An extension of the Lin-Yip estimator for continuous-time capture studies

A unified approach is suggested to estimate the population size for continuous time capture studies with possible removals during the capture process. It extends and improves the Lin-Yip estimator. The usual recaptureand removal models can be shown to be particular cases of the general formulation. A Horvitz-Thompson procedure is used to estimate the population size based on the estimated capture probabilities. The resultant estimators for the regression parameters and population size are consistent and asymptotically normal underappropriateregularity conditions. We assess the properties of the proposed estimators through Monte Carlo simulation. Two examples are given.     

Evaluating estimators of the numbers of females with cubs-of-the-year in the yellowstone grizzly bear population

Current management of the grizzly bear (Ursus arctos) population in Yellowstone National Park and surrounding areas requires annual estimation of the number of adult female bears with cubs-of-the-year. We examined the performance of nine estimators of population size via simulation. Data were simulated using two methods for different combinations of population size, sample size, and coefficient of variation of individual sighting probabilities. We show that the coefficient of variation does not, by itself, adequately describe the effects of capture heterogeneity, because two different distributions of capture probabilities can have the same coefficient of variation. All estimators produced biased estimates of population size with bias decreasing as effort in creased. Based on the simulation results we recommend the Chao estimator for model M _h be used to estimate the number of the female bears with cubs of the year; however, the estimator of Chao and Shen may also be useful depending on the goals of the research.     

A unified approach for estimating population size in capture-recapture studies with arbitrary removals

A unified approach is suggested to estimate the population size for a closed population in discrete time. Individuals can be removed after capture at any time during the experiment. The usual recapture and removal experimentsare shown to be particular cases of the general formulation. The capture probability is assumed to have a logistic function that depends on individual covariates and can be time dependent. The unified approach involves a two-step procedure. A conditional likelihood function is used to estimate the covariates coefficients and a Horvitz-Thompson type estimator to estimate the population size. The asymptotic and small-sample properties of the resulting estimators are in vestigated. A real example is given.     

Persistence of a rare ancient cycad: Effects of environment and demography

Conservation of rare species is frequently complicated by their narrow niches and poor competitive abilities which limit their distribution to a few small populations. We studied how population size and structure of an endemic cycad of northwestern Mexico, Dioon sonorense, known to be threatened by widespread land conversion and plant extraction by humans, varied with 21 environmental variables. We also used matrix population models to show how population structure may affect its long-term persistence. Slope and soil exchangeable potassium explained a moderate proportion of the variance in the abundance of adult individuals, while abundance of adults and slope explained a large proportion of the variance in the abundance of seedlings. The strong non-linear relationships of seedling abundance with slope and adult abundance suggest threshold effects that may limit D. sonorense regeneration in less suitable environments. Modeling suggests that most of the study populations will not be viable in the long-term, except for the largest population. Modeling scenarios with large increments in fecundity and seedling survival produced modest gains in population growth. The long-term persistence of D. sonorense is shown in our study to be further threatened by: (a) small size of adult populations, (b) low-quality habitat in which most populations occur, and potentially, (c) the combined effect of low fecundity and recruitment.     

A new variance estimator for parameters of semiparametric generalized additive models

Generalized additive models (GAMs) have become popular in the air pollution epidemiology literature. Two problems, recently surfaced, concern implementation of these semiparametric models. The first problem, easily corrected, was laxity of the default convergence criteria. The other, noted independently by Klein, Flanders, and Tolbert, and Ramsay, Burnett, and Krewski concerned variance estimates produced by commercially available software. In simulations, they were as much as 50% too small. We derive an expression for a variance estimator for the parametric component of generalized additive models that can include up to three smoothing splines, and show how the standard error (SE) estimated by this method differs from the corresponding SE estimated with error in a study of air pollution and emergency room admissions for cardiorespiratory disease. The derivation is based on asymptotic linearity. Using Monte Carlo experiments, we evaluated performance of the estimator in finite samples. The estimator performed well in Monte Carlo experiments, in the situations considered. However, more work is needed to address performance in additional situations. Using data from our study of air pollution and cardiovascular disease, the standard error estimated using the new method was about 10% to 20% larger than the biased, commercially available standard error estimate.     

Comparison of marker- and pedigree-based methods for estimating heritability in an agroforestry population of Vitellaria paradoxa C.F. Gaertn. (shea tree)

We implemented a regression-based method between pairwise relatedness estimated from markers and phenotypic similarity to estimate heritability of traits related to leaf size and morphology in a wild tree population (Vitellaria paradoxa C.F. Gaertn.: shea tree). We then compared the results with heritability estimated with a classical pedigree-based method. We tested both approaches in an agroforestry population of this tree species, a very important one and abundant in the Sudano-Sahelian zone of Africa. Twelve microsatellite loci were used to estimate pairwise relatedness after selection of estimator coefficients based on Monte Carlo simulation. The regression-based method applied to 200 individuals did not display a significant trend with physical distance between trees for relatedness as well as for actual variance of relatedness. In consequence, estimates of narrow-sense heritability of traits related to leaf size were not significant. The pedigree-based method using a progeny test with 39 families and 15 individuals per family from the same population showed high and significant estimates of narrow-sense heritability for the same traits (h² = 0.36–0.95), demonstrating a marked genetic variation within the population. This discrepancy between methods stresses the poor performance of the molecular marker-based method. This can be explained by the absence of fine-scale structure within the agroforestry population of shea trees, other parameters being consistent with recommended values. The regression-based method does not seem well adapted to the agroforestry tree population. New experiments in tree populations and theoretical approaches are needed to evaluate the real potential of the marker-based methods.     

Can the sample variance estimator be improved by using a covariate?

Variance componentsare quantities of central interest in many applications, e.g., in cultivar yield stability analysis and in the analysis of measurement errors. In some applications, the feasible sample size is rather limited, leading to estimates of variance components that are subject to considerable sampling variation. For example, new crop cultivars are tested in only a few environments before release to the market, so the sample size for the variance across environments is small. Similarly, testing a new measurement instrument for some chemical compound may be costly, allowing only a limited number of replications. This article investigates the potential for improving the usual sample variance estimator by exploiting covariate information. In a cultivartrial, yield data may be available for only a few environments while meteorological data or data on a standard cultivar has been recorded for a very large number of environments. Likewise, in the analysis of measurement errors, there may be long-term data on a standard measurement procedure that can be used as a covariate to improve the variance estimate for a new instrument. It is shown in this article that the gain in accuracy achieved by using a covariate can be considerable, provided there is sufficient correlation between the covariate and the variable of interest.     

Resolving conflicts between butterfly host resource abundance and genet population size estimates for a vegetatively spreading, threatened grassland legume

Population size estimates are an integral part of rare plant conservation, but common abundance measurements such as cover and ramet number may not accurately index genet population size for vegetatively spreading species. Population monitoring of Kincaid’s lupine (threatened species) populations occurs through genet-anonymous leaf cover and raceme counts despite extensive, non-adventitious rhizome growth. While the current monitoring scheme provides important resource abundance measurements for the endangered Fender’s blue butterfly, whose larvae feed on Kincaid’s lupine leaves, the methods are not appropriate for estimating lupine genet number. Major axis regression revealed well supported statistical relationships between cover, raceme, and plantlet (a measurement of modular plant growth) density within six study patches (n = 3 populations) of Kincaid’s lupine (R² > 0.90) and when all patch data were combined (R² > 0.91). Genet population size estimates from genotype only data with ACE (an estimator used to infer species richness) were similar to estimates derived from a combination of plantlet density and genet to plantlet ratios (genotype derived) in small, more thoroughly genotyped lupine patches. However, genet number estimates from ACE were 3–5-fold greater in less intensely genotyped patches. Genet-anonymous plant abundance measurements, such as cover, can be used to estimate genet number in populations of vegetatively spreading plants provided they are calibrated with a unit of modular plant growth. Calibration of vegetative measurements, consistency of between population relationships, and closer scrutiny of highly supported statistical models may be necessary to develop more pertinent monitoring methods for rare, vegetatively spreading plants.     

A gamma-shaped detection function for line-transect surveys with mark-recapture and covariate data

We have developed a procedure for estimating animal population size from aerial survey data collected simultaneously by two observers on the same sighting platform. We used a line transect sample design where transects follow elevation contours in mountainous terrain. Because our 10 data sets from aerial line transect surveys, conducted over a terrestrial environment, consistently show unimodal detection shapes, we chose a gamma-shaped detection function that is unimodal and admits covariates. We fit models separately to data from each observer, and then used all of the data to estimate the probabilities at the apex of the detection curves. We used a Horvitz-Thompson estimator to estimate the population size. We illustrate our procedure on a recently collected brown bear data set.     

Choosing functions for semi-variograms of soil properties and fitting them to sampling estimates

The semi-variogram is central to geostatistics and the single most important tool in geo-statistical applications to soil. Mathematical functions for semi-variograms must be conditional negative semi-definite, and there are only a few families of simple function that meet this demand. These include the transitive models with finite a priori variance deriving from moving average processes. The spherical and exponential schemes are the most often encountered members. The other major group is that of unbounded models in which the variance appears to increase without limit. The linear model is the most common in this group. If more complex models are needed they can be formed by combining two or more simple models. The usual estimator of the semi-variance is often considered inefficient and to be sensitive to departures from normality in the data. It is compared with a robust estimator and shown to be generally preferable in being unbiased and having confidence intervals that are no wider. For routine analysis, fitting models to sample semi-variograms by weighted least squares approximation, with weights proportional to the expected semi-variance, is preferred to the more elaborate and computationally demanding statistical procedures of generalized least squares and maximum likelihood. The Akaike information criterion is recommended for selecting the best model from several plausible ones to describe the observed variation in soil, though for kriging it may be desirable to validate the chosen model. Examples of models fitted to soil semi-variograms are shown and compared.     

Empirical Bayes analysis of variance component models for microarray data

A gene-by-gene mixed model analysis is a useful statistical method for assessing significance for microarray gene differential expression. While a large amount of data on thousands of genes are collected in a microarray experiment, the sample size for each gene is usually small, which could limit the statistical power of this analysis. In this report, we introduce an empirical Bayes (EB) approach for general variance component models applied to microarray data. Within a linear mixed model framework, the restricted maximum likelihood (REML) estimates of variance components of each gene are adjusted by integrating information on variance components estimated from all genes. The approach starts with a series of single-gene analyses. The estimated variance components from each gene are transformed to the “ANOVA components”. This transformation makes it possible to independently estimate the marginal distribution of each “ANOVA component.” The modes of the posterior distributions are estimated and inversely transformed to compute the posterior estimates of the variance components. The EB statistic is constructed by replacing the REML variance estimates with the EB variance estimates in the usual t statistic. The EB approach is illustrated with a real data example which compares the effects of five different genotypes of male flies on post-mating gene expression in female flies. In a simulation study, the ROC curves are applied to compare the EB statistic and two other statistics. The EB statistic was found to be the most powerful of the three. Though the null distribution of the EB statistic is unknown, a t distribution may be used to provide conservative control of the false positive rate.     

Estimating the hidden number of scrapie affected holdings in Great Britain using a simple,truncated count model allowing for heterogeneity

None of the current surveillance streams monitoring the presence of scrapie in Great Britain provide a comprehensive and unbiased estimate of the prevalence of the disease at the holding level. Previous work to estimate the under-ascertainment adjusted prevalence of scrapie in Great Britain applied multiple-list capture-recapture methods. The enforcement of new control measures on scrapie-affected holdings in 2004 has stopped the overlapping between surveillance sources and, hence, the application of multiple-list capture-recapture models. Alternative methods, still under the capture-recapture methodology, relying on repeated entries in one single list have been suggested in these situations. In this article, we apply one-list capture-recapture approaches to data held on the Scrapie Notifications Database to estimate the undetected population of scrapie-affected holdings with clinical disease in Great Britain for the years 2002, 2003, and 2004. For doing so, we develop a new diagnostic tool for indication of heterogeneity as well as a new understanding of the Zelterman and Chao’s lower bound estimators to account for potential unobserved heterogeneity. We demonstrate that the Zelterman estimator can be viewed as a maximum likelihood estimator for a special, locally truncated Poisson likelihood equivalent to a binomial likelihood. This understanding allows the extension of the Zelterman approach by means of logistic regression to include observed heterogeneity in the form of covariates—in case studied here, the holding size and country of origin. Our results confirm the presence of substantial unobserved heterogeneity supporting the application of our two estimators. The total scrapie-affected holding population in Great Britain is around 300 holdings per year. None of the covariates appear to inform the model significantly.     

A unified capture-recapture framework

This article develops a hierarchical framework for capture-recapture data that separates the capture process from the demographic processes of interest, such as birth and survival. This allows users to parameterize in terms of meaningful demographic parameters. The framework is very flexible with many of the current capture-recapture models included as special cases. The hierarchical nature of the model allows natural expression of relationships, both between parameters and between parameters and the realization of random variables, such as population size. Previously, many of these relationships, such as density dependence have been unable to be explored using capture-recapture data. Density dependence, where survival and birth rates depend on the population size, is an interesting special case. We fit a density-dependent model to male Gonodontis bidentata data and report evidence of negative density dependence in percapita birth rates and weak evidence of negative density dependence in survival.     

Cost-effective abundance estimation of rare animals: Testing performance of small-boat surveys for killer whales in British Columbia

Top predators are often rare, subject to anthropogenic mortality, and possess life-history traits that make them inherently vulnerable to extinction. IUCN criteria recognise populations as Critically Endangered when abundance is <250 mature individuals, but estimating abundance of rare species can be more challenging than for common ones. Cost-effective methods are needed to provide robust abundance estimates. In marine environments, small boats are more widely accessible than large ships for researchers conducting sightings surveys with limited funds, but studies are needed into efficacy of small-boat surveys. This study compares line transect and mark-recapture estimates from small-boat surveys in summer 2004 and 2005 for ‘northern resident’ killer whales in British Columbia to true population size, known from censuses conducted by Fisheries and Oceans Canada. The line transect estimate of 195 animals (95% CI 27-559) used model averaging to incorporate uncertainty in the detection function, while the mark-recapture estimate of 239 animals (CI 154-370) used a simple two-sample Chapman estimator. Both methods produced estimates close to the true population size, which numbered 219 animals in 2004 and 235 in 2006, but both suffered from the small sample sizes and violations of some model assumptions that will vex most pilot studies of rare species. Initial abundance estimates from relatively low-cost surveys can be thought of as hypotheses to be tested as new data are collected. For species of conservation concern, any cost-effective attempt to estimate absolute abundance will assist status assessments, as long as estimates are presented with appropriate caveats.     

The ecology of extinction: population fluctuation and decline in amphibians

Even among widespread species with high reproductive potentials and significant dispersal abilities, the probability of extinctions should be correlated both with population size variance and with the extent of population isolation. To address how variation in demographic characteristics and habitat requirements may reflect on the comparative risk of species decline, I examined 617 time series of population census data derived from 89 amphibian species using the normalized estimate of the realized rate of increase, ΔN, and its variance. Amphibians are demonstrably in general decline and exhibit a great range of dispersal abilities, demographic characteristics, and population sizes. I compared species according to life-history characteristics and habitat use. Among the populations examined, census declines outnumbered increases yet the average magnitudes for both declines and increases were not demonstrably different, substantiating findings of amphibian decline. This gives no support for the idea that amphibian population sizes are dictated by regimes featuring relatively rare years of high recruitment offset by intervening years of gradual decline such that declines may outnumber increases without negative effect. For any given population size, those populations living in large streams or in ponds had significantly higher variance than did populations of completely terrestrial or other stream-dwelling amphibians. This could not be related to life-history complexity as all the stream-breeding species examined have larvae and all of the wholly terrestrial species have direct development without a larval stage. Variance in ΔN was highest amongst the smallest populations in each comparison group. Estimated local extinction rates averaged 3.1% among pond-breeding frogs, 2.2% for pond-breeding salamanders, and negligible for both stream-breeding and terrestrial direct-developing species. Recoveries slightly exceeded extinctions among European pond-breeding frogs but not among North American pond-breeding frogs. Less common species had greater negative disparities between extinctions and recoveries. Species with highly fluctuating populations and high frequencies of local extinctions living in changeable environments, such pond- and torrent-breeding amphibians, may be especially susceptible to curtailment of dispersal and restriction of habitat.     

Optimum Sample Size for Estimating Gene Diversity in Wild Wheat using AFLP Markers

Genetic diversity in five wild types of wheat was estimated using Simpson's index (based on heterozygosity) applied to data from AFLP markers. For such studies, the cost of obtaining the required information increases both with the number of samples required to estimate diversity and with the number of markers used. When the population studied is in Hardy–Weinberg equilibrium (HWE), allelic frequencies follow the binomial expansion and parametric methods can be used to calculate the variance of the diversity index in terms of the number of individuals sampled. Inbred species are never in HWE. With regard to such populations, this study addresses the question of the sample size required to estimate gene diversity using a distribution-free re-sampling method. We studied populations of five wild species (Aegilops speltoides, Triticum urartu, Triticum boeoticum, Triticum dicoccoides, and Triticum araraticum) as sources of diversity. We used bootstrap re-sampling with varying sample sizes to develop a relationship between the precision of the diversity estimate and the sample size. Such a relationship was used to determine the samples required for capturing a given amount of diversity and its precision. We found that 5–6 samples are sufficient to obtain a standard error equal to 10% of the diversity in the populations of the species Ae. speltoides, T. dicoccoides and T. araraticum. However, more than 12 samples would be needed for populations of T. urartu and T. boeoticum. The procedure presented here can be used to obtain the optimum sample size for other crop species as well.     

Estimating population size in the subdesert mesite (Monias benschi): new methods and implications for conservation

The subdesert mesite, a terrestrial non-passerine bird endemic to the Mikea Forest of southwest Madagascar, is currently classified as globally threatened (category: Vulnerable). However, accurate assessment of threat in accordance with the IUCN Red List criteria (A, B and C) requires data on effective population size, area of habitat occupied and rate of decline, none of which is available for this species. Here we present the first empirical estimates of its population size using five complementary methods, three incorporating data on territory size and two using data gathered during call-playback surveys conducted throughout its entire global range. Estimates vary from 98,000 to 152,000 individuals, with the most reliable possibly being that generated by distance sampling (115,000). This figure is more than an order of magnitude greater than the only published estimate of <10,000 individuals. By analysing data on forest cover change, we estimate the population of the subdesert mesite to have declined by, at most, 10% in three generations. Although the rate of deforestation in 1994-1999 is double that calculated for 1962-1994, it is deemed unlikely that the population will decline by 20% over the next three generations. As such the species fails to meet criterion A. Although the subdesert mesite's range and area of occupancy are small, they are not fragmented and do not comprise fewer than 10 locations. Consequently, this species does not meet criterion B. Further, the species fails to meet criterion C, for which a maximum of 10,000 mature adults is required. According to IUCN (2000 Red List of Threatened Species) this species should therefore be downlisted in status. We discuss why it still warrants conservation attention and suggest the need for modifications to the criteria thresholds in relation to basic information about the ecology and taxonomic distinctness of species.     

Identifying QTLs and Epistasis in Structured Plant Populations Using Adaptive Mixed LASSO

Association analysis in important crop species has generated heightened interest for its potential in dissecting complex traits by utilizing diverse mapping populations. However, the mixed linear model approach is currently limited to single marker analysis, which is not suitable for studying multiple QTL effects, epistasis and gene by environment interactions. In this paper, we propose the adaptive mixed LASSO method that can incorporate a large number of predictors (genetic markers, epistatic effects, environmental covariates, and gene by environment interactions) while simultaneously accounting for the population structure. We show that the adaptive mixed LASSO estimator possesses the oracle property of adaptive LASSO. Algorithms are developed to iteratively estimate the regression coefficients and variance components. Our results demonstrate that the adaptive mixed LASSO method is very promising in modeling multiple genetic effects when a large number of markers are available and the population structure cannot be ignored. It is expected to be a powerful tool for studying the architecture of complex traits in important plant species. Supplemental materials for this article are available from the journal website.