No fallacies in the formulation of the paternity index

In a recent publication, Li and Chakravarti claim to have shown that the paternity index is not a likelihood ratio. They present a method of estimating the prior probability of paternity from a sample of previous court cases on the basis of exclusions and nonexclusions. They propose calculating the posterior probability on the basis of this estimated prior and the test result expressed as exclusion/nonexclusion. Their claim is wrong--the paternity index is a likelihood-ratio, that is, the ratio of the likelihood of the observation conditional on the two mutually exclusive hypotheses. Their proposed method of estimating the prior has been long known, has been applied to several samples, and is inferior (in terms of variance of the estimate) to maximum likelihood estimation based on all the phenotypic information available. Their proposed "new method" of calculating a posterior probability is based on the use of a less informative likelihood ratio 1/(1-PE) instead of Gürtler's fully informative paternity index X/Y (Acta Med Leg Soc Liege 9:83-93, 1956), but is otherwise identical to the Bayesian approach originally introduced by Essen-M?ller in 1938.     

Basic fallacies in the formulation of the paternity index.

Some basic fallacies in the computation of the paternity index have been pointed out. The general finding that the true fathers' mean paternity index is greater than that of nonfathers is a necessary consequence of an algebraic identity, having nothing to do with paternity or nonpaternity. It has also been shown that the paternity index is not a likelihood ratio as claimed. The fact that a paternity index may frequently take values less than unity leads to absurd conclusions regarding the probability of paternity. A formula relating prior and posterior probabilities of paternity, based solely on genetic marker testing results (exclusion or nonexclusion), is reiterated as a substitute for the current paternity index.     

HLA and the probability of paternity.

In cases of disputed paternity, blood tests are often used to obtain an estimate of the probability that the accused male is the true father. The interpretation of the genetic data is usually based upon a statistic called the paternity index. This paper shows that the paternity index method cannot be applied to data from compound loci in the absence of information on linkage phase. Since phenotypic data from compound loci, such as HLA, MNSs, and Rh, are often useful in disputed paternity proceedings, they should be analyzed with available alternative statistics.     

Some fallacies in the computation of paternity probabilities.

Legal identification of fathers by means of a "paternity probability" has been used in European courts for decades, and has recently been introduced into American courts and accepted by some of them. The voluminous literature on this topic contains virtually no fundamental criticism of the logical basis for the probabilistic computations. Here I suggest that the "paternity probability" suffers from three basic fallacies: (1) contrary to claims, the figure is not, in fact, the probability that the alleged father is the true father, (2) the denominator of the likelihood ratio used in the computation is driven by (sometimes self-contradictory) assumptions and is not based on facts, and (3) post-inclusionary computations are based on speculation about genotypes that does not constitute scientific evidence. It is recommended that pending the resolution of these difficulties "paternity probabilities" should not be computed or introduced as positive evidence of paternity.     

Paternity index and attribution of paternity

If blood typing and similar tests do not exclude a putative father in a paternity case, his probability of paternity can be assessed with the formulae of Essen-M?ller[1938]. Gürtler[1956] uses an alternative route, viz. the paternity index, to reach identical end results. Majumder and Nei [1983] claim that the methods are not powerful enough. This opinion can always be defended, but may have been enhanced by their inadequate computer model. They also contend that current methods may more often than not lead to false attributions of paternity. This is outright erroneous.     

Comparison of Biochemical Polymorphisms and Short Tandem Repeat (STR) DNA Markers for Paternity Testing in Rhesus Monkeys (Macaca mulatta)

Genetic markers are indispensable for molecular and statistical genetic research involving nonhuman primates. Genetic markers must be used to ascertain parentage and to confirm the accuracy of pedigrees based solely on housing or demographic records; otherwise, the results of pedigree, linkage, or quantitative genetic analyses may be unreliable. Until recently, most genetic markers used in nonhuman primates were plasma proteins or isozyme polymorphisms, which were required in large numbers, because levels of genetic variation revealed by these markers were rather low. We compared the newer, PCR-amplified short tandem repeat markers (STRs) with a panel of classical biochemical polymorphic markers, for paternity determination among captive-bred rhesus monkeys. The STR markers exhibited an average genetic diversity of 64% and an expected paternity exclusion probability of 0.443. Both of these were greater than the average 54.5% genetic diversity and 0.298 exclusion probability exhibited by the biochemical markers. The STRs were much more efficient than the biochemical markers for parentage determination, since they required only half the amount of genetic typing data to resolve an average paternity case. Thus, the results of applying these two classes of genetic markers in paternity tests were somewhat different than expected on the basis of theoretical exclusion probabilities. These differences were probably due to inbreeding and other genetic differences among breeding colonies. Because they are more informative and provide rapid and efficient genetic data, STRs are now the method of choice for parentage determination and pedigree corroboration among nonhuman primates.     

Probability and paternity testing.

A probability can be viewed as an estimate of a variable that is sometimes 1 and sometimes 0. To have validity, the probability must equal the expected value of that variable. To have utility, the average squared deviation of the probability from the value of that variable should be small. It is shown that probabilities of paternity calculated by the use of Bayes' theorem under appropriate assumptions are valid, but they can vary in utility. In particular, a recently proposed probability of paternity has less utility than the usual one based on the paternity index. Using an arbitrary prior probability in the calculation cannot lead to a valid probability unless, by chance, the chosen prior probability happens to be appropriate. Appropriate assumptions regarding both the prior probability and gene or genotypic frequencies can be estimated from prior experience.     

The importance of HLA typing in cases of disputed paternity

In dispute paternity, the biologists must reply to two questions: 1. Is the paternity excluded or possible? 2. If it is possible, what is its probability? Valid answers can be given, using several genetic markers, among which HLA genes are specially interesting. Looking at HLA-A, B, C, DR typing of child, mother and presumed father, we propose a method which allows a direct calculation of paternity probability. Crossing over between HLA genes in presumed father and in mother are also considered in this method. In our experience, adding the date provided by the HLA genes and other genetic markers, we obtained, either formal exclusions, or possible paternities with a probability almost always higher than 90%.     

Geographic variation of multiple paternity in the American lobster, Homarus americanus

We studied the frequency of multiple paternity for American lobster (Homarus americanus) at three Canadian sites differing in exploitation rate and mean adult size. The probability of detecting multiple paternity using four microsatellite loci and 100 eggs per female was in excess of 99% under various scenarios of paternal contribution. Overall, 13% of the 108 examined females carried a clutch sired by two or three males. Multiple paternity was observed at the two most exploited sites (11% at Magdalen Islands and 28% at Grand Manan Island), whereas single paternity only was observed at the least exploited site (Anticosti Island). Within populations females with a clutch sired by more than one male tended to be smaller than females with a clutch sired by a single male. Based on these and other findings, we postulate a link between female promiscuity and sperm limitation in the American lobster.     

Paternity testing and forensic DNA typing by multiplex STR analysis using ABI PRISM 310 Genetic Analyzer

Short tandem repeats (STRs) are widespread throughout the human genome and are a rich source of highly polymorphic markers which can be detected by PCR. To gain a better appreciation for how the polymorphism at a particular locus impacts the individual identity, the present study was undertaken to explore the use of 15 STR loci in forensic investigation and paternity testing. Multiplex STR typing was used to study the 15 STR loci (D8S1179, D21S11, D7S820, CSF1PO, D3S1358, TH01, D13S317, D16S539, D2S1338, D19S433, vWA, TPOX, D18S51, D5S818 and FGA) in addition to a gender identification marker, amelogenin, by capillary electrophoresis on 310 Genetic Analyzer. Samples from 85 trio and duo cases of disputed paternity were investigated. The data were analyzed to give information on paternity index, probability of paternity, frequency of number of exclusions and rate of mismatch at each STR locus. The method was also successfully applied to forensic personal identification in theft and murder cases. The results demonstrated that the STR typing is a reliable and robust tool for analyzing the forensic practice as well as for paternity testing. The advantages of using multiplex STR analysis over other conventional methods are discussed.     

Polymorphism profile of nine short tandem repeat Loci in the Han chinese

Nine short tandem repeat (STR) markers (D3S1358, VWA, FGA, THO1, TPOX, CSFIPO, D5S818, D13S317, and D7S820) and a sex-identification marker (Amel-ogenin locus) were amplified with multiplex PCR and were genotyped with a four-color fluorescence method in samples from 174 unrelated Han individuals in North China. The allele frequencies, genotype frequencies, heterozygosity, probability of discrimination powers, probability of paternity exclusion and Hardy-Weinberg equilibrium expectations were determined. The results demonstrated that the genotypes at all these STR loci in Han population conform to Hardy-Weinberg equilibrium expectations. The combined discrimination power (DP) was 1.05×10-10 within nine STR loci analyzed and the probability of paternity exclusion (EPP) was 0.9998. The results indicate that these nine STR loci and the Amelo-genin locus are useful markers for human identification, paternity and maternity testing and sex determination in forensic sciences.     

云南汉族群体FIBRA、DHFRP2、ACTBP2基因座多态性及其法医学应用

采用Amp-FLP分型方法,调查云南汉族群体FIBRA、DHFRP2、ACTBP2基因座的遗传多态性,并将其应用于法医学实践。200份EDTA抗凝血采自昆明地区无血缘关系汉族个体,采用酚—氯仿法提取DNA;法医物证实际检案及亲子鉴定检材取自昆明医学院法医系物证教研室检案,各种动物血痕取自动物中心,采用酚—氯仿法或Chelex法提取DNA,PCR扩增,非变 性聚丙烯酰胺凝胶垂直板电泳,硝酸银染色分型。结果表明,FIBRA、DHFRP2、ACTBP2基因座分别观察到15、7、13个等位基因,基因型数分别是57、25、61。3个STR基因座的杂合度（H）分别为：0.8940、0.8174、0.9130;多态信息容量（PIC）分别是:0.8908、0.8045、0.9117;个人识别力（Dp）分别是：0.9733、0.9416、0.9772;非父排除率（Epp）分别是：0.7994、0.6542、0.8348,基因型频率分布均符合Hardy-Weinberg平衡。20个家系调查结果表明,3个基因组均符合孟德尔遗传规律。 Genetic Polymorphism of FIBRA,DHFRP2 and ACTBP2 and Their Forensic Application in Yunnan Han Population JING Qiang,NIE Sheng-jie Department of Forensic Medicine,Kunming Medical College,Yunnan Province 650031,China Abstract:To investigate the genetic polymorphism of FIBRA,DHFRP2 and ACTBP2 in Yunnan Han population as well as their application in forensic science,EDTA-blood specimens were collected from 200 healthy individuals.The DNA were extracted either by the Chloro form,phenol method or by the Chelex-100 method.The PCR products were analyzed by PAG vertical electrophoresis,following by silver staining.All gene frequencies,discrimination power (DP),exclusion of paternity probability (EPP),heterozygosity (H),polymorphisms information content (PIC),matching probability (PM) as well as the Hardy-Weinberg test were calculated.The obtained data are beneficial in the understanding of population genetics of the three STR loci in Yunnan Han population and the results suggest that these loci are valuable genetic markers for paternity testing and personal identification in forensic science practice. Key words：short tandem repeat; Amp-FLP; genetic polymorphism     

Male-biased mutation rates and the overestimation of extrapair paternity: problem, solution, and illustration using thick-billed murres (Uria lomvia, Alcidae)

The widespread utility of hypervariable loci in genetic studies derives from the high mutation rate, and thus the high polymorphism, of these loci. Recent evidence suggests that mutation rates can be extremely high and may be male biased (occurring in the male germ-line). These two factors combined may result in erroneous overestimates of extrapair paternity, since legitimate offspring with novel alleles will have more mismatches with respect to the biological father than the biological mother. As mutations are male driven, increasing the number of hypervariable loci screened may simply increase the number of mismatches between fathers and their legitimate offspring. Here we describe a simple statistic, the probability of resemblance (PR), to distinguish between mismatches due to parental misassignment versus mutation in either sex or null alleles. We apply this method to parentage data on thick-billed murres (Uria lomvia), and demonstrate that, without considering either mutations or male-biased mutation rates, cases of extrapair paternity (7% in this study) would be grossly overestimated (14.5%-22%). The probability of resemblance can be utilized in parentage studies of any sexually reproducing species when allele or haplotype frequency data are available for putative parents and offspring. We suggest calculating this probability to correctly categorize legitimate offspring when mutations and null alleles may cause mismatches.     

Multiple paternity in an aggregate breeding amphibian: the effect of reproductive skew on estimates of male reproductive success

Aggregate, or explosive, breeding is widespread among vertebrates and likely increases the probability of multiple paternity. We assessed paternity in seven field-collected clutches of the explosively breeding spotted salamander (Ambystoma maculatum) using 10 microsatellite loci to determine the frequency of multiple paternity and the number of males contributing to a female's clutch. Using the Minimum Method of allele counts, multiple paternity was evident in 70% of these egg masses. Simple allele counts underestimate the number of contributing males because this method cannot distinguish multiple fathers with common or similar alleles. Therefore, we used computer simulations to estimate from the offspring genotypes the most likely number of contributing fathers given the distributions of allele frequencies in this population. We determined that two to eight males may contribute to A. maculatum clutches; therefore, multiple paternity is a common strategy in this aggregate breeding species. In aggregate mating systems competition for mates can be intense, thus differential reproductive success (reproductive skew) among males contributing to a female's clutch could be a probable outcome. We use our data to evaluate the potential effect of reproductive skew on estimates of the number of contributing males. We simulated varying scenarios of differential male reproductive success, ranging from equal contribution to high reproductive skew among contributing sires in multiply sired clutches. Our data suggest that even intermediate levels of reproductive skew decrease confidence substantially in estimates of the number of contributing sires when parental genotypes are unknown.     

Bayesian interval estimation of genetic relationships: application to paternity testing.

Using genetic marker data, we have developed a general methodology for estimating genetic relationships between a set of individuals. The purpose of this paper is to illustrate the practical utility of these methods as applied to the problem of paternity testing. Bayesian methods are used to compute the posterior probability distribution of the genetic relationship parameters. Use of an interval-estimation approach rather than a hypothesis-testing one avoids the problem of the specification of an appropriate null hypothesis in calculating the probability of paternity. Monte Carlo methods are used to evaluate the utility of two sets of genetic markers in obtaining suitably precise estimates of genetic relationship as well as the effect of the prior distribution chosen. Results indicate that with currently available markers a "true" father may be reliably distinguished from any other genetic relationship to the child and that with a reasonable number of markers one can often discriminate between an unrelated individual and one with a second-degree relationship to the child.     

Estimation of nonpaternity in the Mexican population of Nuevo Leon: A validation study with blood group markers

A method for estimating the general rate of nonpaternity in a population was validated using phenotype data on seven blood groups (A₁A₂BO, MNSs, Rh, Duffy, Lutheran, Kidd, and P) on 396 mother, child, and legal father trios from Nuevo León, Mexico. In all, 32 legal fathers were excluded as the possible father based on genetic exclusions at one or more loci (combined average exclusion probability of 0.694 for specific mother-child phenotype pairs). The maximum likelihood estimate of the general nonpaternity rate in the population was 0.118 ± 0.020. The nonpaternity rates in Nuevo León were also seen to be inversely related with the socioeconomic status of the families, i.e., the highest in the low and the lowest in the high socioeconomic class. We further argue that with the moderately low (69.4%) power of exclusion for these seven blood group systems, the traditional critical values of paternity index (PI ≥ 19) were not good indicators of true paternity, since a considerable fraction (307/364) of nonexcluded legal fathers had a paternity index below 19 based on the seven markers. Implications of these results in the context of genetic-epidemiological studies as well as for detection of true fathers for child-support adjudications are discussed, implying the need to employ a battery of genetic markers (possibly DNA-based tests) that yield a higher power of exclusion. We conclude that even though DNA markers are more informative, the probabilistic approach developed here would still be needed to estimate the true rate of nonpaternity in a population or to evaluate the precision of detecting true fathers. Am J Phys Anthropol 109:281–293, 1999. © 1999 Wiley-Liss, Inc.     

Hierarchical Patterns of Correlated Mating in Acacia Melanoxylon

Pollen of acacias is transported by insects as polyads, composite pollen grains. The polyad has enough pollen grains to fertilize all ovules within a flower and hence all seed within a pod may be full sibs. Isozyme markers were used to test this hypothesis in two populations of Acacia melanoxylon R.Br. The proportions of fruit pods with multiple paternity detected in two populations were 0.08 and 0.15. The proportions of fullsib pairs within pods estimated by the sibling pair method were 1 and 0.63 for the two populations. Comparison of the diploid paternal genotypes of pods of single paternity showed that the probability of a common pollen source for a pair of pods was high within globular clusters (0.35) or within inflorescences (0.46) but declined to 0.10 or 0.25 within the tree at random. Thus the reproductive system acted to reinforce a hierarchy of paternal correlation within each tree.     

Paternity exclusion and the paternity index for two linked loci

Algebraic expressions for the average exclusion frequency and the paternity index are derived for two linked loci, each with two alleles segregating in a population. The effects of recombination and gametic disequilibrium on these two statistics are discussed. As long as recombination is known to exist, the average exclusion frequency is similar for different recombination fractions. The paternity index, on the other hand, depends very much on both the recombination fraction and gametic disequilibrium. The effects of multiple alleles and dominance on these statistics are also briefly discussed.     

Estimating a Key Parameter of Mammalian Mating Systems: The Chance of Siring Success for a Mated Male

Studies of multiple paternity in mammals and other animal species generally report proportion of multiple paternity among litters, mean litter sizes, and mean number of sires per litter. It is shown how these variables can be used to produce an estimate of the probability of reproductive success for a male that has mated with a female. This estimate of male success is more informative about the mating system that alternative measures, like the proportion of litters with multiple paternity or the mean number of sires per litter. The probability of success for a mated male can be measured both theoretically and empirically, and gives an estimate of the intensity of sperm competition and of a male's “confidence of paternity” upon mating. The probability of success for mated males for ten “exemplar” species of mammals is estimated and they are compared for insights into the functioning of their mating systems.     

Estimations of the efficacy and reliability of paternity assignments from DNA microsatellite analysis of multiple-sire matings

It is important for bovine DNA testing laboratories to provide the cattle industry with accurate estimates of the efficacy and reliability of DNA tests offered so that end users of this technology can adequately assess the cost-benefits of testing. To address these issues for bovine paternity testing, paternity exclusion probability estimates were obtained from breed panel data and were predictive of the efficacy of the DNA tests used in 39 multiple-sire mating groups, involving 5960 calves and 505 bulls. Paternity testing of these mating groups has demonstrated that the majority involve a variable proportion of unknown sires and this impacts on the reliability of sire allocation. Mathematical models based on binomial or beta-binomial probability distributions were used to estimate the reliability of single-sire allocations from multiple-sire matings involving unknown sires. Reliability of 98-99% is achieved when the exclusion probability is 0.99 or greater, after allowing for up to 20% unknown sires. When the exclusion probability drops below 0.90 and there are 20% unknown sires, the reliability is poor, bringing into question the benefits of testing. This highlights the need for DNA testing laboratories to offer paternity tests with an exclusion power of at least 99%.