Statistical Interpretation of Serological Relationship Evidence

Every serological relationship analysis begins with the same step: identifying the allele the child must have inherited from the biological father. A child inherits one allele at each genetic locus from the mother and one from the father. If the mother's genotype is known, her contribution to the child can be subtracted, leaving the allele that had to come from the paternal side. This is the obligatory paternal allele (OPA).

For the ABO system, consider a child typed as blood group A (genotype AO) and a mother typed as group O (genotype OO). The mother can only have contributed O. The child's A allele must therefore have come from the father. Any man who cannot contribute an A allele is excluded. A man typed as blood group O (genotype OO) carries no A allele and is excluded. A man typed as blood group A or AB carries an A allele and cannot be excluded on the basis of ABO alone.

When the mother's blood group is not available, the analysis is less precise but not impossible. If the child is group AB, the father must have contributed either A or B (since no individual is AA or BB for ABO). A man typed as group O contributes neither A nor B and is excluded regardless of the mother's type. The absence of maternal typing reduces the strength of inclusions but rarely affects exclusions at systems where the alleles are well characterised.

Exclusion probability (W, sometimes called the probability of exclusion or PE) is a population-level statistic. It answers the question: if a randomly chosen unrelated man from the reference population were tested, what is the probability he would be excluded by this marker panel? A panel with W = 0.99 would exclude 99% of unrelated men. The remaining 1% would share the obligatory allele(s) by chance and could not be excluded.

For a single biallelic system with alleles p and q (where p + q = 1), the probability that a random man carries the obligatory allele depends on allele frequencies and Hardy-Weinberg equilibrium. If the OPA is allele A with frequency p, a random man can have genotype AA (frequency p squared) or AO (frequency 2pq), so the probability of him carrying at least one copy of A is p squared + 2pq = p(p + 2q). The exclusion probability for this locus is 1 minus that value: 1 minus p(p + 2q).

Combined exclusion probability across multiple independent systems is calculated by multiplying the non-exclusion probabilities for each system and subtracting from 1. If system 1 has a non-exclusion probability of 0.10 and system 2 has a non-exclusion probability of 0.05, the combined non-exclusion probability is 0.10 times 0.05 = 0.005, and the combined exclusion probability is 0.995. This is the figure most often cited in court reports from the pre-DNA era: the test panel can exclude 99.5% of unrelated men.

The paternity index (PI) for a single marker system is a likelihood ratio: the probability of the observed child genotype given that the alleged father is the biological father, divided by the probability of the observed child genotype given that a random unrelated man from the population is the biological father. When PI is greater than 1, the observed results are more consistent with the alleged father being the true father than with a random man. When PI equals 0, the alleged father is excluded.

In the table, AF denotes the alleged father and p is the population frequency of the obligatory paternal allele. The numerator is the probability that the alleged father transmitted the OPA given his typed genotype. The denominator is the probability that a random man from the population transmitted the same allele. If p is small (the allele is rare), the PI is high: finding a rare allele in both the child and the alleged father is strong evidence that he contributed it. If p is large (the allele is common), the PI is closer to 1: finding a common allele tells you less.

Population allele frequencies are the critical input. Laboratories maintained frequency tables for the populations they served, sometimes stratified by ethnic group or region. German courts in the 1970s relied on frequencies measured in German blood donor cohorts; US laboratories used frequencies measured in European-American, African-American, or Hispanic cohorts depending on the case. Using the wrong frequency table for the case population was recognised as a source of error, and challenged in several appellate decisions in the US during the 1980s and 1990s.

The combined paternity index (CPI) is the product of the individual paternity indices across all marker systems tested, provided those systems are independent. Blood group systems at different chromosomal loci satisfy this requirement. Red cell enzyme systems and serum protein systems tested in the same era were also assumed to be independent of the major blood group loci. The CPI is still a likelihood ratio: it compares the probability of all observed results together under the two competing hypotheses.

Converting the CPI to a probability of paternity requires Bayes' theorem and a prior probability. Let H1 be the hypothesis that the alleged father is the biological father and H2 be the hypothesis that a random unrelated man is. The prior probability of H1 is set before considering the genetic evidence. Most forensic laboratories adopted the convention of a neutral prior of 0.5, treating the two hypotheses as equally likely before testing. The posterior probability of H1, called the probability of paternity (W%), is then CPI divided by (CPI + 1). This is the Essen-Moller formula.

Typical reporting thresholds varied by country. Germany required a probability of paternity of at least 99.73% before a court could order a declaration of paternity, a threshold sometimes called the 'practically proven' standard. US courts used no uniform threshold; some accepted results in the 90% range, others required values above 99%. UK courts took a similar non-prescriptive approach, treating the probability of paternity as one item of evidence among several. Indian courts under the Indian Evidence Act 1872 (now replaced by the Bharatiya Sakshya Adhiniyam 2023) treated blood group exclusions as strong negative evidence but were cautious about affirmative paternity probability claims from serological testing alone.

Serological paternity reports typically followed a standard structure. The laboratory identified the marker systems used, the typed results for the mother, child, and alleged father, the obligatory paternal allele at each system, and whether the alleged father was excluded or included at each. If not excluded, the report gave the individual PI for each system, the CPI, and the probability of paternity calculated with a stated prior. Many reports also included the combined exclusion probability for the panel as a measure of the test's discriminating power.

Courts in common law jurisdictions heard challenges to three aspects of these reports. First, whether the population frequency data was appropriate for the individuals tested. Second, whether the prior probability of 0.5 was a legitimate scientific choice or an assumption that had improperly loaded the result. Third, whether the probability of paternity was a statement about the defendant in particular or a statement about a class of similar defendants. These were not merely theoretical objections; they were raised in appellate courts in the US (People v. Macedonio, 1992; State v. Frye issues around novel scientific evidence) and the UK, and they produced a body of case law that shaped how probabilistic evidence was presented in courts through the 1990s.

The transition to DNA profiling from the late 1980s onward did not eliminate these debates; it intensified them. STR profiling produces much higher likelihood ratios than serological systems, but the conceptual structure is the same: an obligatory allele, a population frequency, a per-locus likelihood ratio, a combined likelihood ratio across loci, and a prior-dependent probability statement. The debates over population databases, mixture interpretation, and prior probability that dominated forensic DNA cases in the 1990s were extensions of the same questions that arose in serological paternity testing two decades earlier.

DNA short tandem repeat (STR) profiling surpassed serological typing for relationship testing in high-resource laboratories during the 1990s. STR profiles at 15 to 20 loci routinely produce CPIs in the billions, generating probabilities of paternity that approach certainty with a neutral prior. They are also more discriminating for exclusions: a single STR mismatch between child and alleged father at multiple loci is an unambiguous exclusion that does not depend on blood group inheritance assumptions.

Serological methods remain relevant in three settings. First, in jurisdictions where DNA infrastructure is not available or not affordable, ABO, Rh, and MNS typing continues to be used as a first-pass screen. A clear exclusion from blood group typing alone ends a case without the cost of DNA analysis. Second, in historical archive cases where blood group typing records exist but no biological material survives for DNA extraction, interpreting the statistical weight of those old records still requires understanding the original framework. Third, the conceptual framework (OPA, PI, CPI, Essen-Moller conversion) is the same framework used in DNA profiling, so serological relationship statistics serve as a pedagogically clear introduction to probabilistic forensic evidence.