Hierarchical spatial modeling for estimation of population size

Estimation of population size has traditionally been viewed from a finite population sampling perspective. Typically, the objective is to obtain an estimate of the total population count of individuals within some region. Often, some stratification scheme is used to estimate counts on subregions, whereby the total count is obtained by aggregation with weights, say, proportional to the areas of the subregions. We offer an alternative to the finite population sampling approach for estimating population size. The method does not require that the subregions on which counts are available form a complete partition of the region of interest. In fact, we envision counts coming from areal units that are small relative to the entire study region and that the total area sampled is a very small proportion of the total study area. In extrapolating to the entire region, we might benefit from assuming that there is spatial structure to the counts. We implement this by modeling the intensity surface as a realization from a spatially correlated random process. In the case of multiple population or species counts, we use the linear model of coregionalization to specify a multivariate process which provides associated intensity surfaces hence association between counts within and across areal units. We illustrate the method of population size estimation with simulated data and with tree counts from a Southwestern pinyon-juniper woodland data set.     

Quantifying the influence of the cultivated plant species on the occurrence of carabid beetles within certain species using contingency table analysis

The influence of the cultivated plant stand, subject to the timing of sampling, on the occurrence of selected carabid species is investigated for three years running. Due to the nominal character of the factors (cultivar, timing, carabid species and year) we used log-linear models of four dimensional contingency tables under certain hypotheses of independence. A coefficient of contingency is calculated for characterizing the partial association between two factors respectively. The difference in G 2 's between two nested models is used to test if there are differences between the three years concerning the strength of partial association between the number of beetles in the selected species and the cultivar or the sampling time respectively.     

An incidence-based richness estimator for quadrats sampled without replacement

Most richness estimators currently in use are derived from models that consider sampling with replacement or from the assumption of infinite populations. Neither of the assumptions is suitable for sampling sessile organisms such as plants where quadrats are often sampled without replacement and the area of study is always limited. In this paper, we propose an incidence-based parametric richness estimator that considers quadrat sampling without replacement in a fixed area. The estimator is derived from a zero-truncated binomial distribution for the number of quadrats containing a given species (e.g., species i) and a modified beta distribution for the probability of presence-absence of a species in a quadrat. The maximum likelihood estimate of richness is explicitly given and can be easily solved. The variance of the estimate is also obtained. The performance of the estimator is tested against nine other existing incidence-based estimators using two tree data sets where the true numbers of species are known. Results show that the new estimator is insensitive to sample size and outperforms the other methods as judged by the root mean squared errors. The superiority of the new method is particularly noticeable when large quadrat size is used, suggesting that a few large quadrats are preferred over many small ones when sampling diversity.     

Evaluation of sampling methods for the estimation of structural indices in forest stands

The paper is about the accurate (i.e. unbiased and precise) and efficient estimation of structural indices in forest stands. We present SIAFOR, a computer programme for the calculation of four nearest-neighbour indices, which describe the spatial arrangement of tree positions, the distribution pattern of species, and the size differentiation between trees. The study uses SIAFOR as a sampling simulator in eight completely stem-mapped forest stands of varying area and structural complexity. We statistically evaluate two sample types (distance and plot sampling), comparing sampling error, bias and minimum sample size for index estimation. We introduce the concepts of measurement expansion factor (MEF) and design expansion factor (DEF) for the technical evaluation of sample type efficiency (optimal sample type). Results indicate that sampling error can reach high levels and that minimum sample sizes for index estimation often amply exceed the limit of 20% of tree density or 20 trees per species per hectare, that we set as the highest feasible sample size in normal situations. We found clear feasibility limits (in terms of minimal tree densities and reachable accuracy levels) for the estimation of all investigated indices. Generally, equal or higher sample sizes are needed for plot sampling than for distance sampling to reach equal accuracy levels. Nevertheless, plot sampling resulted more efficient for the estimation of tree size differentiation at low to medium accuracy levels. For all other investigated indices distance sampling resulted more efficient than plot sampling. Minimum sample size increases with accuracy and is negatively correlated with tree density. At a given accuracy level minimum sample size is highest for the estimation of relative mingling and lowest for tree size differentiation; furthermore it is generally lower in large stands than in small ones. Because of the consistency of our conclusions in all of the investigated stands, we think they apply in most stands of similar area (between 1 and 10 ha) and species diversity (not more than four species).     

Estimating Population Size with Noninvasive Capture-Mark-Recapture Data

Abstract:  Estimating population size of elusive and rare species is challenging. The difficulties in catching such species has triggered the use of samples collected noninvasively, such as feces or hair, from which genetic analysis yields data similar to capture-mark-recapture (CMR) data. There are, however, two differences between classical CMR and noninvasive CMR. First, capture and recapture data are gathered over multiple sampling sessions in classical CMR, whereas in noninvasive CMR they can be obtained from a single sampling session. Second, because of genotyping errors and unlike classical CMR, there is no simple relationship between (genetic) marks and individuals in noninvasive CMR. We evaluated, through simulations, the reliability of population size estimates based on noninvasive CMR. For equal sampling efforts, we compared estimates of population size N obtained from accumulation curves, a maximum likelihood, and a Bayesian estimator. For a closed population and without sampling heterogeneity, estimates obtained from noninvasive CMR were as reliable as estimates from classical CMR. The sampling structure (single or multiple session) did not alter the results, the Bayesian estimator in the case of a single sampling session presented the best compromise between low mean squared error and a 95% confidence interval encompassing the parametric value of N in most simulations. Finally, when suitable field and lab protocols were used, genotyping errors did not substantially bias population size estimates (bias < 3.5% in all simulations). The ability to reliably estimate population size from noninvasive samples taken during a single session offers a new and useful technique for the management and conservation of elusive and rare species.     

Species abundance estimation using point-to-plant sampling in a design-based setting

A new species abundance estimator is proposed when point-to-plant sampling is adopted in a design-based framework. The method is based on the relationship between each species abundance and the probability density function of the relative squared point-to-plant distance. Using this result, a kernel estimator for species abundance is provided and the nearest neighbor method is suggested for bandwidth selection. The proposed estimator requires no assumptions about the species point patterns nor corrections for sampling near the edges of the study region. Moreover, the estimator shows suitable statistical properties as well as good practical performance as is shown in a simulation study.     

Estimating the complexity of bird song by using capture-recapture approaches from community ecology

Repertoire size, the number of unique song or syllable types in the repertoire, is a widely used measure of song complexity in birds, but it is difficult to calculate this exactly in species with large repertoires. A new method of repertoire size estimation applies species richness estimation procedures from community ecology, but such capture-recapture approaches have not been much tested. Here, we establish standardized sampling schemes and estimation procedures using capture-recapture models for syllable repertoires from 18 bird species, and suggest how these may be used to tackle problems of repertoire estimation. Different models, with different assumptions regarding the heterogeneity of the use of syllable types, performed best for different species with different song organizations. For most species, models assuming heterogeneous probability of occurrence of syllables (so-called detection probability) were selected due to the presence of both rare and frequent syllables. Capture-recapture estimates of syllable repertoire size from our small sample did not differ significantly from previous estimates using larger samples of count data. However, the enumeration of syllables in 15 songs yielded significantly lower estimates than previous reports. Hence, heterogeneity in detection probability of syllables should be addressed when estimating repertoire size. This is neglected using simple enumeration procedures, but is taken into account when repertoire size is estimated by appropriate capture-recapture models adjusted for species-specific song organization characteristics. We suggest that such approaches, in combination with standardized sampling, should be applied in species with potentially large repertoire size. On the other hand, in species with small repertoire size and homogenous syllable usage, enumerations may be satisfactory. Although researchers often use repertoire size as a measure of song complexity, listeners to songs are unlikely to count entire repertoires and they may rely on other cues, such as syllable detection probability.Communicated by A. Cockburn     

Effects of Detectability on Estimates of Geographic Range Size in Bignonieae

Abstract: Extinction risk has not been evaluated for 96% of all described plant species. Given that the Global Strategy for Plant Conservation proposes preliminary conservation assessments of all described plant species by 2010, herbarium specimens (i.e., primary occurrence data) are increasingly being used to infer threat components from estimates of geographic range size. Nevertheless, estimates of range size based on herbarium data may be inaccurate due to collection bias associated with interspecific variation in detectability. We used data on 377 species of Bignonieae to test the hypothesis that there is a positive relationship between detectability and estimates of geographic range size derived from herbarium specimens. This relationship is expected if the proportion of the true geographic range size of a species that is documented by herbarium specimens is given by the product of the true geographic range size and the detectability of the species, assuming no relationship between true geographic range size and detectability. We developed 4 measures of detectability that can be estimated from herbarium data and examined the relationship between detectability and 2 types of estimates of geographic range size: area of occupancy and extent of occurrence. Our results from regressing estimates of extent of occurrence and area of occupancy on detectability across genera provided no support for this hypothesis. The same was true for regressions of estimated extent of occurrence on detectability across species within genera. Nevertheless, regressions of estimated area of occupancy on detectability across species within genera provided partial support for our hypothesis. We considered 3 possible explanations for this mixed outcome: violation of the assumption of no relationship between true geographic range size and detectability; the relationships between estimated geographic range size and detectability may be an artifact of a negative relationship between estimated area of occupancy and the sampling variance of detectability; detectability may have had 2 opposite effects on estimated species range sizes: one determines the proportion of the true range of a species documented by herbarium specimens and the other determines the distribution of true range size for the species actually observed with herbarium data. Our findings should help improve understanding of the potential biases incurred with the use of herbarium data.     

Species abundance distributions result from body size-energetics relationships

Abundance distributions are a central characteristic of ecosystems. Certain distributions have been derived from theoretical models of community organization, and therefore the fit of data to these distributions has been proposed as a test of these theories. However, it is shown here that the geometric sequence distribution can be derived directly from the empirical relationship between population density and body size, with the assumption of random or uniform body size distributions on a log scale (as holds at local scales). The geometric sequence model provides a good to excellent fit to empirical data. The presence of noise in the relationship between population density and body size creates a curve that begins to approximate a lognormal species abundance distribution as the noise term increases. For continental-scale data in which the body size distribution is not flat, the result of sampling tends again toward the lognormal. Repeat sampling over time smooths out species population fluctuations and damps out the noise, giving a more precise geometric sequence abundance distribution. It is argued that the direct derivation of this distribution from empirical relationships gives it priority over distributions derived from complex theoretical community models.     

The Botanist Effect Revisited: Plant Species Richness, County Area, and Human Population Size in the United States

Abstract:  The "botanist effect" is thought to be the reason for higher plant species richness in areas where botanists are disproportionately present as an artefactual consequence of a more thorough sampling. We examined whether this was the case for U.S. counties. We collated the number of species of vascular plants, human population size, and the area of U.S. counties. Controlling for spatial autocorrelation and county area, plant species richness increased with human population size and density in counties with and without universities and/or botanical gardens, with no significant differences in the relation between the two subsets. This is consistent with previous findings and further evidence of a broad-scale positive correlation between species richness and human population presence, which has important consequences for the experience of nature by inhabitants of densely populated regions. Combined with the many reports of a negative correlation between the two variables at a local scale, the positive relation between plant species richness in U.S. counties and human population presence stresses the need for the conservation of seminatural areas in urbanized ecosystems and for the containment of urban and suburban sprawl.     

Untested assumptions about within-species sample size and missing data in interspecific studies

Phylogenetic comparative studies rely on species-specific data that often contain missing values and/or differ in sample size among species. These phenomena may violate statistical assumptions about the non-random variance component in sampling effort. A major reason why this assumption is often not fulfilled is because the probability of being sampled (i.e., being captured or observed) may depend on species-specific characteristics. Here, we test this assumption by using information on within-species sample sizes and missing data from five independent comparative datasets of European birds. First, we show that the two estimates of data availability (missing values and within-species sample size) are positively correlated and are associated with research effort in general (the number of papers published). Second, we demonstrate biologically meaningful relationships between data availability and phenotypic traits. For example, population size, risk-taking, and habitat specialization independently predicted within-species sample size. The key determinants of missing data were population size and distribution range. However, data availability was not structured by phylogenetic relationships. These results indicate that the accuracy of sampling is repeatable and distributed non-randomly among species, as several species-specific attributes determined the probability of observation. Therefore, data availability seems to be a species-specific trait that can be shaped by ecology, life history, and behavior. Such relationships raise issues about non-random sampling, which requires attention in comparative studies.     

Disentangling the effects of heterogeneity, stochastic dynamics and sampling in a community of aquatic insects

A dynamic and heterogeneous species abundance model generating the lognormal species abundance distribution is fitted to time series of species data from an assemblage of stoneflies and mayflies (Plecoptera and Ephemeroptera) of an aquatic insect community collected over a period of 15 years. In each year except one, we analyze 5 parallel samples taken at the same time of the season giving information about the over-dispersion in the sampling relative to the Poisson distribution. Results are derived from a correlation analysis, where the correlation in the bivariate normal distribution of log abundance is used as measurement of similarity between communities. The analysis enables decomposition of the variance of the lognormal species abundance distribution into three components due to heterogeneity among species, stochastic dynamics driven by environmental noise, and over-dispersion in sampling, accounting for 62.9, 30.6 and 6.5% of the total variance, respectively. Corrected for sampling the heterogeneity and stochastic components accordingly account for 67.3 and 32.7% of the among species variance in log abundance. By using this method, it is possible to disentangle the effect of heterogeneity and stochastic dynamics by quantifying these components and correctly remove sampling effects on the observed species abundance distribution.     

A hidden species-area curve

The species-area curve, which describes the relationship between the number of observed species in a geographical region and the area of the region, plays a central role in biogeography. Beyond its scientific importance, the species-area curve is commonly used to assess the loss of species due to habitat loss. When the species-area curve is estimated from spatial samples, the existence of species with low or highly spatially variable abundance exaggerates the true rate at which species accumulate with area. Here, a hidden species-area curve is defined that accounts for this sampling effect and its estimation by maximum likelihood is outlined. Both the species-area curve and the hidden species-area curve are conditioned on the observed species list; thus the analysis does not depend on the total number of observed and unobserved species, that is, species richness. The method is tested by sub-sampling some tropical forest data and found to work well. It is also applied to a classic data set from the deep sea. For these data, accounting for this sampling effect has a large impact.     

Habitat association studies in conjunction with adaptive cluster samples

Habitat association studies investigate the relationships between habitat characteristics and animal usage of study regions. These studies are often conducted in conjunction with surveys designed primarily to estimate population totals. This paper shows that habitat association studies may proceed from surveys using adaptive cluster sampling. The manner in which units appear in the sample turns out not be relevant to the habitat association study, which proceeds as though the units came from a simple random sample. However, it is also shown that the information about the habitat association parameters is greater than one would expect from a simple random sample of the same general size.     

土壤种子库研究方法评述

土壤种子库（soil seed bank,SSB）是指存在于土壤表面和土壤中的全部存活种子的总和。开展对土壤种子库的研究,对于分析了解过去地上植被植物群落和预示未来植被发展方向具有重要的指导意义,而对于土壤种子库研究方法的探讨则是开展土壤种子库基础实验研究的基础和支撑。本文从土壤种子库的取样时间、取样方法、取样大小和数量、鉴定方法四个方面归纳了土壤种子库的研究方法,理论分析了同位素应用于土壤种子库中种子年龄结构测定的可行性,以及植冠种子库的研究方法,并探讨了提高土壤种子库中种子萌发率的新方法。综合分析来看：（1）土壤种子库的取样时间、取样方法、取样大小和数量、鉴定方法是土壤种子库研究方法需要关注的四大关键问题,对土壤种子库的实验结果具有显著的直接影响。不同的取样时间代表的土壤种子库的内容和意义不同,取样方法、大小和数量则应根据研究区和研究内容来确定,鉴定方法也应根据具体实验区研究的物种数来考量;（2）理论上在利用同位素来测定土壤种子库中种子年龄结构是可行的,但若想广泛应用于实验中仍需进一步的深入研究;（3）目前对植冠种子库的研究方法也仅仅局限于直接计数法和间接计数法,对植冠种子库的研究方法的探讨则急需加强;（4）同时总结和提出了4种提高土壤种子库中种子萌发率的方法,未来在利用种子萌发法时,应加强注意和应用这4种方法来测定土壤种子库。     

Optimization of capture–recapture monitoring of elusive species illustrated with a threatened grasshopper

Information on population sizes and trends of threatened species is essential for their conservation, but obtaining reliable estimates can be challenging. We devised a method to improve the precision of estimates of population size obtained from capture–recapture studies for species with low capture and recapture probabilities and short seasonal activity, illustrated with population data of an elusive grasshopper (Prionotropis rhodanica). We used data from 5 capture–recapture studies to identify methodological and environmental factors affecting capture and recapture probabilities and estimates of population size. In a simulation, we used the population size and capture and recapture probability estimates obtained from the field studies to identify the minimum number of sampling occasions needed to obtain unbiased and robust estimates of population size. Based on these results we optimized the capture–recapture design, implemented it in 2 additional studies, and compared their precision with those of the nonoptimized studies. Additionally, we simulated scenarios based on thresholds of population size in criteria C and D of the International Union for Conservation of Nature (IUCN) Red List to investigate whether estimates of population size for elusive species can reliably inform red-list assessments. Identifying parameters that affect capture and recapture probabilities (for the grasshopper time since emergence of first adults) and optimizing field protocols based on this information reduced study effort (−6% to −27% sampling occasions) and provided more precise estimates of population size (reduced coefficient of variation) compared with nonoptimized studies. Estimates of population size from the scenarios based on the IUCN thresholds were mostly unbiased and robust (only the combination of very small populations and little study effort produced unreliable estimates), suggesting capture–recapture can be considered reliable for informing red-list assessments. Although capture–recapture remains difficult and costly for elusive species, our optimization procedure can help determine efficient protocols to increase data quality and minimize monitoring effort.     

Determining association networks in social animals: choosing spatial–temporal criteria and sampling rates

Social Network Analysis has become an important methodological tool for advancing our understanding of human and animal group behaviour. However, researchers tend to rely on arbitrary distance and time measures when defining ‘contacts’ or ‘associations’ between individuals based on preliminary observation. Otherwise, criteria are chosen on the basis of the communication range of sensor devices (e.g. bluetooth communication ranges) or the sampling frequencies of collection devices (e.g. Global Positioning System devices). Thus, researchers lack an established protocol for determining both relevant association distances and minimum sampling rates required to accurately represent the network structure under investigation. In this paper, we demonstrate how researchers can use experimental and statistical methods to establish spatial and temporal association patterns and thus correctly characterise social networks in both time and space. To do this, we first perform a mixing experiment with Merino sheep (Ovis aries) and use a community detection algorithm that allows us to identify the spatial and temporal distance at which we can best identify clusters of previously familiar sheep. This turns out to be within 2–3 m of each other for at least 3 min. We then calculate the network graph entropy rate—a measure of ease of spreading of information (e.g. a disease) in a network—to determine the minimum sampling rate required to capture the variability observed in our sheep networks during distinct activity phases. Our results indicate the need for sampling intervals of less than a minute apart. The tools that we employ are versatile and could be applied to a wide range of species and social network datasets, thus allowing an increase in both the accuracy and efficiency of data collection when exploring spatial association patterns in gregarious species.     

Chinook salmon use of spawning patches: relative roles of habitat quality, size, and connectivity.

Declines in many native fish populations have led to reassessments of management goals and shifted priorities from consumptive uses to species preservation. As management has shifted, relevant environmental characteristics have evolved from traditional metrics that described local habitat quality to characterizations of habitat size and connectivity. Despite the implications this shift has for how habitats may be prioritized for conservation, it has been rare to assess the relative importance of these habitat components. We used an information-theoretic approach to select the best models from sets of logistic regressions that linked habitat quality, size, and connectivity to the occurrence of chinook salmon (Oncorhynchus tshawytscha) nests. Spawning distributions were censused annually from 1995 to 2004, and data were complemented with field measurements that described habitat quality in 43 suitable spawning patches across a stream network that drained 1150 km2 in central Idaho. Results indicated that the most plausible models were dominated by measures of habitat size and connectivity, whereas habitat quality was of minor importance. Connectivity was the strongest predictor of nest occurrence, but connectivity interacted with habitat size, which became relatively more important when populations were reduced. Comparison of observed nest distributions to null model predictions confirmed that the habitat size association was driven by a biological mechanism when populations were small, but this association may have been an area-related sampling artifact at higher abundances. The implications for habitat management are that the size and connectivity of existing habitat networks should be maintained whenever possible. In situations where habitat restoration is occurring, expansion of existing areas or creation of new habitats in key areas that increase connectivity may be beneficial. Information about habitat size and connectivity also could be used to strategically prioritize areas for improvement of local habitat quality, with areas not meeting minimum thresholds being deemed inappropriate for pursuit of restoration activities.     

Obtaining species: sample size considerations

Suppose fish are to be sampled from a stream. A fisheries biologist might ask one of the following three questions: ‘How many fish do I need to catch in order to see all of the species?’, ‘How many fish do I need to catch in order to see all species whose relative frequency is more than 5%?’, or ‘How many fish do I need to catch in order to see a member from each of the species A, B, and C?’. This paper offers a practical solution to such questions by setting a target sample size designed to achieve desired results with known probability. We present three sample size methods, one we call ‘exact’ and the others approximate. Each method is derived under assumed multinomial sampling, and requires (at least approximate) independence of draws and (usually) a large population. The minimum information needed to compute one of the approximate methods is the estimated relative frequency of the rarest species of interest. Total number of species is not needed. Choice of a sample size method depends largely on available computer resources. One approximation (called the ‘Monte Carlo approximation’) gets within ±6 units of exact sample size, but usually requires 20–30 minutes of computer time to compute. The second approximation (called the ‘ratio approximation’) can be computed manually and has relative error under 5% when all species are desired, but can be as much as 50% or more too high when exact sample size is small. Statistically, this problem is an application of the ‘sequential occupancy problem’. Three examples are given which illustrate the calculations so that a reader not interested in technical details can apply our results.     

A method for assigning species into groups based on generalized Mahalanobis distance between habitat model coefficients

Habitat association models are commonly developed for individual animal species using generalized linear modeling methods such as logistic regression. We considered the issue of grouping species based on their habitat use so that management decisions can be based on sets of species rather than individual species. This research was motivated by a study of western landbirds in northern Idaho forests. The method we examined was to separately fit models to each species and to use a generalized Mahalanobis distance between coefficient vectors to create a distance matrix among species. Clustering methods were used to group species from the distance matrix, and multidimensional scaling methods were used to visualize the relations among species groups. Methods were also discussed for evaluating the sensitivity of the conclusions because of outliers or influential data points. We illustrate these methods with data from the landbird study conducted in northern Idaho. Simulation results are presented to compare the success of this method to alternative methods using Euclidean distance between coefficient vectors and to methods that do not use habitat association models. These simulations demonstrate that our Mahalanobis-distance-based method was nearly always better than Euclidean-distance-based methods or methods not based on habitat association models. The methods used to develop candidate species groups are easily explained to other scientists and resource managers since they mainly rely on classical multivariate statistical methods.