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.     

A note on positive identification of paternity by using genetic markers

To see the efficiency of the statistical methods for positive identification of fathers using genetic markers, the statistical properties of the paternity index are studied algebraically and numerically. It is found that the currently used statistical methods are not powerful enough to discriminate between true fathers and non-excluded non-fathers, and, more often than not, may lead to false attributions of paternity. It is, therefore, suggested that exclusion of paternity is the only conclusive evidence that can be accepted by courts of law until better methods are devised.     

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.     

Paternity testing and civil affairs

Paternity testing is based on the genetic analysis of human polymorphism markers. Recently, studies of short tandem repeats (STR) after DNA amplification have led to increased standardisation and performance of the tests and have become the reference method. Thus, the results expressed in terms of probability achieve a high level of certitude, for example the probability of paternity can exceed 99.999%. All these analyses, which are performed according to a specific quality assurance program, are a very useful tool for courts of justice.     

Genetic Patterns of Paternity and Testes Size in Mammals

Background

Testes size is used as a proxy of male intrasexual competition, with larger testes indicative of greater competition. It has been shown that in some taxa, social mating systems reflect variance in testes size, but results are not consistent, and instead it has been suggested that genetic patterns of mating may reflect testes size. However, there are different measures of genetic patterns of mating. Multiple paternity rates are the most widely used measure but are limited to species that produce multi-offspring litters, so, at least for group living species, other measures such as loss of paternity to males outside the social group (extra group paternity) or the proportion of offspring sired by the dominant male (alpha paternity) might be appropriate. This study examines the relationship between testes size and three genetic patterns of mating: multiple paternity, extragroup paternity and alpha paternity.

Methodology/Principal Findings

Using data from mammals, phylogenetically corrected general linear models demonstrate that both multiple paternity and alpha paternity, but not extra group paternity, relate to testes size. Testes size is greater in species with high multiple paternity rates, whereas the converse is found for alpha paternity. Additionally, length of mating season, ovulation mode and litter size significantly influenced testes size in one model.

Conclusions/Significance

These results demonstrate that patterns of mating (multiple paternity and alpha paternity rates) determined by genetic analysis can provide reliable indicators of male postcopulatory intrasexual competition (testes size), and that other variables (length of mating season, ovulation mode, litter size) may also be important.     

Faithful without care: the evolution of monogyny

The traditional paradigm of male polygamy and female monogamy has been replaced by the recognition that both sexes may typically mate with several partners. As a consequence, much attention has focused on the evolution of polyandry, while the evolutionary significance of monogyny (male monogamy) remains poorly understood. Monogyny, a taxonomically widespread mating system that includes dramatic examples of male self-sacrifice, is predicted when the benefits of paternal investment exceed those of searching for additional mates. However, monogyny also occurs in animals lacking paternal investment, instead representing a form of paternity protection. It has been suggested that such mating systems are expected where the costs of mate search for males are high. However, this argument fails to recognize that if there is a low probability of a male finding a mate, then there may be a high probability that he will not need to defend his paternity. Using a mathematical model, we show that monogyny as a means of increasing paternity is favored when the sex ratio is male biased, but not necessarily by high search costs. The importance of a male-biased sex ratio for the evolution of monogyny is supported by various empirical studies.     

Fecundity and MHC affects ejaculation tactics and paternity bias in sand lizards

We demonstrate that extending copulation enhances probability of paternity in sand lizards and that determinants of copulation duration depend on a males' mating order (first or second). First males, with no information on presence of rivals, extend copulation when mating with a more fecund female. Second males, however, adjust copula duration in relation to a first male's relatedness with his female, which there is reason to believe can be deduced from the MHC-related odor of the copulatory plug. Male-female relatedness negatively influences a male's probability of paternity, and when second males are in a favored role (i.e., the first male is the one more closely related to the female), second males transfer larger ejaculates, resulting in higher probability of paternity. This result corroborates predictions from recent theoretical models on sperm expenditure theory incorporating cryptic female choice and sexual conflict. More specifically, the results conform to a "random roles" model, which depicts males as being favored by some females and disfavored by others, but not to a "constant-type" model, in which a male is either favored or disfavored uniformly by all females in a population.     

Exclusion probabilities for pedigree testing farm animals

Pedigree testing, using genetic markers, may be undertaken for a variety of situations, of which the classical paternity testing is only one. This has not always been made clear in the literature. Exclusion probabilities associated with various testing situations, including the use of autosomal or X-linked codominant marker systems with any number of alleles, are presented. These formulae can be used to determine the appropriate exclusion probability for the situation being investigated. One such situation is where sire groups of progeny are to be verified without knowledge of the dams' genotypes, in which case the classical paternity exclusion probability is too high, and if used may result in an optimistic declaration about the progeny that have not been excluded. On the other hand, if mating pairs are known then incorrect progeny can be excluded at a higher rate than suggested by paternity exclusion calculations. The formulae also assist in determining the usefulness of X-linked markers, particularly if the pedigree checks involve progeny of only one sex. A system of notation that is useful for the algebraic manipulation of genetic probabilities, including exclusion probabilities as presented here, is also given.     

Exclusion probabilities for single-locus paternity analysis when related males compete for matings

The statistical power of single-locus paternity analyses has previously been assessed by calculating an expected exclusion probability ( E ), the probability of excluding a randomly chosen nonfather. This E -statistic assumes that putative sires are a random selection of individuals from a panmictic study population. In species that display male natal philopatry, closely related individuals may be the principal competitors for paternity. In such structured populations, the E statistic will overestimate exclusion probability because males competing for paternity are more closely related than males chosen randomly from the population. A suite of loci thought to be sufficient for a panmictic population may frequently incorrectly assign paternity to close relatives of true sires. This study provides equations for calculating the expected probability of excluding a close male relative of the genetic sire ( Erel ) for any genotyping system that uses codominant markers. We also describe the use of Monte Carlo modelling to estimate exclusion probabilities when multiple male relatives compete for paternity. We show that the utility of a set of codominant markers will depend on the breeding behaviour and social system of the species in question.     

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.     

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%.     

Empirical validation of the Essen-Möller probability of paternity.

The validity of the Essen-Möller formulation probability of paternity is supported by demonstrating its correctness in a model genetic system--the ABO system. An analysis was made of 1,393 paternity cases typed uniformly for HLA-A and -B, ABO, Rh, and MNSs, in which the mother named one man only as the child''s father and in which both mother and putative father identified themselves as Caucasian. For purposes of analysis, putative fathers not excluded from paternity by the four systems tested were regarded as actual fathers. The joint distribution of observed triplets of ABO phenotypes is shown to be statistically consistent with expected values, and the fractions of "true" fathers for a given triplet closely approximated the probability of paternity calculated using a realistic prior probability. Recent allegations of fallaciousness of the method by Li and Chakravarty and Aickin are discussed in terms of the results presented.     

An expository review of two methods of calculating the paternity probability.

There are two methods for calculating the posttests probability of paternity, viz., the nonexclusion probability method (E method) and the paternity index method (lambda method). This report reviews these two methods and explains the reasons behind them, in the hope that it might alleviate the current controversy between the advocates of these two methods. The emphasis throughout the paper is on exposition, using simple examples to illustrate certain principles or properties. A discussion follows the presentation of the two methods. The calculation of the paternity index is based on the genotype (or phenotype) of the accused man; and the value of the paternity index remains the same whether the accusation itself is true or false.     

Rapid and efficient resolution of parentage by amplification of short tandem repeats.

Short tandem repeat (STR) loci are highly informative polymorphic loci that are gaining popularity for identity testing. We have conducted parentage testing by using nine STR loci on 50 paternity trios that had been previously tested using VNTR loci. These nine unlinked STR loci are amplified in three multiplex reactions and, when examined for genetic informativeness, provide a combined average power of exclusion of 99.73% (Caucasian data). The informative value of the selected loci is based on extensive STR typing of four racial/ethnic populations. In 37 of the 50 cases, paternity could not be excluded by any of the loci. In the remaining 13 cases, paternity was excluded by at least two of the STR markers. The probability of paternity calculated for the alleged father of each matching trio was > 99% in 36 of the 37 inclusion cases. All data agreed with the results reported using VNTR loci and conventional Southern technology. Our studies validate the use of DNA typing with STR loci for parentage testing, thus providing an accurate, highly sensitive, and rapid assay.     

社会性单配制鸟类婚外父权的发生和影响因素

一雌一雄单配制鸟类中,雌性个体与配偶外雄性发生交配的行为称为婚外交配,继而导致了婚外受精产生婚外子代的现象称为产生了婚外父权。婚外父权广泛存在于鸟类中,针对其发生和影响因素已经成为了鸟类行为生态学研究的热点。本文收集了近十年社会性单配制鸟类婚外父权方面的研究文献,从婚外父权的发生及其影响因素两个方面综述了单配制鸟类婚外父权的研究进展。婚外父权发生原因的探讨主要包括：1、从两性的角度探讨雌雄两性在婚外行为中不同的进化繁殖策略。雄性策略旨在增加自身的繁殖输出;有关雌性策略则提出了确保受精假说、食物供给假说、遗传利益假说等,但目前尚存争议;2、在遗传利益假说中较常见的又分为3个假说：“优秀基因”假说、“遗传相容性”假说和“遗传多样性”假说,该三种假说是针对雌性从遗传方面获得的利益而提出的,不断有报道指出雌性配偶选择会被潜在的雄性遗传特性所影响;3、非遗传利益——母系效应影响婚外父权的进化。一些研究指出遗传质量参数,如体重、身体大小、存活率和免疫应答等方面可能会存在母系效应。婚外父权发生的影响因素这里主要指环境因素,包括繁殖同步性、繁殖密度、栖息地环境、产卵及孵化时机等。由于物种不同,受到环境压力不同,导致婚外父权发生率千差万别。最后本文针对未来的研究方向做出了展望。尽管近十年的研究进一步解释了鸟类婚外父权现象,但是该领域仍然存在并且产生了许多新的未解决的问题,而相关实验操作和理论的完善是深入探讨这些问题的关键。     

HOW WELL DO MATING FREQUENCY AND DURATION PREDICT PATERNITY SUCCESS IN THE POLYGYNANDROUS WATER STRIDER AQUARIUS REMIGIS?

Abstract.— The relationship between mating success and paternity success is a key component of sexual selection but has seldom been estimated for species in which both sexes mate with many partners (polygynandry). We used a modification of Parker's sterile male technique to measure this relationship for the water strider Aquarius remigis in 47 laboratory populations simulating natural conditions of polygynandry. We also tested the hypothesis that prolonged copulation, a characteristic of this species, enhances paternity success. Mating behavior and paternity success were assayed for four days while males and females freely interacted. Paternity success was also assayed for an additional 7 days when females were isolated from males. Mating success significantly predicted paternity success and accounted for ≤ 36% of the variance. Copulation duration was negatively related to both mating success and paternity success and did not explain any of the residual variance in paternity success. Thus, we found no evidence that prolonged copulation functions as a paternity assurance strategy in this species. Comparisons of sterile and fertile males suggested that paternity success is directly influenced by the quantity of sperm transferred. Our results support previous studies that have used mating success to estimate sexual selection, but also highlight the potential importance of sperm competition and other postinsemination processes.     

Identification of visual paternity cues in humans

Understanding how individuals identify their relatives has implications for the evolution of social behaviour. Kinship cues might be based on familiarity, but in the face of paternity uncertainty and costly paternal investment, other mechanisms such as phenotypic matching may have evolved. In humans, paternal recognition of offspring and subsequent discriminative paternal investment have been linked to father–offspring facial phenotypic similarities. However, the extent to which paternity detection is impaired by environmentally induced facial information is unclear. We used 27 portraits of fathers and their adult sons to quantify the level of paternity detection according to experimental treatments that manipulate the location, type and quantity of visible facial information. We found that (i) the lower part of the face, that changes most with development, does not contain paternity cues, (ii) paternity can be detected even if relational information within the face is disrupted and (iii) the signal depends on the presence of specific information rather than their number. Taken together, the results support the view that environmental effects have little influence on the detection of paternity using facial similarities. This suggests that the cognitive dispositions enabling the facial detection of kinship relationships ignore genetic irrelevant facial information.     

Individual specific DNA fingerprints from a hypervariable region probe: alpha-globin 3′HVR

Summary A probe detecting a hypervariable region (HVR) 3 to the alpha globin locus on chromosome 16 has been used to produce DNA fingerprints. Segregation analysis has revealed multiple, randomly dispersed DNA fragments inherited in a Mendelian fashion with minimal allelism and linkage. The fingerprints are highly polymorphic (probability of chance association between random individuals «10^-14). The probe is, therefore, a powerful discriminating tool: it is envisaged that this probe will have forensic applications, including paternity cases, and will be informative in linkage analysis.     

Male investment under changing conditions among chacma baboons at Suikerbosrand.

Male investment in infant baboons was measured by frequency of carrying from 1978 through 1985. A series of hypotheses was generated and tested with the carrying data, based on the assumptions that: male baboons have some capacity to estimate likelihood of paternity; where paternity probability is greater, males will invest more, where potential benefits to males or infants are higher, males will invest more. Carrying was affected by probability of paternity, availability of estrous females, season of conception and season of carrying, infant age, subtrooping, and predation risk. Infants were carried by probable fathers, siblings, mothers' siblings, and unrelated "suitors." Male investment increased female reproductive fitness: carried infants were more likely to survive, and mothers of carried infants had shorter interbirth intervals. Males appeared to estimate paternity both by observed copulations by other males and by the likelihood that copulations could have occurred without being observed. Male care of infant baboons may also be affected by female choice among males, the distribution of probable infants in time, male tenure at alpha rank, the number of males per troop, the probability of infanticide, and energy demands. Subtrooping seems to be historically crucial, by initially creating a situation in which some males have high paternity certainty.     

Gene flow estimation with microsatellites in a Malagasy seed orchard of Eucalyptus grandis

Eucalyptus grandis has a mixed-mating reproductive system. Malagasy Eucalyptus seed orchards were established 15 years ago with two aims both based on panmixia: open-pollinated seed production and genetic improvement. The panmixia hypothesis has never been confirmed in the seed orchard. From a seedling seed-orchard stand comprising 349 trees and using data obtained with six selected microsatellite markers, paternity analysis was performed for 724 offspring collected on 30 adult trees. Paternity assignment, based on exclusion procedures and likelihood-ratio method, was achieved with high accuracy; the exclusion probability value was 0.997. The outcrossing rate was very high (96.7%). More than 50% of potential male trees (199 out of 349) in the seed orchard contributed to pollination for 440 offspring of 30 progenies (8.6% of the basic population). The pollination rate from outside the seed orchard was high (39.2%), but might be due to the small size of this seed orchard. This study showed that "panmixia-like pollination" can be assumed.