The Planning of Sample Sizes for Maximum Likelihood Estimation of the Parameters of the Weibull Distribution

Based on maximum likelihood estimators the problem of setting confidence limits for the scale and shape parameters of the Weibull distribution with two parameters is considered. Approximation formulae for the determination of sample sizes for the parameter estimation are given for complete and type II censored samples.     

The Use of a Symbol Processing Computer Language in Point Estimation of Parameters and in Construction of Confidence Intervals

This paper presents a method to generate automatically computer programs which are necessary for parameter estimation, hypothesis tests and construction of confidence intervals by the maximum likelihood method. The spectral or density function of the random variable is arbitrary, but must be known and given in closed form. The programming language used is the symbol processing language LIBAFORM, whose statements are interpreted by a package of LISP-routines. The application of the method is illustrated by the analysis of a linear model whose residuals follow a logarithmic F-distribution, and the analysis of a dose-response curve.     

Accuracy of coalescent likelihood estimates: do we need more sites, more sequences, or more loci?

A computer simulation study has been made of the accuracy of estimates of Theta = 4Nemu from a sample from a single isolated population of finite size. The accuracies turn out to be well predicted by a formula developed by Fu and Li, who used optimistic assumptions. Their formulas are restated in terms of accuracy, defined here as the reciprocal of the squared coefficient of variation. This should be proportional to sample size when the entities sampled provide independent information. Using these formulas for accuracy, the sampling strategy for estimation of Theta can be investigated. Two models for cost have been used, a cost-per-base model and a cost-per-read model. The former would lead us to prefer to have a very large number of loci, each one base long. The latter, which is more realistic, causes us to prefer to have one read per locus and an optimum sample size which declines as costs of sampling organisms increase. For realistic values, the optimum sample size is 8 or fewer individuals. This is quite close to the results obtained by Pluzhnikov and Donnelly for a cost-per-base model, evaluating other estimators of Theta. It can be understood by considering that the resources spent collecting larger samples prevent us from considering more loci. An examination of the efficiency of Watterson's estimator of Theta was also made, and it was found to be reasonably efficient when the number of mutants per generation in the sequence in the whole population is less than 2.5.     

Likelihood‐based inference of population history from low‐coverage de novo genome assemblies

Short‐read sequencing technologies have in principle made it feasible to draw detailed inferences about the recent history of any organism. In practice, however, this remains challenging due to the difficulty of genome assembly in most organisms and the lack of statistical methods powerful enough to discriminate between recent, nonequilibrium histories. We address both the assembly and inference challenges. We develop a bioinformatic pipeline for generating outgroup‐rooted alignments of orthologous sequence blocks from de novo low‐coverage short‐read data for a small number of genomes, and show how such sequence blocks can be used to fit explicit models of population divergence and admixture in a likelihood framework. To illustrate our approach, we reconstruct the Pleistocene history of an oak‐feeding insect (the oak gallwasp Biorhiza pallida), which, in common with many other taxa, was restricted during Pleistocene ice ages to a longitudinal series of southern refugia spanning the Western Palaearctic. Our analysis of sequence blocks sampled from a single genome from each of three major glacial refugia reveals support for an unexpected history dominated by recent admixture. Despite the fact that 80% of the genome is affected by admixture during the last glacial cycle, we are able to infer the deeper divergence history of these populations. These inferences are robust to variation in block length, mutation model and the sampling location of individual genomes within refugia. This combination of de novo assembly and numerical likelihood calculation provides a powerful framework for estimating recent population history that can be applied to any organism without the need for prior genetic resources.     

Estimating population divergence time and phylogeny from single-nucleotide polymorphisms data with outgroup ascertainment bias

The inference of population divergence times and branching patterns is of fundamental importance in many population genetic analyses. Many methods have been developed for estimating population divergence times, and recently, there has been particular attention towards genome-wide single-nucleotide polymorphisms (SNP) data. However, most SNP data have been affected by an ascertainment bias caused by the SNP selection and discovery protocols. Here, we present a modification of an existing maximum likelihood method that will allow approximately unbiased inferences when ascertainment is based on a set of outgroup populations. We also present a method for estimating trees from the asymmetric dissimilarity measures arising from pairwise divergence time estimation in population genetics. We evaluate the methods by simulations and by applying them to a large SNP data set of seven East Asian populations.     

A likelihood-based comparison of population histories in a parasitoid guild

Little is known about the stability of trophic relationships in complex natural communities over evolutionary timescales. Here, we use sequence data from 18 nuclear loci to reconstruct and compare the intraspecific histories of major Pleistocene refugial populations in the Middle East, the Balkans and Iberia in a guild of four Chalcid parasitoids (Cecidostiba fungosa, Cecidostiba semifascia, Hobbya stenonota and Mesopolobus amaenus) all attacking Cynipid oak galls. We develop a likelihood method to numerically estimate models of divergence between three populations from multilocus data. We investigate the power of this framework on simulated data, and—using triplet alignments of intronic loci—quantify the support for all possible divergence relationships between refugial populations in the four parasitoids. Although an East to West order of population divergence has highest support in all but one species, we cannot rule out alternative population tree topologies. Comparing the estimated times of population splits between species, we find that one species, M. amaenus, has a significantly older history than the rest of the guild and must have arrived in central Europe at least one glacial cycle prior to other guild members. This suggests that although all four species may share a common origin in the East, they expanded westwards into Europe at different times.     

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.     

Bayes or bootstrap? A simulation study comparing the performance of Bayesian Markov chain Monte Carlo sampling and bootstrapping in assessing phylogenetic confidence

Bayesian Markov chain Monte Carlo sampling has become increasingly popular in phylogenetics as a method for both estimating the maximum likelihood topology and for assessing nodal confidence. Despite the growing use of posterior probabilities, the relationship between the Bayesian measure of confidence and the most commonly used confidence measure in phylogenetics, the nonparametric bootstrap proportion, is poorly understood. We used computer simulation to investigate the behavior of three phylogenetic confidence methods: Bayesian posterior probabilities calculated via Markov chain Monte Carlo sampling (BMCMC-PP), maximum likelihood bootstrap proportion (ML-BP), and maximum parsimony bootstrap proportion (MP-BP). We simulated the evolution of DNA sequence on 17-taxon topologies under 18 evolutionary scenarios and examined the performance of these methods in assigning confidence to correct monophyletic and incorrect monophyletic groups, and we examined the effects of increasing character number on support value. BMCMC-PP and ML-BP were often strongly correlated with one another but could provide substantially different estimates of support on short internodes. In contrast, BMCMC-PP correlated poorly with MP-BP across most of the simulation conditions that we examined. For a given threshold value, more correct monophyletic groups were supported by BMCMC-PP than by either ML-BP or MP-BP. When threshold values were chosen that fixed the rate of accepting incorrect monophyletic relationship as true at 5%, all three methods recovered most of the correct relationships on the simulated topologies, although BMCMC-PP and ML-BP performed better than MP-BP. BMCMC-PP was usually a less biased predictor of phylogenetic accuracy than either bootstrapping method. BMCMC-PP provided high support values for correct topological bipartitions with fewer characters than was needed for nonparametric bootstrap.     

The estimation and applicability of confidence intervals for Stander's Similarity Index (SIMI) in algal assemblage comparisons

Stander's Similarity Index (SIMI) has become a popular measure for comparing algal assemblages. The interpretation of the value produced by this index, however, has been highly variable between studies. Using replicate sampling of natural algal assemblages and computer simulation techniques, a method by which confidence intervals could be applied to SIMI was developed. This paper presents that method and gives an example using hypothetical species counts of two algal assemblages. Since a more realistic range of the actual similarity between two assemblages is produced, the estimation of confidence intervals when using SIMI is recommended.