Precision of the mean and the design of benthos sampling programmes: caution advised

To calculate the number of samples required to estimate population density to a specified precision, a prior estimate of the sample variance is needed. Using data from the freshwater benthic literature, Downing (1979, 1980a) calculated a regression equation to predict sample variance from sampler size and population density. He used predicted sample variance to calculate the number of samples, of a range of sizes, required to estimate a range of population densities to a specified precision. He concludes that massive savings (1300 to 5000%) of total surface area sampled may be achieved by using sample units of small surface area. These conclusions are misleading. The data set used for the regression does not adequately cover the combination of a low-density population sampled by a device of small surface area. The benthic community of Belhaven Bay, East Lothian, Scotland was sampled in 1982 with a 0.1 m² grab and a 0.0018 m² corer, providing 112 sets of replicate data which were used to test the hypothesis that for a specified precision of the mean a considerable saving of total area sampled may be obtained by sampling with a device of small surface area. The benthos of Loch Creran, Argyll, Scotland was sampled with contiguous corer samples on four occasions in 1980 and 1981, providing 234 independent sets of replicate data. Contiguous samples were grouped to form several simulated series of samples of increasing surface area. A sampler of small surface area provided a saving of total area sampled of about 20%. Whether such a small saving is justifiable will depend on the extra field expenses incurred by taking many small samples.     

Can modeling improve estimation of desert tortoise population densities?

The federally listed desert tortoise (Gopherus agassizii) is currently monitored using distance sampling to estimate population densities. Distance sampling, as with many other techniques for estimating population density, assumes that it is possible to quantify the proportion of animals available to be counted in any census. Because desert tortoises spend much of their life in burrows, and the proportion of tortoises in burrows at any time can be extremely variable, this assumption is difficult to meet. This proportion of animals available to be counted is used as a correction factor (g0) in distance sampling and has been estimated from daily censuses of small populations of tortoises (6-12 individuals). These censuses are costly and produce imprecise estimates of go due to small sample sizes. We used data on tortoise activity from a large (N = 150) experimental population to model activity as a function of the biophysical attributes of the environment, but these models did not improve the precision of estimates from the focal populations. Thus, to evaluate how much of the variance in tortoise activity is apparently not predictable, we assessed whether activity on any particular day can predict activity on subsequent days with essentially identical environmental conditions. Tortoise activity was only weakly correlated on consecutive days, indicating that behavior was not repeatable or consistent among days with similar physical environments.     

Adaptive two-stage one-per-stratum sampling

We briefly describe adaptive cluster sampling designs in which the initial sample is taken according to a Markov chain one-per-stratum design (Breidt, 1995) and one or more secondary samples are taken within strata if units in the initial sample satisfy a given condition C. An empirical study of the behavior of the estimation procedure is conducted for three small artificial populations for which adaptive sampling is appropriate. The specific sampling strategy used in the empirical study was a single random-start systematic sample with predefined systematic samples within strata when the initially sampled unit in that stratum satisfies C. The bias of the Horvitz-Thompson estimator for this design is usually very small when adaptive sampling is conducted in a population for which it is suited. In addition, we compare the behavior of several alternative estimators of the standard error of the Horvitz-Thompson estimator of the population total. The best estimator of the standard error is population-dependent but it is not unreasonable to use the Horvitz-Thompson estimator of the variance. Unfortunately, the distribution of the estimator is highly skewed hence the usual approach of constructing confidence intervals assuming normality cannot be used here.     

Statistical inference using stratified judgment post-stratified samples from finite populations

This paper develops statistical inference for population mean and total using stratified judgment post-stratified (SJPS) samples. The SJPS design selects a judgment post-stratified sample from each stratum. Hence, in addition to stratum structure, it induces additional ranking structure within stratum samples. SJPS is constructed from a finite population using either a with or without replacement sampling design. Inference is constructed under both randomization theory and a super population model. In both approaches, the paper shows that the estimators of population mean and total are unbiased. The paper also constructs unbiased estimators for the variance (mean square prediction error) of the sample mean (predictor of population mean), and develops confidence and prediction intervals for the population mean. The empirical evidence shows that the proposed estimators perform better than their competitors in the literature.     

Effects of sampling error and temporal correlations in population growth on process variance estimators

Estimates of a population’s growth rate and process variance from time-series data are often used to calculate risk metrics such as the probability of quasi-extinction, but temporal correlations in the data from sampling error, intrinsic population factors, or environmental conditions can bias process variance estimators and detrimentally affect risk predictions. It has been claimed (McNamara and Harding, Ecol Lett 7:16–20, 2004) that estimates of the long-term variance that incorporate observed temporal correlations in population growth are unaffected by sampling error; however, no estimation procedures were proposed for time-series data. We develop a suite of such long-term variance estimators, and use simulated data with temporally autocorrelated population growth and sampling error to evaluate their performance. In some cases, we get nearly unbiased long-term variance estimates despite ignoring sampling error, but the utility of these estimators is questionable because of large estimation uncertainty and difficulties in estimating correlation structure in practice. Process variance estimators that ignored temporal correlations generally gave more precise estimates of the variability in population growth and of the probability of quasi-extinction. We also found that the estimation of probability of quasi-extinction was greatly improved when quasi-extinction thresholds were set relatively close to population levels. Because of precision concerns, we recommend using simple models for risk estimates despite potential biases, and limiting inference to quantifying relative risk; e.g., changes in risk over time for a single population or comparative risk among populations.     

Empirical likelihood confidence intervals for adaptive cluster sampling

Adaptive cluster sampling (ACS) is an efficient sampling design for estimating parameters of rare and clustered populations. It is widely used in ecological research. The modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators based on small samples under ACS have often highly skewed distributions. In such situations, confidence intervals based on traditional normal approximation can lead to unsatisfactory results, with poor coverage properties. Christman and Pontius (Biometrics 56:503–510, 2000) showed that bootstrap percentile methods are appropriate for constructing confidence intervals from the HH estimator. But Perez and Pontius (J Stat Comput Simul 76:755–764, 2006) showed that bootstrap confidence intervals from the HT estimator are even worse than the normal approximation confidence intervals. In this article, we consider two pseudo empirical likelihood functions under the ACS design. One leads to the HH estimator and the other leads to a HT type estimator known as the Hájek estimator. Based on these two empirical likelihood functions, we derive confidence intervals for the population mean. Using a simulation study, we show that the confidence intervals obtained from the first EL function perform as good as the bootstrap confidence intervals from the HH estimator but the confidence intervals obtained from the second EL function perform much better than the bootstrap confidence intervals from the HT estimator, in terms of coverage rate.     

Addressing onsite sampling in recreation site choice models

Independent experts and politicians have criticized statistical analyses of recreation behavior, which rely upon onsite samples due to their potential for biased inference. The use of onsite sampling usually reflects data or budgetary constraints, but can lead to two primary forms of bias in site choice models. First, the strategy entails sampling site choices rather than sampling individuals—a form of bias called endogenous stratification. Under these conditions, sample choices may not reflect the site choices of the true population. Second, exogenous attributes of the individuals sampled onsite may differ from the attributes of individuals in the population—the most common form in recreation demand is avidity bias. We propose addressing these biases by combining two the existing methods: Weighted Exogenous Stratification Maximum Likelihood estimation and propensity score estimation. We use the National Marine Fisheries Service's Marine Recreational Fishing Statistics Survey to illustrate methods of bias reduction, employing both simulated and empirical applications. We find that propensity score based weights can significantly reduce bias in estimation. Our results indicate that failure to account for these biases can overstate anglers' willingness to pay for improvements in fishing catch, but weighted models exhibit higher variance of parameter estimates and willingness to pay.     

Spreading of oil on water in the surface-tension regime

The third stage of oil spreading on water, in which surface-tension force promotes spreading against the resisting viscous effect, is investigated using a similarity solution in combination with an integral boundary-layer technique to solve the unidirectional oil-spreading dynamics problem in the last stage of spreading. The thin layer is assumed to be supplied by oil from a bulk boundary. Using a constitutive equation for oil-film surface tension versus oil-film thickness, analytical solutions near the bulk boundary and near the edge are developed. Using the asymptotic solutions to initiate integration, the differential equations for the oil thickness, oil velocity, and boundary-layer profiles are integrated starting from the leading edge and bulk boundary, which after matching provide a complete solution. The results for the spreading-law prefactors are found to differ by about 10 % from published theoretical results using the same constitutive equation. Using an empirical constitutive equation for oil-film surface tension versus distance from the bulk boundary leads to a spreading-law prefactor that is in excellent agreement with the published experimental result and published theoretical work providing and using the same empirical constitutive equation.     

Integral projection models for finite populations in a stochastic environment

Continuous types of population structure occur when continuous variables such as body size or habitat quality affect the vital parameters of individuals. These structures can give rise to complex population dynamics and interact with environmental conditions. Here we present a model for continuously structured populations with finite size, including both demographic and environmental stochasticity in the dynamics. Using recent methods developed for discrete age-structured models we derive the demographic and environmental variance of the population growth as functions of a continuous state variable. These two parameters, together with the expected population growth rate, are used to define a one-dimensional diffusion approximation of the population dynamics. Thus, a substantial reduction in complexity is achieved as the dynamics of the complex structured model can be described by only three population parameters. We provide methods for numerical calculation of the model parameters and demonstrate the accuracy of the diffusion approximation by computer simulation of specific examples. The general modeling framework makes it possible to analyze and predict future dynamics and extinction risk of populations with various types of structure, and to explore consequences of changes in demography caused by, e.g., climate change or different management decisions. Our results are especially relevant for small populations that are often of conservation concern.     

Estimating survival rates of uncatchable animals: the myth of high juvenile mortality in reptiles

Survival rates of juvenile reptiles are critical population parameters but are difficult to obtain through mark-recapture programs because these small, secretive animals are rarely caught. This scarcity has encouraged speculation that survival rates of juveniles are very low, and we test this prediction by estimating juvenile survival rates indirectly. A simple mathematical model calculates the annual juvenile survival rate needed to maintain a stable population size, using published data on adult survival rates, reproductive output, and ages at maturity in 109 reptile populations encompassing 57 species. Counter to prediction, estimated juvenile survival rates were relatively high (on average, only about 13% less than those of conspecific adults) and highly correlated with adult survival rates. Overall, survival rates during both juvenile and adult life were higher in turtles than in snakes, and higher in snakes than in lizards. As predicted from life history theory, rates of juvenile survival were higher in species that produce large offspring, and higher in viviparous squamates than in oviparous species. Our analyses challenge the widely held belief that juvenile reptiles have low rates of annual survival and suggest instead that sampling problems and the elusive biology of juvenile reptiles have misled researchers in this respect.     

Optimizing two- and three-stage designs for spatial inventories of natural resources by simulated annealing

This paper reviews design-based estimators for two- and three-stage sampling designs to estimate the mean of finite populations. This theory is then extended to spatial populations with continuous, infinite populations of sampling units at the latter stages. We then assume that the spatial pattern is the result of a spatial stochastic process, so the sampling variance of the estimators can be predicted from the variogram. A realistic cost function is then developed, based on several factors including laboratory analysis, time of fieldwork, and numbers of samples. Simulated annealing is used to find designs with minimum sampling variance for a fixed budget. The theory is illustrated with a real-world problem dealing with the volume of contaminated bed sediments in a network of watercourses. Primary sampling units are watercourses, secondary units are transects perpendicular to the axis of the watercourse, and tertiary units are points. Optimal designs had one point per transect, from one to three transects per watercourse, and the number of watercourses varied depending on the budget. However, if laboratory costs are reduced by grouping all samples within a watercourse into one composite sample, it appeared to be efficient to sample more transects within a watercourse.     

Mutation and Conservation

Mutation can critically affect the viability of small populations by causing inbreeding depression, by maintaining potentially adaptive genetic variation in quantitative characters, and through the erosion of fitness by accumulation of mildly detrimental mutations. I review and integrate recent empirical and theoretical work on spontaneous mutation and its role in population viability and conservation planning. I analyze both the maintenance of potentially adaptive genetic variation in quantitative characters and the role of detrimental mutations in increasing the extinction risk of small populations. Recent experiments indicate that the rate of production of quasineutral, potentially adaptive genetic variance in quantitative characters is an order of magnitude smaller than the total mutational variance because mutations with large phenotypic effects tend to be strongly detrimental. This implies that, to maintain normal adaptive potential in quantitative characters under a balance between mutation and random genetic drift (or among mutation, drift, and stabilizing natural selection), the effective population size should be about 5000 rather than 500 (the Franklin-Soulé number). Recent theoretical results suggest that the risk of extinction due to the fixation of mildly detrimental mutations may be comparable in importance to environmental stochasticity and could substantially decrease the long-term viability of populations with effective sizes as large as a few thousand. These findings suggest that current recovery goals for many threatened and endangered species are inadequate to ensure long-term population viability.     

Population Demographics and Genetic Diversity in Remnant and Translocated Populations of Sea Otters

Abstract: The effects of small population size on genetic diversity and subsequent population recovery are theoretically predicted, but few empirical data are available to describe those relations. We use data from four remnant and three translocated sea otter ( Enhydra lutris ) populations to examine relations among magnitude and duration of minimum population size, population growth rates, and genetic variation. Mitochondrial (mt)DNA haplotype diversity was correlated with the number of years at minimum population size ( r _s = −0.741, p = 0.038) and minimum population size ( r _s = 0.709, p = 0.054). We found no relation between population growth and haplotype diversity, although growth was significantly greater in translocated than in remnant populations. Haplotype diversity in populations established from two sources was higher than in a population established from a single source and was higher than in the respective source populations. Haplotype frequencies in translocated populations of founding sizes of 4 and 28 differed from expected, indicating genetic drift and differential reproduction between source populations, whereas haplotype frequencies in a translocated population with a founding size of 150 did not. Relations between population demographics and genetic characteristics suggest that genetic sampling of source and translocated populations can provide valuable inferences about translocations.     

Randomized graph sampling

Randomized graph sampling (RGS) is an approach for sampling populations associated with or describable as graphs, when the structure of the graph is known and the parameter of interest is the total weight of the graph. RGS is related to, but distinct from, other graph-based approaches such as snowball and network sampling. Graph elements are clustered into walks that reflect the structure of the graph, as well as operational constraints on sampling. The basic estimator in RGS can be constructed as a Horvitz-Thompson estimator. I prove it to be design-unbiased, and also show design-unbiasedness of an estimator of the sample variance when walks are sampled with replacement. Covariates can be employed for variance reduction either through improved assignment of selection probabilities to walks in the design step, or through the use of alternative estimators during analysis. The approach is illustrated with a trail maintenance example, which demonstrates that complicated approaches to assignment of selection probabilities can be counterproductive. I describe conditions under which RGS may be efficient in practice, and suggest possible applications.     

Sedimentary Pigments and Organic Matter in Relation to Benthic Fauna in the Tan-Shui Estuary, Taiwan

The benthic ecological structure of the Tan-Shui estuary, Taiwan is changed due to long term effects of dumping of urban wastewater and of engineering actions. to monitor these changes, we sampled and analyzed benthos and sediment from 12 stations on the estuary.

The composition of the dominant species of benthos varied seasonally, with molluscan and crustacean species having greater numbers and higher frequencies of occurrence than other species. the dominant taxa during winter were Nassarius sp. and Maldanidae at two stations. Analysis using Simpson's index and Shannon's index showed the benthic community varied more in coastal areas than in offshore areas.

Physicochemical analysis showed that most of the Tan-Shui estuary consisted of sandy sediment. the variations in concentrations of organic carbon and total nitrogen at each station were small. Although the concentrations of chlorophyll-a and carotenoid at all stations were generally low, the two stations had the highest concentrations, and we concluded that the concentration of pigments in these sediments was related to the abundance of benthos. the community structure of the benthos reflected the characteristics of the sediments, and benthic species exhibited selection of and adaptation to specific sedimentary environments.     

Sedimentary Pigments and Organic Matter in Relation to Benthic Fauna in the Tan-Shui Estuary,Taiwan

The benthic ecological structure of the Tan-Shui estuary, Taiwan is changed due to long term effects of dumping of urban wastewater and of engineering actions. to monitor these changes, we sampled and analyzed benthos and sediment from 12 stations on the estuary.

The composition of the dominant species of benthos varied seasonally, with molluscan and crustacean species having greater numbers and higher frequencies of occurrence than other species. the dominant taxa during winter were Nassarius sp. and Maldanidae at two stations. Analysis using Simpson's index and Shannon's index showed the benthic community varied more in coastal areas than in offshore areas.

Physicochemical analysis showed that most of the Tan-Shui estuary consisted of sandy sediment. the variations in concentrations of organic carbon and total nitrogen at each station were small. Although the concentrations of chlorophyll-a and carotenoid at all stations were generally low, the two stations had the highest concentrations, and we concluded that the concentration of pigments in these sediments was related to the abundance of benthos. the community structure of the benthos reflected the characteristics of the sediments, and benthic species exhibited selection of and adaptation to specific sedimentary environments.     

Solitary nesting and reproductive success in the paper wasp Polistes aurifer

Female Polistes paper wasps are capable of independent nesting, yet many populations demonstrate a mixture of solitary and cooperative nest foundation. Previous studies of Polistes have found survival and/or productivity advantages of cooperative nest foundation compared to solitary nesting, and reproductive skew models have been designed to predict the dynamics of such flexible cooperation. In this paper, we examine the success of different nesting strategies in a previously unstudied population of Polistes aurifer in southern California. The colony cycle of this population is less synchronous than that of other temperate species, and the frequency of solitary nesting averages 86.2%. Our data suggest that this low rate of cooperative nest founding is adaptive, as demonstrated by the lack of survival or productivity advantages for cooperative foundress associations. Due to foundress turnover and nest foundation later in the season, many nests produce only one set of offspring. This results in a loss of the eusocial nature of some nests in the population. Data from a small sample of multifoundress nests show significant positive reproductive skew, despite concession model predictions that skew should be low in populations with low ecological constraints on independent nesting. This lack of support for the concessions skew model reflects a diminished incentive for cooperation.Communicated by L. Keller     

Importance of biotic and abiotic regulators of abundance of the intertidal amphipod Calliopius laeviusculus (Amphipoda: Gammaridae) and assessment of the accuracy and precision of sampling methods for detecting abundance changes

We examined seasonal patterns of abundance for the intertidal amphipod Calliopius laeviusculus (Amphipoda: Gammaridae). Amphipods were sampled with an epibenthic sled during the daytime high tide period from 18 May to 8 August 1988. Amphipod density increased from May to June and reached maxima in both late June and early August. Amphipod density was unrelated to any abiotic component measured in the intertidal community. These variables included sampling location, wave height, water column height, water temperature, salinity and cloud cover. Changes in abundance were related with sampling date and with the onset of capelin (Mallotus villosus) spawning activity in the intertidal. Capelin eggs are an important food item for amphipods. The accuracy of density estimates obtained with the epibenthic sled was assessed through comparison with densities obtained with a more efficient quadrat sampler. Sled samples consistently sampled ca. 1% of the amphipod population. We found that a large portion of the amphipod population burrowed into the sediment and was not effectively sampled by the sled. Sled sampler precision was roughly equivalent to that of quadrats with D (precision) ranging from 0.26 to 0.42 for sled samples and D=0.29 for quadrat samples. Although sample collection with the epibenthic sled was achieved more quickly and under a wider range of weather conditions than was possible with the quadrat sampler, the serious underestimate of amphipod density based on sled samples alone indicates that both sled and quadrat samples, obtained in concert, are required to obtain accurate measures of daily variation in C. laeviusculus abundance.Contribution to the program of GIROQ (Groupe interuniversitaire de recherches oceanographique du Quebec)     

Predicting the Deleterious Effects of Mutation Load in Fragmented Populations

Abstract:  Human-induced habitat fragmentation constitutes a major threat to biodiversity. Both genetic and demographic factors combine to drive small and isolated populations into extinction vortices. Nevertheless, the deleterious effects of inbreeding and drift load may depend on population structure, migration patterns, and mating systems and are difficult to predict in the absence of crossing experiments. We performed stochastic individual-based simulations aimed at predicting the effects of deleterious mutations on population fitness (offspring viability and median time to extinction) under a variety of settings (landscape configurations, migration models, and mating systems) on the basis of easy-to-collect demographic and genetic information. Pooling all simulations, a large part (70%) of variance in offspring viability was explained by a combination of genetic structure ( F_ST ) and within-deme heterozygosity ( H_S ). A similar part of variance in median time to extinction was explained by a combination of local population size ( N ) and heterozygosity ( H_S ). In both cases the predictive power increased above 80% when information on mating systems was available. These results provide robust predictive models to evaluate the viability prospects of fragmented populations.