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.     

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.     

Measuring the genetic structure of the pollen pool as the probability of paternal identity

Contemporary pollen flow in forest plant species is measured by the probability of paternal identity (PPI) for two randomly sampled offspring, drawn from a single female, and contrasting that with PPI for two random offspring, drawn from different females. Two different estimation strategies have emerged: (a) an indirect approach, using the 'genetic structure' of the pollen received by different mothers and (b) a direct approach, based on parentage analysis. The indirect strategy is somewhat limited by the assumptions, but is widely useful. The direct approach is most appropriate where a large majority of the true fathers can be identified exactly, which is sometimes possible with high-resolution SSR markers. Using the parentage approach, we develop estimates of PPI, showing that the obvious estimates are severely biased, and providing an unbiased alternative. We then illustrate the methods with SSR data from a 36-tree isolated population of Pinus sylvestris from the Meseta region of Spain, for which categorical paternity assignment was available for over 95% of offspring. For all the females combined, we estimate that PPI=0.0425, indicating uneven male reproductive contributions. Different (but overlapping) arrays of males pollinate different females, and for the average female, PPI=0.317, indicating substantial 'pollen structure' for the population. We also relate the direct measures of PPI to those available from indirect approaches, and show that they are generally comparable.     

A bayesian framework for parentage analysis: the value of genetic and other biological data

We develop fractional allocation models and confidence statistics for parentage analysis in mating systems. The models can be used, for example, to estimate the paternities of candidate males when the genetic mother is known or to calculate the parentage of candidate parent pairs when neither is known. The models do not require two implicit assumptions made by previous models, assumptions that are potentially erroneous. First, we provide formulas to calculate the expected parentage, as opposed to using a maximum likelihood algorithm to calculate the most likely parentage. The expected parentage is superior as it does not assume a symmetrical probability distribution of parentage and therefore, unlike the most likely parentage, will be unbiased. Second, we provide a mathematical framework for incorporating additional biological data to estimate the prior probability distribution of parentage. This additional biological data might include behavioral observations during mating or morphological measurements known to correlate with parentage. The value of multiple sources of information is increased accuracy of the estimates. We show that when the prior probability of parentage is known, and the expected parentage is calculated, fractional allocation provides unbiased estimates of the variance in reproductive success, thereby correcting a problem that has previously plagued parentage analyses. We also develop formulas to calculate the confidence interval in the parentage estimates, thus enabling the assessment of precision. These confidence statistics have not previously been available for fractional models. We demonstrate our models with several biological examples based on data from two fish species that we study, coho salmon (Oncorhychus kisutch) and bluegill sunfish (Lepomis macrochirus). In coho, multiple males compete to fertilize a single female's eggs. We show how behavioral observations taken during spawning can be combined with genetic data to provide an accurate calculation of each male's paternity. In bluegill, multiple males and multiple females may mate in a single nest. For a nest, we calculate the fertilization success and the 95% confidence interval of each candidate parent pair.     

Comparing two methods of calculating the probability of absence: the Negative Predictive Value, and a Credible Interval

If sampling fails to reveal the presence of an invasive species with potential to actually be present, how may we calculate the probability that it is truly absent, e.g. didymo (Didymosphenia geminate) in New Zealand’s North Island. In statistical terms this is a Bayesian question, concerning the probability of a hypothesis (presence/absence), given the obtained data (all results negative). “Classical” theory doesn’t answer this question, because it inverts the required considerations by calculating the probability of all samples being absent if the invasive was actually present. Accordingly, the Bayesian view of “probability” must be adopted in order to answer the question. That definition differs from classical probability in that it always includes an element of subjective belief, particularly in the choice of an appropriate “prior probability” (this is our belief as to the presence of the invasive organism before collecting new data). Bayesian methods can therefore be somewhat controversial – but we seem forced to use them. One Bayesian approach is to use the “Negative Predictive Value”, in which a point estimate of the probability of presence (or absence) prior to sample collection (the “prior probability”) is updated using data once collected using Bayes’ rule. This is in common use in medical studies, where the prior probability is the background disease prevalence, which is generally well understood. It is sometimes used in environmental ‘hot-spot’ investigations. An alternative approach is to recognise the uncertainty in the prior belief by using a distribution of prior probability and updating that using data once collected to give a Credible Interval in which the probability of presence (or absence) should lie – if all our assumptions have been satisfied. We will compare the merits of these approaches considering didymo, southern salt marsh mosquito (Aedes camptorhychus) and the sea squirt Styela clava.     

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.     

Genetic relatedness to sisters' children has been underestimated

Males of many species help in the care and provisioning of offspring, and these investments often correlate with genetic relatedness. For example, many human males invest in the children of sisters, and this is especially so where men are less likely to share genes with children of wives. Although this makes qualitative sense, it has been difficult to support quantitatively. The prevailing model predicts investment in children of sisters only when paternity confidence falls below 0.268. This value is often seen as too low to be credible; so investment in sisters'' children represents an unsolved problem. I show here that the prevailing model rests on a series of restrictive assumptions that underestimate relatedness to sisters'' children. For this reason, it understates the fitness payoff to men who invest in these children. This effect can be substantial, especially in societies with low confidence in paternity. But this effect cannot be estimated solely from confidence in paternity. One must also estimate the probability that two siblings share the same father.     

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.     

A simple Bayesian analysis of misclassified binary data with a validation substudy

A two-stage Bayesian method is presented for analyzing case-control studies in which a binary variable is sometimes measured with error but the correct values of the variable are known for a random subset of the study group. The first stage of the method is analytically tractable and MCMC methods are used for the second stage. The posterior distribution from the first stage becomes the prior distribution for the second stage, thus transferring all relevant information between the stages. The method makes few distributional assumptions and requires no asymptotic approximations. It is computationally fast and can be run using standard software. It is applied to two data sets that have been analyzed by other methods, and results are compared.     

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.     

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%.     

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.     

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.     

Simulation analysis to test the influence of model adequacy and data structure on the estimation of genetic parameters for traits with direct and maternal effects

Simulations were used to study the influence of model adequacy and data structure on the estimation of genetic parameters for traits governed by direct and maternal effects. To test model adequacy, several data sets were simulated according to different underlying genetic assumptions and analysed by comparing the correct and incorrect models. Results showed that omission of one of the random effects leads to an incorrect decomposition of the other components. If maternal genetic effects exist but are neglected, direct heritability is overestimated, and sometimes more than double. The bias depends on the value of the genetic correlation between direct and maternal effects. To study the influence of data structure on the estimation of genetic parameters, several populations were simulated, with different degrees of known paternity and different levels of genetic connectedness between flocks. Results showed that the lack of connectedness affects estimates when flocks have different genetic means because no distinction can be made between genetic and environmental differences between flocks. In this case, direct and maternal heritabilities are under-estimated, whereas maternal environmental effects are overestimated. The insufficiency of pedigree leads to biased estimates of genetic parameters.     

The statistical basis of public policy: a paradigm shift is overdue.

The recent controversy over the increased risk of venous thrombosis with third generation oral contraceptives illustrates the public policy dilemma that can be created by relying on conventional statistical tests and estimates: case-control studies showed a significant increase in risk and forced a decision either to warn or not to warn. Conventional statistical tests are an improper basis for such decisions because they dichotomise results according to whether they are or are not significant and do not allow decision makers to take explicit account of additional evidence--for example, of biological plausibility or of biases in the studies. A Bayesian approach overcomes both these problems. A Bayesian analysis starts with a "prior" probability distribution for the value of interest (for example, a true relative risk)--based on previous knowledge--and adds the new evidence (via a model) to produce a "posterior" probability distribution. Because different experts will have different prior beliefs sensitivity analyses are important to assess the effects on the posterior distributions of these differences. Sensitivity analyses should also examine the effects of different assumptions about biases and about the model which links the data with the value of interest. One advantage of this method is that it allows such assumptions to be handled openly and explicitly. Data presented as a series of posterior probability distributions would be a much better guide to policy, reflecting the reality that degrees of belief are often continuous, not dichotomous, and often vary from one person to another in the face of inconclusive evidence.     

Exclusion probabilities obtainable by biochemical polymorphisms in dogs

General formulae are given to calculate the exclusion probabilities in false paternity and parentage cases by means of gene loci with an arbitrary number of alleles whereas in paternity cases an arbitrary number of offspring per litter is considered additionally.
By aid of these formulae and on the basis of the allele frequencies of four blood protein and enzyme systems the probabilities of excluding incorrect paternity and parentage are calculated in seven German dog breeds. The results are tabulated and discussed.
It can be shown that the exclusion probability in false paternity cases increases distinctly with an increasing number of offspring per litter and its maximum is nearly attained if 5 offspring are examined. Therefore it is of value to consider entire litters in paternity controls in dogs.     

Tail paradox, partial identifiability, and influential priors in Bayesian branch length inference

Recent studies have observed that Bayesian analyses of sequence data sets using the program MrBayes sometimes generate extremely large branch lengths, with posterior credibility intervals for the tree length (sum of branch lengths) excluding the maximum likelihood estimates. Suggested explanations for this phenomenon include the existence of multiple local peaks in the posterior, lack of convergence of the chain in the tail of the posterior, mixing problems, and misspecified priors on branch lengths. Here, we analyze the behavior of Bayesian Markov chain Monte Carlo algorithms when the chain is in the tail of the posterior distribution and note that all these phenomena can occur. In Bayesian phylogenetics, the likelihood function approaches a constant instead of zero when the branch lengths increase to infinity. The flat tail of the likelihood can cause poor mixing and undue influence of the prior. We suggest that the main cause of the extreme branch length estimates produced in many Bayesian analyses is the poor choice of a default prior on branch lengths in current Bayesian phylogenetic programs. The default prior in MrBayes assigns independent and identical distributions to branch lengths, imposing strong (and unreasonable) assumptions about the tree length. The problem is exacerbated by the strong correlation between the branch lengths and parameters in models of variable rates among sites or among site partitions. To resolve the problem, we suggest two multivariate priors for the branch lengths (called compound Dirichlet priors) that are fairly diffuse and demonstrate their utility in the special case of branch length estimation on a star phylogeny. Our analysis highlights the need for careful thought in the specification of high-dimensional priors in Bayesian analyses.     

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.     

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.     

Paternity and paternal effort in the pumpkinseed sunfish

Theoretical models suggest that males should adjust their parentaleffort according to paternity when parental effort is costly,paternity varies among clutches, and males have a cue to assesspaternity. To date, nearly all tests of this theory have beenconducted using birds as model organisms. In this study we examinedthese three factors and the relationship between paternity andmale parental care in a fish system. In the pumpkinseed sunfish(Lepomis gibbosus), parental care is provided exclusively bymales (parentals), but some males (sneakers) parasitize othersby sneaking fertilizations. Parental males significantly lostweight during the parental care period. Clutch size and amountof parental effort did not affect a male's probability of obtainingmore eggs. Paternity was variable among broods. The proportionof young sired by a parental male was not associated with frequencyof fanning eggs or defense of hatched young, but was positivelycorrelated with levels of nest defense during the egg stage.Egg survivorship might restrict an adjustment of fanning behavior,and a general decline in parental behavior (with brood age)might explain the lack of adjustment once the eggs hatch. Parentalmales did not adjust their care when we experimentally manipulatedone possible cue of paternity. Together, these results indicatethat male pumpkinseeds do adjust their care in relation to paternity,but the cues used to assess paternity are not clear.