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.     

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.     

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.     

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.     

Extrapair paternity and the evolution of bird song

Bird song is usually considered to have evolved in the contextof sexual selection. Because extrapair paternity is a majorcomponent of sexual selection, mating advantages at the sociallevel for males that produce songs of high quality may be transformedinto higher success in extrapair paternity. Therefore, maleswith longer and more complex songs should suffer less from extrapairpaternity intraspecifically, whereas species with high ratesof extrapair paternity, reflecting intense sperm competition,should produce more elaborate songs. Although some intraspecificstudies demonstrated a negative link between features of songsand extrapair paternity in own nest, others failed to detectsuch a relationship. Contrary to expectation, a meta-analysisof all studies revealed no significant intraspecific evidencefor songs being associated with extrapair paternity. In addition,in comparative analyses based on generalized least squares (GLS)models, we found that no measures of song complexity and temporaloutput were significantly related to extrapair paternity interspecifically,even when potentially confounding factors such as social matingsystem, life history, migration, habitat, or sexual dichromatismwere held constant. Only plumage dichromatism was significantlyrelated to extrapair paternity. The absence of both intra- andinterspecific relationships between measures of song variabilityand extrapair paternity suggests that factors other than postmatingsexual selection have been the important evolutionary forcesshaping differences in song.     

Colour bands, mate choice and paternity in the bluethroat

Studies of several bird species have shown that coloured leg bands may affect a male's success in mate attraction and/or mating competition. From a colour band experiment in the field, we have previously reported that male bluethroats, Luscinia s. svecica, with blue and orange bands (BO males) guarded their mates less intensely at the peak of female fertility, and spent more time advertising for additional mates, than males banded with non-BO colours. These responses indicated that BO males experienced less threat to their paternity than did non-BO males, possibly mediated through an increased attractiveness. Here we present paternity analyses of the broods from the field study and test whether there were differences between the two male groups in within-pair or extrapair paternity. There were no significant differences between the two groups of males in paternity, suggesting effective male protection of paternity. However, extrapair paternity was infrequent in the 2 years of the field experiment; hence, the power in detecting effects on paternity does not allow a definitive conclusion on this issue. We also conducted an aviary experiment in which females were given the choice between a BO male and a non-BO male, to test whether females had preferences for particular colour bands. Females did not associate more with BO males, as would have been expected if these males were more attractive in social mate choice. Our results suggest that the effects of colour bands on social mate choice and paternity are, at best, weak. Copyright 2000 The Association for the Study of Animal Behaviour.     

Statistical approaches to paternity analysis in natural populations and applications to the North Atlantic humpback whale

We present a new method for paternity analysis in natural populations that is based on genotypic data that can take the sampling fraction of putative parents into account. The method allows paternity assignment to be performed in a decision theoretic framework. Simulations are performed to evaluate the utility and robustness of the method and to assess how many loci are necessary for reliable paternity inference. In addition we present a method for testing hypotheses regarding relative reproductive success of different ecologically or behaviorally defined groups as well as a new method for estimating the current population size of males from genotypic data. This method is an extension of the fractional paternity method to the case where only a proportion of all putative fathers have been sampled. It can also be applied to provide abundance estimates of the number of breeding males from genetic data. Throughout, the methods were applied to genotypic data collected from North Atlantic humpback whales (Megaptera novaeangliae) to test if the males that appear dominant during the mating season have a higher reproductive success than the subdominant males.     

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.     

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。     

Inheritance of the cephalic index

Using data from 462 paternity examinations, it was found that, (1) The cephalic index of an infant does not correlate at all closely to those of his or her parents as a pair, nor with that of the like sexed parent, but (2) The cephalic index of an infant, in a majority of cases, falls within a single percentage point of one of the two parents. Because of this it is hypothesized that this aspect of skullshape in inherited in a unitary fashion.     

Female infidelity and genetic compatibility in birds: the role of the genetically loaded raffle in understanding the function of extrapair paternity

Despite two decades of research into over one hundred species, the function of extrapair paternity to female birds remains unclear. Recent studies have demonstrated patterns between extrapair paternity and the genetic similarity of females with social partners and extrapair males. We believe that selection on females to gain genetically compatible fathers for their offspring offers a possible general explanation for the function of extrapair paternity. The idea of sexual selection being driven by genetic compatibility is widely considered by workers on other taxa but has been largely ignored by studies of birds. Genetic compatibility could be optimised by females through a behavioural process before copulation or through a postcopulatory process. Postcopulatory processes such as cryptic female choice have been recently demonstrated in birds and would allow female birds to use a 'genetically loaded raffle' to target compatible genes through sperm competition. We discuss the general weaknesses of studies of extrapair paternity to date and suggest a number of avenues for future research that will help to elucidate the function of extrapair paternity and widespread genetic polyandry in birds.     

Extra-pair paternity and song characteristics in the willow warbler Phylloscopus trochilus

Complexity in bird song is often argued to be advantageous in processes of sexual selection, and numerous studies show that characteristics of song are associated with increased success in territory defence or mate attraction. Less evidence exists on the relationship between bird song characteristics and patterns of extra-pair paternity. We tested whether males that suffered from extra-pair paternity differed in song characteristics from males with no extra pair paternity in the willow warbler Phylloscopus trochilus . In the Scottish population that we studied, we found that 23.5% of young were not related to the social father, and that 47% of nests contained at least one extra pair young. Older males were less likely to suffer paternity loss. While song repertoire size was not related to loss of paternity, males with short songs suffered higher paternity loss than males with long songs. Although arrival date is a good correlate of social mate choice in this population, it was not related to extra-pair paternity. These results suggest that females use song length, or a trait correlated with this song characteristic, as a cue for choosing extra-pair partners in this species, or alternatively that variance in the success of mate guarding or female coercion is related to this song variable.     

A self-consistent approach to paternity and parental effort

We review the relationship between optimal parental effort and paternity, and emphasize the need for a self-consistent approach. A fundamental consistency condition is what we refer to as the conservation of paternity. Every offspring has exactly one father. If a male has a paternity of less than unity, then another male or other males must have gained the lost paternity. Our approach also emphasizes that paternity emerges as the result of interactions between males and females. From this viewpoint, if paternity changes it is because some aspect of the interaction changes, and the correlation between effort and paternity depends on the aspect that has changed. This has implications for comparative analyses of paternity. The conclusions that are drawn about the correlation between effort and paternity within a population depend on, for example, the types of male in the population and how their abilities are correlated. It is easy to construct models that predict negative correlations between effort and paternity.     

Female mate choice predicts paternity success in the absence of additive genetic variance for other female paternity bias mechanisms in Drosophila serrata

After choosing a first mate, polyandrous females have access to a range of opportunities to bias paternity, such as repeating matings with the preferred male, facilitating fertilization from the best sperm or differentially investing in offspring according to their sire. Female ability to bias paternity after a first mating has been demonstrated in a few species, but unambiguous evidence remains limited by the access to complex behaviours, sperm storage organs and fertilization processes within females. Even when found at the phenotypic level, the potential evolution of any mechanism allowing females to bias paternity other than mate choice remains little explored. Using a large population of pedigreed females, we developed a simple test to determine whether there is additive genetic variation in female ability to bias paternity after a first, chosen, mating. We applied this method in the highly polyandrous Drosophila serrata, giving females the opportunity to successively mate with two males ad libitum. We found that despite high levels of polyandry (females mated more than once per day), the first mate choice was a significant predictor of male total reproductive success. Importantly, there was no detectable genetic variance in female ability to bias paternity beyond mate choice. Therefore, whether or not females can bias paternity before or after copulation, their role on the evolution of sexual male traits is likely to be limited to their first mate choice in D. serrata.     

Certainty of paternity and paternal effort in the collared flycatcher

Models of optimal parental investment predict that variationin certainty of paternity can affect the optimal level of paternalinvestment when a male's expected paternity in different nestingattempts is not fixed throughout his lifetime. Several attemptsto test this prediction experimentally in monogamous birds havefailed to induce a reduction in care by males. This may be becausethe method used, detaining males, is a poor model for what happenswhen a male's certainly of paternity is naturally reduced. Wecaught and detained female collared flycatchers Ficedula albicollisfor 1 h immediately after laying on one or two occasions inan attempt to induce variation in certainty of paternity forthe males they were mated to. By capturing females immediatelyafter laying we hoped to exploit the existence of an "inseminationwindow" since males should be very sensitive to female absenceduring this period. The general effect of the experimental manipulationwas consistent with reduced certainty of paternity: males respondedby reducing their level of paternal care to nestlings, and malesmated to females that had been caught on one morning fed nestlingssignificantly less often and made a smaller share of feedingvisits than males mated to control females. The effects of theexperiment were generally weak, however, and we argue that certaintyof paternity may be fixed well before egg laying, in which caseexperimental manipulations are unlikely to have large effects.It is difficult to predict die effects of natural variationin certainty of paternity on levels of male paternal care becausedifferential allocation by females mated to attractive malesmay act in the opposite direction     

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 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.     

Sperm length variation as a predictor of extrapair paternity in passerine birds

Background

The rate of extrapair paternity is a commonly used index for the risk of sperm competition in birds, but paternity data exist for only a few percent of the approximately 10400 extant species. As paternity analyses require extensive field sampling and costly lab work, species coverage in this field will probably not improve much in the foreseeable future. Recent findings from passerine birds, which constitute the largest avian order (∼5 900 species), suggest that sperm phenotypes carry a signature of sperm competition. Here we examine how well standardized measures of sperm length variation can predict the rate of extrapair paternity in passerine birds.

Methodology/Principal Findings

We collected sperm samples from 55 passerine species in Canada and Europe for which extrapair paternity rates were already available from either the same (n = 24) or a different (n = 31) study population. We measured the total length of individual spermatozoa and found that both the coefficient of between-male variation (CV_bm) and within-male variation (CV_wm) in sperm length were strong predictors of the rate of extrapair paternity, explaining as much as 65% and 58%, respectively, of the variation in extrapair paternity among species. However, only the CV_bm predictor was independent of phylogeny, which implies that it can readily be converted into a currency of extrapair paternity without the need for phylogenetic correction.

Conclusion/Significance

We propose the CV_bm index as an alternative measure to extrapair paternity for passerine birds. Given the ease of sperm extraction from male birds in breeding condition, and a modest number of sampled males required for a robust estimate, this new index holds a great potential for mapping the risk of sperm competition across a wide range of passerine birds.     

Demographic correlates of paternity confidence and pregnancy outcomes among Albuquerque men

We examine the demographic correlates of paternity confidence, or men's assessment of the likelihood that they are the genetic father of a particular child. Evolutionary theory predicts that men will provide less parental investment for putative genetic offspring who are unlikely to be their actual offspring, but confidence of paternity has not been as extensively examined as its importance would merit. Using self-reported data on paternity confidence in 3,360 pregnancies reported by men living in Albuquerque, New Mexico, we find that low paternity confidence is more common among unmarried couples and for unplanned pregnancies. We also find that men are more likely not to state paternity confidence (i.e., they refuse to answer the question) if a pregnancy is unplanned. We additionally examine the pregnancy outcomes associated with confidence of paternity. We find that low paternity confidence pregnancies are significantly more likely to be aborted, and pregnancies for which paternity confidence is unstated are more likely to be aborted or to miscarry. Both abortion and miscarriage are associated with unmarried couples, with unplanned pregnancies, and with couples who have fewer children together.     

Matrilateral biases in the investment of aunts and uncles

Gaulin, McBurney, and Brakeman-Wartell (1997) found that college students reported both matrilateral and sex biases in the investment of aunts and uncles (aunts invested more than uncles). They interpreted the matrilateral bias as a consequence of paternity uncertainty. We replicated that study with Orthodox Jewish college students, selected because they come from a population we presume to have higher paternity certainty than the general population. The Orthodox sample also showed matrilateral and sex biases. Comparing the two data sets, the Orthodox sample reported more investment, and slightly less matrilateral and sex biases, but the differences were not statistically significant. We did find an interaction between sex of relative and group membership, resulting from greater investment by Orthodox uncles. We interpret the results as reflecting the operation of a facultative investment mechanism whose upper limit is tuned to the maximum levels of paternity certainty found in ancestral environments. Lack of a difference in matrilateral bias between groups may result from levels of paternity certainty near to, or above, that maximum in both groups.