This approach, called the combined probability of inclusion (CPI), has been criticised for being insufficiently discriminating: it assigns the same RMP to a weak inclusion as to a strong one. The current direction, now standard in the US (PCAST report, 2016) and UK, is probabilistic genotyping, in which software models the mixture statistically, considers all possible contributor combinations given the observed peak heights, and produces a LR for each candidate contributor.

For partial profiles, the LR is calculated only over the loci that could be typed. A partial profile produces a less discriminating LR because fewer loci are included. For low-template samples, probabilistic genotyping systems such as STRmix, TrueAllele, and ArmedXpert model dropout and drop-in probabilities explicitly, producing a LR that accounts for stochastic uncertainty. The validation requirements for these systems have been the subject of extensive court scrutiny in the US, UK, and Australia. The 2016 PCAST report recommended that probabilistic genotyping software be treated as a form of feature-comparison method requiring foundational validity studies before admission.

The question of how to present a DNA LR or RMP to a jury has generated a large body of case law. In the United Kingdom, the Court of Appeal in R v T [2010] EWCA Crim 2439 held that experts must not present a bare LR number without explaining what it means and that presenting a Bayesian calculation involving the prior probability of guilt was inappropriate in a jury trial. In R v Adams [1996] and [1998], the Court of Appeal rejected a defence attempt to have the jury perform a formal Bayesian calculation. These decisions shape how UK forensic scientists phrase their conclusions in DNA cases.

In the United States, individual state courts govern DNA evidence admissibility. Daubert v Merrell Dow Pharmaceuticals (1993) set the federal standard for expert scientific testimony: the method must be tested, peer-reviewed, have a known error rate, and be generally accepted. Courts applying Daubert to DNA statistics have accepted both the product rule with theta correction and probabilistic genotyping, though challenges to specific software implementations are ongoing. Frye jurisdictions, which retain the older general-acceptance standard, have also admitted these methods consistently.

The role of statistics in evidence evaluation is inseparable from the legal context: see Role of Statistics in Evidence Evaluation for a broader treatment. The history of how courts came to accept or resist probabilistic evidence is traced in History of Statistical Evidence in Courts. Under the Bharatiya Sakshya Adhiniyam 2023 in India, Section 57 (formerly Section 45 of the Indian Evidence Act) governs the admissibility of expert opinions including forensic DNA statistics; the court may accept an expert's LR statement but is not bound by it.