LIKELIHOOD-BASED INFERENCE IN ISOLATION-BY-DISTANCE MODELS USING THE SPATIAL DISTRIBUTION OF LOW-FREQUENCY ALLELES

Estimating dispersal distances from population genetic data provides an important alternative to logistically taxing methods for directly observing dispersal. Although methods for estimating dispersal rates between a modest number of discrete demes are well developed, methods of inference applicable to "isolation-by-distance" models are much less established. Here, we present a method for estimating ρσ², the product of population density (ρ) and the variance of the dispersal displacement distribution (σ²). The method is based on the assumption that low-frequency alleles are identical by descent. Hence, the extent of geographic clustering of such alleles, relative to their frequency in the population, provides information about ρσ². We show that a novel likelihood-based method can infer this composite parameter with a modest bias in a lattice model of isolation-by-distance. For calculating the likelihood, we use an importance sampling approach to average over the unobserved intraallelic genealogies, where the intraallelic genealogies are modeled as a pure birth process. The approach also leads to a likelihood-ratio test of isotropy of dispersal, that is, whether dispersal distances on two axes are different. We test the performance of our methods using simulations of new mutations in a lattice model and illustrate its use with a dataset from Arabidopsis thaliana .     

Intra-sexual female agonistic behaviour of the South American sea lion (Otaria flavescens) in two colonies with different breeding substrates

Female agonistic behaviour during pregnancy and lactation is common in several pinnipeds and has been associated to pup or territory defence. Previous studies on female agonistic behaviour in pinniped breeding groups have not usually considered the number of females per breeding group, which could influence the degree of aggressiveness. We compared female agonistic behaviour (open-mouth displays and bites) within two colonies with different breeding substrates (homogeneous, Punta Norte; heterogeneous, Pirámide) of South American sea lions using two methods: weighted means and general linear models. We found that both aggressive interaction rates were significantly affected by female density, which accounted for a greater proportion on the variability in bite rates than in open-mouth rates. Controlling for the number of females, we found with both methods that open-mouth rates were higher than bite rates within the Pirámide colony; however, no differences were found within the Punta Norte colony. In Pirámide, open-mouth rates could be used more often as threats to minimise the chances of more severe aggressions. The conclusion is that females vary their use of agonistic interactions in relation to their density in the colony with heterogeneous substrate, which may be related to the presence of limited numbers of tide pools that heightens competition for thermoregulatory resources.     

Modeling nonhomogeneous Markov processes via time transformation

Summary .   Longitudinal studies are a powerful tool for characterizing the course of chronic disease. These studies are usually carried out with subjects observed at periodic visits giving rise to panel data. Under this observation scheme the exact times of disease state transitions and sequence of disease states visited are unknown and Markov process models are often used to describe disease progression. Most applications of Markov process models rely on the assumption of time homogeneity, that is, that the transition rates are constant over time. This assumption is not satisfied when transition rates depend on time from the process origin. However, limited statistical tools are available for dealing with nonhomogeneity. We propose models in which the time scale of a nonhomogeneous Markov process is transformed to an operational time scale on which the process is homogeneous. We develop a method for jointly estimating the time transformation and the transition intensity matrix for the time transformed homogeneous process. We assess maximum likelihood estimation using the Fisher scoring algorithm via simulation studies and compare performance of our method to homogeneous and piecewise homogeneous models. We apply our methodology to a study of delirium progression in a cohort of stem cell transplantation recipients and show that our method identifies temporal trends in delirium incidence and recovery.     

Estimating average annual percent change for disease rates without assuming constant change

The annual percent change (APC) is often used to measure trends in disease and mortality rates, and a common estimator of this parameter uses a linear model on the log of the age-standardized rates. Under the assumption of linearity on the log scale, which is equivalent to a constant change assumption, APC can be equivalently defined in three ways as transformations of either (1) the slope of the line that runs through the log of each rate, (2) the ratio of the last rate to the first rate in the series, or (3) the geometric mean of the proportional changes in the rates over the series. When the constant change assumption fails then the first definition cannot be applied as is, while the second and third definitions unambiguously define the same parameter regardless of whether the assumption holds. We call this parameter the percent change annualized (PCA) and propose two new estimators of it. The first, the two-point estimator, uses only the first and last rates, assuming nothing about the rates in between. This estimator requires fewer assumptions and is asymptotically unbiased as the size of the population gets large, but has more variability since it uses no information from the middle rates. The second estimator is an adaptive one and equals the linear model estimator with a high probability when the rates are not significantly different from linear on the log scale, but includes fewer points if there are significant departures from that linearity. For the two-point estimator we can use confidence intervals previously developed for ratios of directly standardized rates. For the adaptive estimator, we show through simulation that the bootstrap confidence intervals give appropriate coverage.     

The evolution of genomic base composition in bacteria

Abstract. Guanine plus cytosine (GC) content ranges broadly among bacterial genomes. In this study, we explore the use of a Brownian-motion model for the evolution of GC content over time. This model assumes that GC content varies over time in a continuous and homogeneous manner. Using this model and a maximum-likelihood approach, we analyzed the evolution of GC content across several bacterial phylogenies. Using three independent tests, we found that the observed divergence in GC content was consistent with a homogeneous Brownian-motion model. For example, similar rates of GC content evolution were inferred in several different bacterial subclades, indicating that there is relatively little rate heterogeneity in GC content evolution over broad evolutionary time scales. We thus argue that the homogeneous Brownian-motion model provides a good working model for GC content evolution. We then use this model to determine the overall rate of GC content evolution among eubacteria. We also determine the time frame over which GC content remains similar in related taxa, using a flexible definition for "similarity" in GC content so that, depending on the context, more or less stringent criteria may be applied. Our results have implications for models of sequence evolution, including those used for phylogenetic reconstruction and for inferring unusual changes in GC content.     

Taxon sampling effects in molecular clock dating: an example from the African Restionaceae

Three commonly used molecular dating methods for correction of variable rates (non-parametric rate smoothing, penalized likelihood, and Bayesian rate correction) as well as the assumption of a global molecular clock were tested for sensitivity to taxon sampling. The test dataset of 6854 basepairs for 300 terminals includes a nearly complete sample of the Restio-clade of the African Restionaceae (272 of the 288 species), as well as 26 outgroup species. Of this, nested subsets of 35, 51, 80, 120, 150, and the full 300 species were used. Molecular dating experiments with these datasets showed that all methods are sensitive to undersampling, but that this effect is more severe in analyses that use more extreme rate smoothing. Additionally, the undersampling effect is positively related to distance from the calibration node. The combined effect of undersampling and distance from the calibration node resulted in up to threefold differences in the age estimation of nodes from the same dataset with the same calibration point. We suggest that the most suitable methods are penalized likelihood and Bayesian when a global clock assumption has been rejected, as these methods are more successful at finding optimal levels of smoothing to correct for rate heterogeneity, and are less sensitive to undersampling.     

ESTIMATING SPECIATION AND EXTINCTION RATES FROM DIVERSITY DATA AND THE FOSSIL RECORD

Understanding the processes that underlie biodiversity requires insight into the evolutionary history of the taxa involved. Accurate estimation of speciation, extinction, and diversification rates is a prerequisite for gaining this insight. Here, we develop a stochastic birth–death model of speciation and extinction that predicts the probability distribution of both extinct and extant numbers of species in a clade. We present two estimation methods based on this model given data on the number of extinct species (from the fossil record) and extant species (from diversity assessments): a multivariate method of moments approach and a maximum-likelihood approach. We show that, except for some special cases, the two estimation methods produce very similar estimates. This is convenient, because the usually preferred maximum-likelihood approach is much more computationally demanding, so the method of moments can serve as a proxy. Furthermore, we introduce a correction for possible bias that can arise by the mere fact that we will normally only consider extant clades. We find that in some cases the bias correction affects the estimates profoundly. Finally, we show how our model can be extended to incorporate incomplete preservation. Preservation rates can, however, not be reliably estimated on the basis of numbers of extant and extinct species alone.     

Testing the relationship between morphological and molecular rates of change along phylogenies

Abstract.— Molecular evolution has been considered to be essentially a stochastic process, little influenced by the pace of phenotypic change. This assumption was challenged by a study that demonstrated an association between rates of morphological and molecular change estimated for "total-evidence" phylogenies, a finding that led some researchers to challenge molecular date estimates of major evolutionary radiations. Here we show that Omland's (1997) result is probably due to methodological bias, particularly phylogenetic nonindependence, rather than being indicative of an underlying evolutionary phenomenon. We apply three new methods specifically designed to overcome phylogenetic bias to 13 published phylogenetic datasets for vertebrate taxa, each of which includes both morphological characters and DNA sequence data. We find no evidence of an association between rates of molecular and morphological rates of change.     

Evaluating the performance of likelihood methods for detecting population structure and migration

A plethora of statistical models have recently been developed to estimate components of population genetic history. Very few of these methods, however, have been adequately evaluated for their performance in accurately estimating population genetic parameters of interest. In this paper, we continue a research program of evaluation of population genetic methods through computer simulation. Specifically, we examine the software MIGRATEE-N 1.6.8 and test the accuracy of this software to estimate genetic diversity (Theta), migration rates, and confidence intervals. We simulated nucleotide sequence data under a neutral coalescent model with lengths of 500 bp and 1000 bp, and with three different per site Theta values of (0.00025, 0.0025, 0.025) crossed with four different migration rates (0.0000025, 0.025, 0.25, 2.5) to construct 1000 evolutionary trees per-combination per-sequence-length. We found that while MIGRATEE-N 1.6.8 performs reasonably well in estimating genetic diversity (Theta), it does poorly at estimating migration rates and the confidence intervals associated with them. We recommend researchers use this software with caution under conditions similar to those used in this evaluation.     

A note on plating efficiency in fluctuation experiments

In estimating mutation rates using the Luria-Delbrück experimental protocol, it is often assumed that all mutant cells survive the plating procedure to form visible mutant colonies. This assumption of perfect plating efficiency may not hold in certain circumstances, but none of the existing estimation methods that adjust for plating efficiency is strictly based on the likelihood principle. To ameliorate this situation, we propose likelihood based algorithms for computing point and interval estimates of mutation rates.     

Bayesian Reliability Estimation of a Two-parameter Cauchy Distribution

This paper gives an approximate Bayes procedure for the estimation of the reliability function of a two-parameter Cauchy distribution using Jeffreys' non-informative prior with a squared-error loss function, and with a log-odds ratio squared-error loss function. Based on a Monte Carlo simulation study, two such Bayes estimators of the reliability are compared with the maximum likelihood estimator.     

A Spatial Dirichlet Process Mixture Model for Clustering Population Genetics Data

Summary Identifying homogeneous groups of individuals is an important problem in population genetics. Recently, several methods have been proposed that exploit spatial information to improve clustering algorithms. In this article, we develop a Bayesian clustering algorithm based on the Dirichlet process prior that uses both genetic and spatial information to classify individuals into homogeneous clusters for further study. We study the performance of our method using a simulation study and use our model to cluster wolverines in Western Montana using microsatellite data.     

A burden from birth? Non‐invasive prenatal testing and the stigmatization of people with disabilities

The notion of being a burden to others is mostly discussed in the context of care‐intensive diseases or end‐of‐life decisions. But the notion is also crucial in decision‐making at the beginning of life, namely regarding prenatal testing. Ever more sophisticated testing methods, especially non‐invasive prenatal testing (NIPT), allow the detection of genetic traits in the unborn child that may cause disabilities. A positive result often influences the decision of the pregnant women towards a termination of the pregnancy. Thus, critics claim that these testing methods send a negative message to people with disabilities. At the core of this is what we call the burden assumption. This assumption claims that children with disabilities are necessarily a burden to others, especially to their parents and other family members. In this paper, we discuss what being a burden to others means in this context and how such an attitude can be avoided without restraining reproductive autonomy. A closer examination shows that the burden assumption is mostly based on misinformation and a false model of disability. Empirical studies as well as narrative evidence from parents who raise a child with disabilities show that the burden assumption is wrong. Raising a child with disabilities does not necessarily mean a decrease in the quality of life. We show how the burden assumption can be challenged through an advanced genetic counselling that combines empirical evidence with narratives from a first‐person perspective.     

Parameter redundancy in discrete state‐space and integrated models

Discrete state‐space models are used in ecology to describe the dynamics of wild animal populations, with parameters, such as the probability of survival, being of ecological interest. For a particular parametrization of a model it is not always clear which parameters can be estimated. This inability to estimate all parameters is known as parameter redundancy or a model is described as nonidentifiable. In this paper we develop methods that can be used to detect parameter redundancy in discrete state‐space models. An exhaustive summary is a combination of parameters that fully specify a model. To use general methods for detecting parameter redundancy a suitable exhaustive summary is required. This paper proposes two methods for the derivation of an exhaustive summary for discrete state‐space models using discrete analogues of methods for continuous state‐space models. We also demonstrate that combining multiple data sets, through the use of an integrated population model, may result in a model in which all parameters are estimable, even though models fitted to the separate data sets may be parameter redundant.     

Nonstochastic variation of species-level diversification rates within angiosperms

Variations in the origination and extinction rates of species over geological time often are linked with a range of factors, including the evolution of key innovations, changes in ecosystem structure, and environmental factors such as shifts in climate and physical geography. Before hypothesizing causality of a single factor, it is critical to demonstrate that the observed variation in diversification is significantly greater than one would expect due to natural stochasticity in the evolutionary branching process. Here, we use a likelihood-ratio test to compare taxonomic rate heterogeneity to a neutral birth-death model, using data on well-supported sister pairs of taxa and their species richness. We test the likelihood that the distribution of extant species among angiosperm genera and families could be the result of constant diversification rates. Results strongly support the conclusion that there is significantly more heterogeneity in diversity at the species level within angiosperms than would be expected due to stochastic processes. This result is consistent in datasets of genus pairs and family pairs and is not affected significantly by degrading pairs to simulate inaccuracy in the assumption of simultaneous origin of sister taxa. When we parse taxon pairs among higher groups of angiosperms, results indicate that a constant rates model is not rejected by rosid and basal eudicot pairs but is rejected by asterid and eumagnoliid pairs. These results provide strong support for the hypothesis that species-level rates of origination and/or extinction have varied nonrandomly within angiosperms and that the magnitude of heterogeneity varies among major groups within angiosperms.     

Restoration Ecology: Two-Sex Dynamics and Cost Minimization

We model a spatially detailed, two-sex population dynamics, to study the cost of ecological restoration. We assume that cost is proportional to the number of individuals introduced into a large habitat. We treat dispersal as homogeneous diffusion in a one-dimensional reaction-diffusion system. The local population dynamics depends on sex ratio at birth, and allows mortality rates to differ between sexes. Furthermore, local density dependence induces a strong Allee effect, implying that the initial population must be sufficiently large to avert rapid extinction. We address three different initial spatial distributions for the introduced individuals; for each we minimize the associated cost, constrained by the requirement that the species must be restored throughout the habitat. First, we consider spatially inhomogeneous, unstable stationary solutions of the model’s equations as plausible candidates for small restoration cost. Second, we use numerical simulations to find the smallest rectangular cluster, enclosing a spatially homogeneous population density, that minimizes the cost of assured restoration. Finally, by employing simulated annealing, we minimize restoration cost among all possible initial spatial distributions of females and males. For biased sex ratios, or for a significant between-sex difference in mortality, we find that sex-specific spatial distributions minimize the cost. But as long as the sex ratio maximizes the local equilibrium density for given mortality rates, a common homogeneous distribution for both sexes that spans a critical distance yields a similarly low cost.     

The influence of noninstantaneous mixing on the measurement of fluxes of solute and water across a biological membrane

The usual method of tracer analysis for calculating the flow across a biological membrance is based on the assumption that the compartments on either side are well-stirred. Thus, the validity of the rate of flow determination is questionable for cases where the distribution of tracer is not homogeneous. In this study, a mathematical model is developed for the purpose of estimating the effect of slow mixing on the calculation of the flow rate. The model is applied to the measurement of the rate of flow of aqueous humor through the living eye by use of a fluorescent dye as a tracer. A transit time of several minutes for the passage of fluorescein through the posterior chamber and an extended period of nonuniform distribution of fluorescein in the anterior chamber was observed. The effect of slow mixing on the calculated flow rate is compared to rates derived from equations based on the assumption of rapid mixing. Aqueous flow rates determined by the two methods were found to agree to within ≈20%.     

Hierarchical models for the probabilities of conception

In the last thirty years, there has been considerable interest in finding better models to fit data for probabilities of conception. An important early model was proposed by Barrett and Marshall (1969) and extended by Schwartz, MacDonald and Heuchel (1980). Recently, researchers have further extended these models by adding covariates. However, the increasingly complicated models are challenging to analyze with frequentist methods such as the EM algorithm. Bayesian models are more feasible, and the computation can be done via Markov chain Monte Carlo (MCMC). We consider a Bayesian model with an effect for protected intercourse to analyze data from the California Women's Reproductive Health Study and assess the effects of water contaminants and hormones. There are two main contributions in the paper. (1) For protected intercourse, we propose modeling the ratios of daily conception probabilities with protected intercourse to corresponding daily conception probabilities with unprotected intercourse. Due to the small sample size of our data set, we assume the ratios are the same for each day but unknown. (2) We consider Bayesian analysis under a unimodality assumption where the probabilities of conception increase before ovulation and decrease after ovulation. Gibbs sampling is used for finding the Bayesian estimates. There is some evidence that the two covariates affect fecundability.     

Stability and stabilization for models of chemostats with multiple limiting substrates

We study chemostat models in which multiple species compete for two or more limiting nutrients. First, we consider the case where the nutrient flow and species removal rates and input nutrient concentrations are all given as positive constants. In that case, we use Brouwer degree theory to give conditions guaranteeing that the models admit globally asymptotically stable componentwise positive equilibrium points, from all componentwise positive initial states. Then we use the results to develop stabilization theory for a class of controlled chemostats with two or more limiting nutrients. For cases where the dilution rate and input nutrient concentrations can be selected as controls, we prove that many different componentwise positive equilibria can be made globally asymptotically stable. This extends the existing control results for chemostats with one limiting nutrient. We demonstrate our methods in simulations.     

Retrospective versus prospective score tests for genetic association with case‐control data

Since the seminal work of Prentice and Pyke, the prospective logistic likelihood has become the standard method of analysis for retrospectively collected case‐control data, in particular for testing the association between a single genetic marker and a disease outcome in genetic case‐control studies. In the study of multiple genetic markers with relatively small effects, especially those with rare variants, various aggregated approaches based on the same prospective likelihood have been developed to integrate subtle association evidence among all the markers considered. Many of the commonly used tests are derived from the prospective likelihood under a common‐random‐effect assumption, which assumes a common random effect for all subjects. We develop the locally most powerful aggregation test based on the retrospective likelihood under an independent‐random‐effect assumption, which allows the genetic effect to vary among subjects. In contrast to the fact that disease prevalence information cannot be used to improve efficiency for the estimation of odds ratio parameters in logistic regression models, we show that it can be utilized to enhance the testing power in genetic association studies. Extensive simulations demonstrate the advantages of the proposed method over the existing ones. A real genome‐wide association study is analyzed for illustration.