MAXIMUM-LIKELIHOOD ESTIMATION OF POPULATION DIVERGENCE TIMES AND POPULATION PHYLOGENY IN MODELS WITHOUT MUTATION

In this paper we present a method for estimating population divergence times by maximum likelihood in models without mutation. The maximum-likelihood estimator is compared to a commonly applied estimator based on Wright's F_ST statistic. Simulations suggest that the maximum-likelihood estimator is less biased and has a lower variance than the F_ST-based estimator. The maximum-likelihood estimator provides a statistical framework for the analysis of population history given genetic data. We demonstrate how maximum-likelihood estimates of the branching pattern of divergence of multiple populations may be obtained. We also describe how the method may be applied to test hypotheses such as whether populations have maintained equal population sizes. We illustrate the method by applying it to two previously published sets of human restriction fragment length polymorphism (RFLP) data.     

Measurement error and estimates of population extinction risk

It is common to estimate the extinction probability for a vulnerable population using methods that are based on the mean and variance of the long‐term population growth rate. The numerical values of these two parameters are estimated from time series of population censuses. However, the proportion of a population that is registered at each census is typically not constant but will vary among years because of stochastic factors such as weather conditions at the time of sampling. Here, we analyse how such sampling errors influence estimates of extinction risk and find sampling errors to produce two opposite effects. Measurement errors lead to an exaggerated overall variance, but also introduce negative autocorrelations in the time series (which means that estimates of annual growth rates tend to alternate in size). If time series data are treated properly these two effects exactly counter balance. We advocate routinely incorporating a measure of among year correlations in estimating population extinction risk.     

ESTIMATION OF GENE FLOW FROM F-STATISTICS

We present theory clarifying the general behavior of F_ST-based and G_ST-based estimators of gene flow, and confirm these predictions with simulations. In particular, we use the correlation of genes within groups within populations to define an estimator. The theoretical value of the correlation doe not depend on the number of groups in a population, and properties of the estimated correlation do not depend on the number of groups sampled or the number of individuals sampled per group. This invariance is in contrast to properties of G_ST. For a complete census of a population, bias and variance considerations would suggest the use of the G_ST-based estimator of gene flow, but lack of knowledge of population size or group number in practice suggests preference be given to the correlation-based estimator. We acknowledge that these estimators require that several conditions of a population-genetic model be met, since they do not make use of direct observations on the flow of genes. Our results differ from some of those based on simulation in a series of recent papers by M. Slatkin.     

Sampling distributions, biases, variances, and confidence intervals for genetic correlations

Genetic correlations are frequently estimated from natural and experimental populations, yet many of the statistical properties of estimators of are not known, and accurate methods have not been described for estimating the precision of estimates of Our objective was to assess the statistical properties of multivariate analysis of variance (MANOVA), restricted maximum likelihood (REML), and maximum likelihood (ML) estimators of by simulating bivariate normal samples for the one-way balanced linear model. We estimated probabilities of non-positive definite MANOVA estimates of genetic variance-covariance matrices and biases and variances of MANOVA, REML, and ML estimators of and assessed the accuracy of parametric, jackknife, and bootstrap variance and confidence interval estimators for MANOVA estimates of were normally distributed. REML and ML estimates were normally distributed for but skewed for and 0.9. All of the estimators were biased. The MANOVA estimator was less biased than REML and ML estimators when heritability (H), the number of genotypes (n), and the number of replications (r) were low. The biases were otherwise nearly equal for different estimators and could not be reduced by jackknifing or bootstrapping. The variance of the MANOVA estimator was greater than the variance of the REML or ML estimator for most H, n, and r. Bootstrapping produced estimates of the variance of close to the known variance, especially for REML and ML. The observed coverages of the REML and ML bootstrap interval estimators were consistently close to stated coverages, whereas the observed coverage of the MANOVA bootstrap interval estimator was unsatisfactory for some H, n, and r. The other interval estimators produced unsatisfactory coverages. REML and ML bootstrap interval estimates were narrower than MANOVA bootstrap interval estimates for most H, and r. Received: 6 July 1995 / Accepted: 8 March 1996     

Quantile regression estimates of animal population trends

Ecologists often estimate population trends of animals in time series of counts using linear regression to estimate parameters in a linear transformation of multiplicative growth models, where logarithms of rates of change in counts in time intervals are used as response variables. We present quantile regression estimates for the median (0.50) and interquartile (0.25, 0.75) relationships as an alternative to mean regression estimates for common density-dependent and density-independent population growth models. We demonstrate that the quantile regression estimates are more robust to outliers and require fewer distributional assumptions than conventional mean regression estimates and can provide information on heterogeneous rates of change ignored by mean regression. We provide quantile regression trend estimates for 2 populations of greater sage-grouse (Centrocercus urophasianus) in Wyoming, USA, and for the Crawford population of Gunnison sage-grouse (Centrocercus minimus) in southwestern Colorado, USA. Our selected Gompertz models of density dependence for both populations of greater sage-grouse had smaller negative estimates of density-dependence terms and less variation in corresponding predicted growth rates (λ) for quantile than mean regression models. In contrast, our selected Gompertz models of density dependence with piecewise linear effects of years for the Crawford population of Gunnison sage-grouse had predicted changes in λ across years from quantile regressions that varied more than those from mean regression because of heterogeneity in estimated λs that were both less and greater than mean estimates. Our results add to literature establishing that quantile regression provides better behaved estimates than mean regression when there are outlying growth rates, including those induced by adjustments for zeros in the time series of counts. The 0.25 and 0.75 quantiles bracketing the median provide robust estimates of population changes (λ) for the central 50% of time series data and provide a 50% prediction interval for a single new prediction without making parametric distributional assumptions or assuming homogeneous λs. Compared to mean estimates, our quantile regression trend estimates for greater sage-grouse indicated less variation in density-dependent λs by minimizing sensitivity to outlying values, and for Gunnison sage-grouse indicated greater variation in density-dependent λs associated with heterogeneity among quantiles.     

Estimation of effective population size of HIV-1 within a host: a pseudomaximum-likelihood approach

Using pseudomaximum-likelihood approaches to phylogenetic inference and coalescent theory, we develop a computationally tractable method of estimating effective population size from serially sampled viral data. We show that the variance of the maximum-likelihood estimator of effective population size depends on the serial sampling design only because internal node times on a coalescent genealogy can be better estimated with some designs than with others. Given the internal node times and the number of sequences sampled, the variance of the maximum-likelihood estimator is independent of the serial sampling design. We then estimate the effective size of the HIV-1 population within nine hosts. If we assume that the mutation rate is 2.5 x 10(-5) substitutions/generation and is the same in all patients, estimated generation lengths vary from 0.73 to 2.43 days/generation and the mean (1.47) is similar to the generation lengths estimated by other researchers. If we assume that generation length is 1.47 days and is the same in all patients, mutation rate estimates vary from 1.52 x 10(-5) to 5.02 x 10(-5). Our results indicate that effective viral population size and evolutionary rate per year are negatively correlated among HIV-1 patients.     

Grizzly bear population vital rates and trend in the Northern Continental Divide Ecosystem,Montana

We estimated grizzly bear (Ursus arctos) population vital rates and trend for the Northern Continental Divide Ecosystem (NCDE), Montana, between 2004 and 2009 by following radio-collared females and observing their fate and reproductive performance. Our estimates of dependent cub and yearling survival were 0.612 (95% CI = 0.300–0.818) and 0.682 (95% CI = 0.258–0.898). Our estimates of subadult and adult female survival were 0.852 (95% CI = 0.628–0.951) and 0.952 (95% CI = 0.892–0.980). From visual observations, we estimated a mean litter size of 2.00 cubs/litter. Accounting for cub mortality prior to the first observations of litters in spring, our adjusted mean litter size was 2.27 cubs/litter. We estimated the probabilities of females transitioning from one reproductive state to another between years. Using the stable state probability of 0.322 (95% CI = 0.262–0.382) for females with cub litters, our adjusted fecundity estimate (m_x) was 0.367 (95% CI = 0.273–0.461). Using our derived rates, we estimated that the population grew at a mean annual rate of approximately 3% (λ = 1.0306, 95% CI = 0.928–1.102), and 71.5% of 10,000 Monte Carlo simulations produced estimates of λ > 1.0. Our results indicate an increasing population trend of grizzly bears in the NCDE. Coupled with concurrent studies of population size, we estimate that over 1,000 grizzly bears reside in and adjacent to this recovery area. We suggest that monitoring of population trend and other vital rates using radioed females be continued. © 2011 The Wildlife Society.     

Accounting for Sampling Error When Inferring Population Synchrony from Time-Series Data: A Bayesian State-Space Modelling Approach with Applications

Background

Data collected to inform time variations in natural population size are tainted by sampling error. Ignoring sampling error in population dynamics models induces bias in parameter estimators, e.g., density-dependence. In particular, when sampling errors are independent among populations, the classical estimator of the synchrony strength (zero-lag correlation) is biased downward. However, this bias is rarely taken into account in synchrony studies although it may lead to overemphasizing the role of intrinsic factors (e.g., dispersal) with respect to extrinsic factors (the Moran effect) in generating population synchrony as well as to underestimating the extinction risk of a metapopulation.

Methodology/Principal findings

The aim of this paper was first to illustrate the extent of the bias that can be encountered in empirical studies when sampling error is neglected. Second, we presented a space-state modelling approach that explicitly accounts for sampling error when quantifying population synchrony. Third, we exemplify our approach with datasets for which sampling variance (i) has been previously estimated, and (ii) has to be jointly estimated with population synchrony. Finally, we compared our results to those of a standard approach neglecting sampling variance. We showed that ignoring sampling variance can mask a synchrony pattern whatever its true value and that the common practice of averaging few replicates of population size estimates poorly performed at decreasing the bias of the classical estimator of the synchrony strength.

Conclusion/Significance

The state-space model used in this study provides a flexible way of accurately quantifying the strength of synchrony patterns from most population size data encountered in field studies, including over-dispersed count data. We provided a user-friendly R-program and a tutorial example to encourage further studies aiming at quantifying the strength of population synchrony to account for uncertainty in population size estimates.     

Inference from single occasion capture experiments using genetic markers

Accurate estimation of the size of animal populations is an important task in ecological science. Recent advances in the field of molecular genetics researches allow the use of genetic data to estimate the size of a population from a single capture occasion rather than repeated occasions as in the usual capture–recapture experiments. Estimating the population size using genetic data also has sometimes led to estimates that differ markedly from each other and also from classical capture–recapture estimates. Here, we develop a closed form estimator that uses genetic information to estimate the size of a population consisting of mothers and daughters, focusing on estimating the number of mothers, using data from a single sample. We demonstrate the estimator is consistent and propose a parametric bootstrap to estimate the standard errors. The estimator is evaluated in a simulation study and applied to real data. We also consider maximum likelihood in this setting and discover problems that preclude its general use.     

An Evaluation of Sex-Age-Kill (SAK) Model Performance

ABSTRACT The sex-age-kill (SAK) model is widely used to estimate abundance of harvested large mammals, including white-tailed deer (Odocoileus virginianus). Despite a long history of use, few formal evaluations of SAK performance exist. We investigated how violations of the stable age distribution and stationary population assumption, changes to male or female harvest, stochastic effects (i.e., random fluctuations in recruitment and survival), and sampling efforts influenced SAK estimation. When the simulated population had a stable age distribution and λ > 1, the SAK model underestimated abundance. Conversely, when λ < 1, the SAK overestimated abundance. When changes to male harvest were introduced, SAK estimates were opposite the true population trend. In contrast, SAK estimates were robust to changes in female harvest rates. Stochastic effects caused SAK estimates to fluctuate about their equilibrium abundance, but the effect dampened as the size of the surveyed population increased. When we considered both stochastic effects and sampling error at a deer management unit scale the resultant abundance estimates were within ±121.9% of the true population level 95% of the time. These combined results demonstrate extreme sensitivity to model violations and scale of analysis. Without changes to model formulation, the SAK model will be biased when λ ≠ 1. Furthermore, any factor that alters the male harvest rate, such as changes to regulations or changes in hunter attitudes, will bias population estimates. Sex-age-kill estimates may be precise at large spatial scales, such as the state level, but less so at the individual management unit level. Alternative models, such as statistical age-at-harvest models, which require similar data types, might allow for more robust, broad-scale demographic assessments.     

Assessing the spatial variability of density dependence in waterfowl populations

Population models commonly assume that the demographic parameters are spatially invariant, but there is considerable evidence that population growth rate (r) and the strength of density dependence (β) can vary over a species' range. To address this issue we developed a spatially explicit Gompertz population model based on the spatially varying coefficients approach to assess the spatial variation in population drivers. The model was fit to spatially stratified time series population estimates of the mallard Anas platyrhynchos in western North America. We included precipitation during the previous year and spring maximum temperature in the current year as environmental factors in the density dependent population model. Because density dependent models can give biased estimates for time series of abundance data, we fit a naïve model without informative priors and a model where we constrained the mean and variance of r to biologically realistic values that were derived via a comparative demography approach. In the naïve model, r and β were not separately identifiable and their values were overestimated, leading to unrealistic population growth. The naïve model also implied spatial variation in population r and the return time to equilibrium [1?(– β)] across the survey area. In contrast, in the informative model, r and the return time to equilibrium did not vary markedly among populations and were generally equal across populations. The effects of the climatic factors were similar across models. Population growth rates in the Prairie‐pothole region were positively correlated with precipitation, while in Alaska rates were positively correlated with spring temperature. Although it has been argued in the past that adding ecological realism could help avoid the pitfalls associated with density dependent models, our results demonstrate that imposing constraints on the population parameters is still the best course of action.     

Density dependence and population dynamics of black rhinos (Diceros bicornis michaeli) in Kenya’s rhino sanctuaries

Density‐dependent feedback mechanisms provide insights into the population dynamics and interactions of large herbivores with their ecosystem. Sex ratio also has particularly important implications for growth rates of many large mammal populations through its influence on reproductive potential. Therefore, the interrelationships between density‐dependent factors, comprising density, sex ratio and underlying growth rates (r) were examined for the Eastern black rhino (Diceros bicornis michaeli) living in three rhino sanctuaries in Kenya using four population models. The exponential and logistic models gave similar results and the former were accepted because they better portrayed the actual situation on the ground. Sex ratios in all sanctuary populations were positively correlated with r but interpreted with realization of other factors also affecting r. We caution that the results of population models should be interpreted alongside ground‐truthed observations. We recommend that future translocation strategies should take into account sex and age structures of the donor population, while future studies of density dependence should take into account both biotic and abiotic factors.     

Estimation of Finite Population Variance Using Ratio and Product Methods of Estimation

For estimating finite population variance σ_y² of a character y under our study, estimators using auxiliary information on a character x in the form of ratio, product, ratio-type or product-type estimators have been suggested, and their comparative study with the conventional unbiased estimator s_y² of σ_y² has been made in simple random sampling with replacement. A generalized estimator representing a class of estimators for the finite populations variance, has also been studied.     

Impact of Spatial and Temporal Variation in Calf Survival on the Growth of Elk Populations

Abstract: The realized impact of a vital rate on population growth (λ) is determined by both the relative influence of the vital rate on λ (elasticity) and its magnitude of variability. We estimated mean survival and reproductive rates in elk (Cervus elaphus) and spatial and temporal variation in these rates from 37 sources located primarily across the Rocky Mountain region and northwestern United States. We removed sampling variance from estimates of process variance both within and across vital-rate data sets using the variance discounting method developed by White (2000). Deterministic elasticities calculated from a population matrix model parameterized with these mean vital rates ranked adult female survival (e_Scow = 0.869) much higher than calf survival (e_Scalf = 0.131). However, process variance in calf survival was >11 times greater than process variance in female survival across data sets and 10 times greater on average within studies. We conducted Life-Stage Simulation Analysis to incorporate both vital-rate elasticity patterns and empirical estimates of variability to identify those vital rates most influential in elk population dynamics. The overwhelming magnitude of variation in calf survival explained 75% of the variation in the population growth rates generated from 1,000 matrix replicates, compared to just 16% of the variation in λ explained by variation in female survival. Variation in calf survival greatly impacts elk population growth and calls into question the utility of classical elasticity analysis alone for guiding elk management. These results also suggest that the majority of interannual variability that wildlife managers document in late-winter and spring elk surveys is attributable to variation in calf survival over the previous year and less influenced by variation in the harvest of females during the preceding autumn. To meet elk population size objectives, managers should consider the inherent variation in calf survival, and its apparent sensitivity to management, in addition to female harvest.     

Criticisms Biologically Unwarranted and Analytically Irrelevant: Reply to Rominger et al.

ABSTRACT The criticisms of Rominger et al. (2008) of our retrospective analysis of desert bighorn sheep (DBS; Ovis canadensis mexicana) dynamics in the San Andres Mountains of south-central New Mexico, USA, contained many biological errors and analytical oversights. Herein, we show that Rominger et al. (2008) 1) overstated both magnitude and potential effect of predator removal; 2) incorrectly claimed that our total precipitation (TP) model did not fit the data when TP correctly classed ≥66% of subsequent population increases and declines (P ≥0.063); 3) presented a necessary prerequisite of the exponential model (serial correlation between N_t and N_t+1) as the key relationship in the DBS data, when it merely reflected that DBS are strongly K-selected and was irrelevant to our hypothesis tests specific to factors affecting the instantaneous rate of population increase (r); 4) greatly oversimplified relationships among precipitation, arid environments, and DBS; and 5) advocated a time for collection of lamb/female (L/F) ratio data that was unrelated to any meaningful period in the biological year of DBS and consequently presented L/F ratio data unrelated to observed dynamics of DBS. In contrast, the L/F ratios used in Bender and Weisenberger (2005) correctly predicted annual changes and were correlated with long-term population rates of change.     

Evaluating methods for estimating local effective population size with and without migration

Effective population size is a fundamental parameter in population genetics, evolutionary biology, and conservation biology, yet its estimation can be fraught with difficulties. Several methods to estimate N_e from genetic data have been developed that take advantage of various approaches for inferring N_e. The ability of these methods to accurately estimate N_e, however, has not been comprehensively examined. In this study, we employ seven of the most cited methods for estimating N_e from genetic data (Colony2, CoNe, Estim, MLNe, ONeSAMP, TMVP, and NeEstimator including LDNe) across simulated datasets with populations experiencing migration or no migration. The simulated population demographies are an isolated population with no immigration, an island model metapopulation with a sink population receiving immigrants, and an isolation by distance stepping stone model of populations. We find considerable variance in performance of these methods, both within and across demographic scenarios, with some methods performing very poorly. The most accurate estimates of N_e can be obtained by using LDNe, MLNe, or TMVP; however each of these approaches is outperformed by another in a differing demographic scenario. Knowledge of the approximate demography of population as well as the availability of temporal data largely improves N_e estimates.     

When are genetic methods useful for estimating contemporary abundance and detecting population trends?

The utility of microsatellite markers for inferring population size and trend has not been rigorously examined, even though these markers are commonly used to monitor the demography of natural populations. We assessed the ability of a linkage disequilibrium estimator of effective population size (N_e) and a simple capture-recapture estimator of abundance (N) to quantify the size and trend of stable or declining populations (true N = 100–10,000), using simulated Wright–Fisher populations. Neither method accurately or precisely estimated abundance at sample sizes of S = 30 individuals, regardless of true N. However, if larger samples of S = 60 or 120 individuals were collected, these methods provided useful insights into abundance and trends for populations of N = 100–500. At small population sizes (N = 100 or 250), precision of the N_e estimates was improved slightly more by a doubling of loci sampled than by a doubling of individuals sampled. In general, monitoring N_e proved a more robust means of identifying stable and declining populations than monitoring N over most of the parameter space we explored, and performance of the N_e estimator is further enhanced if the N_e/N ratio is low. However, at the largest population size (N = 10,000), N estimation outperformed N_e. Both methods generally required ≥ 5 generations to pass between sampling events to correctly identify population trend.     

Phylogenetic prediction of the maximum per capita rate of population growth

The maximum per capita rate of population growth, r, is a central measure of population biology. However, researchers can only directly calculate r when adequate time series, life tables and similar datasets are available. We instead view r as an evolvable, synthetic life-history trait and use comparative phylogenetic approaches to predict r for poorly known species. Combining molecular phylogenies, life-history trait data and stochastic macroevolutionary models, we predicted r for mammals of the Caniformia and Cervidae. Cross-validation analyses demonstrated that, even with sparse life-history data, comparative methods estimated r well and outperformed models based on body mass. Values of r predicted via comparative methods were in strong rank agreement with observed values and reduced mean prediction errors by approximately 68 per cent compared with two null models. We demonstrate the utility of our method by estimating r for 102 extant species in these mammal groups with unknown life-history traits.     

LDJump: Estimating variable recombination rates from population genetic data

As recombination plays an important role in evolution, its estimation and the identification of hotspot positions is of considerable interest. We propose a novel approach for estimating population recombination rates based on genotyping or sequence data that involves a sequential multiscale change point estimator. Our method also permits demography to be taken into account. It uses several summary statistics within a regression model fitted on suitable scenarios. Our proposed method is accurate, computationally fast, and provides a parsimonious solution by ensuring a type I error control against too many changes in the recombination rate. An application to human genome data suggests a good congruence between our estimated and experimentally identified hotspots. Our method is implemented in the R ‐package LDJump, which is freely available at https://github.com/PhHermann/LDJump .     

A Class of Almost Unbiased Estimators for Finite Populations Mean Using Two Auxiliary Variables

For estimating finite population mean -Y₀ of study character y₀, a class of almost unbiased estimators applying jackknife technique envisaged by Quenouille (1956) is derived. Optimum unbiased estimator (OUE) is also investigated with its variance formula. An empirical study is carried out to demonstrate the performance of the constructed estimator over the usual unbiased estimator, Srivastava (1965), Singh (1967), Singh and Biradar (1992), Tracy , Singh , and Singh (1996) and other almost unbiased estimators.