Variances and variance estimators of the improved ratio estimators under adaptive cluster sampling

Although not design-unbiased, the ratio estimator is recognized as more efficient when a certain degree of correlation exists between the variable of primary interest and the auxiliary variable. Meanwhile, the Rao–Blackwell method is another commonly used procedure to improve estimation efficiency. Various improved ratio estimators under adaptive cluster sampling (ACS) that make use of the auxiliary information together with the Rao–Blackwellized univariate estimators have been proposed in past research studies. In this article, the variances and the associated variance estimators of these improved ratio estimators are proposed for a thorough framework of statistical inference under ACS. Performance of the proposed variance estimators is evaluated in terms of the absolute relative percentage bias and the empirical mean-squared error. As expected, results show that both the absolute relative percentage bias and the empirical mean-squared error decrease as the initial sample size increases for all the variance estimators. To evaluate the confidence intervals based on these variance estimators and the finite-population Central Limit Theorem, the coverage rate and the interval width are used. These confidence intervals suffer a disadvantage similar to that of the conventional ratio estimator. Hence, alternative confidence intervals based on a certain type of adjusted variance estimators are constructed and assessed in this article.     

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.     

Community structure of the benthos in Scottish sea-lochs. III. Further studies on patchiness

Large-scale patchiness in the distribution of the benthic fauna was investigated in Loch Etive (Scottish west coast) by two series of van Veen grab hauls. Each series was taken along a traverse across the width of the loch, with sampling points about 100 m apart. One sample series was taken on sandy mud and the other, at a greater depth, on soft mud. Two approaches in data processing were applied: (1) The variance: mean ratio and the Morisita I tests for significant aggregation were applied to the species abundances in the sample series; (2) 3 measures of sample homogeneity, each involving the calculation of an index of faunal similarity, were applied to the separate samples. The results, however, could not show significant differences between the two series of samples. These findings were compared to results obtained from (a) a previous study, where differences in patchiness between areas were shown for a smaller scale of sampling design in Loch Etive and neighbouring areas, and to which the I method is also applied here; and (b) the results of applying the variance: mean ratio and I tests to data published by Holme (1953). The present results indicate greater aggregation at the present scale of sampling than for the previous, smaller scale of sampling, or for the comparable scale of Holme's sampling. The degree of concordance shown by the values of I for the circular-design sampling to the values of the 3 measures of patchiness applied previously, and to the means of the species abundances, was measured by Spearman's rank correlation coefficient. The results clearly demonstrated the I values, unlike the others measured, to be almost completely independent of the mean. It was concluded that, for comparing the pattern from benthic samples using standard-size bottom samplers, where the mean may vary widely between each set of samples, the I method is probably most useful.     

On the meta-analysis of response ratios for studies with correlated and multi-group designs

A common effect size metric used to quantify the outcome of experiments for ecological meta-analysis is the response ratio (RR): the log proportional change in the means of a treatment and control group. Estimates of the variance of RR are also important for meta-analysis because they serve as weights when effect sizes are averaged and compared. The variance of an effect size is typically a function of sampling error; however, it can also be influenced by study design. Here, I derive new variances and covariances for RR for several often-encountered experimental designs: when the treatment and control means are correlated; when multiple treatments have a common control; when means are based on repeated measures; and when the study has a correlated factorial design, or is multivariate. These developments are useful for improving the quality of data extracted from studies for meta-analysis and help address some of the common challenges meta-analysts face when quantifying a diversity of experimental designs with the response ratio.     

Iterating Masuyama’s method to reduce sampling variation

Masuyama in Sankhya 14(3):181–186, 1954 gave a method for addressing the boundary overlap problem that arises when sampling in a delineated study region. Here we propose a general sampling framework that allows one to iterate Masuyama’s method in a way that provides a smaller sampling variance. Results of simulations comparing the proposed method to existing approaches for addressing the boundary overlap problem are given. Variations, including systematic sampling, are also discussed.     

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.     

Etude de la migration verticale quotidienne des larves de Mesochaetopterus sagittarius à Nosy-Bé (Madagascar)

The vertical distribution of the planktonic larvae of Mesochaetopterus sagittarius (Claparède, 1870) has been studied on the basis of samples taken with closing nets every 3 or 4 h, for periods of at least 24 h. The daily migration does not show the typical pattern. In the evening, the larvae begin to descend from the surface layer; they reach a maximum depth around midnight. The subsequent ascent ends at about noon, when the concentration again attains its maximum close to the surface. The biometric analysis of the samples shows that the migration range depends on the size of the larvae: the bigger they are, the higher is the respective amplitude of migration. Furthermore, there exists a size gradient independent of the time of capture which is most distinct during night time; at night, average size, as well as total number of larvae, decrease in the surface layer and, accordingly, increase in the greater depths. The techniques used allow a high degree of significance to be attached to the results obtained. The standard deviation calculated for each sampling series is lower when two parallel samples are taken at each depth, and in the same water mass, by using a parachute drogue set at 20 m depth and followed during the series.     

Differences in stable isotopes in blood and feathers of seabirds are consistent across species, age and latitude: implications for food web studies

Stable isotopes of growing feathers and blood both represent assimilated diet, and both tissues are used to study the diet and foraging distribution of marine and terrestrial birds. Although most studies have assumed that both tissues represent a difference of one trophic level to diet, the enrichment factors of blood and feathers may differ, especially where endogenous reserves are used as precursors during feather synthesis. In this study, we compare carbon and nitrogen stable isotopes of blood and simultaneously growing feathers of five species of Procellariiformes, representing five genera, different geographical regions and different life stages (chicks and adults). In all species, feathers were enriched in ¹⁵N and ¹³C compared with blood. Isotopic values of carbon and nitrogen were correlated in different tissues growing simultaneously for most species analyzed, suggesting that mathematical corrections could be used to compare different tissues. Our results imply that more care needs to be taken when comparing stable isotope signatures across studies assuming different tissues are equivalent indicators of trophic ecology.     

Quantifying the efficiency of soil sampling designs: A multivariate approach

The objective of this paper is to quantify and compare the loss functions of the standard two-stage design and its composite sample alternative in the context of multivariate soil sampling. The loss function is defined (conceptually) as the ratio of cost over information and measures design inefficiency. The efficiency of the design is the reciprocal of the loss function. The focus of this paper is twofold: (a) we define a measure of multivariate information using the Kullback–Leibler distance, and (b) we derive the variance-covariance structure for two soil sampling designs: a standard two-stage design and its composite sample counterpart. Randomness in the mass of soil samples is taken into account in both designs. A pilot study in Slovenia is used to demonstrate the calculations of the loss function and to compare the efficiency of the two designs. The results show that the composite sample design is more efficient than the two-stage design. The efficiency ratio is 1.3 for pH, 2.0 for C, 2.1 for N, and 2.5 for CEC. The multivariate efficiency ratio is 2.3. These ratios primarily reflect cost ratios; influence of the information is small.     

Comparison of distribution function estimators for forestry applications following variable probability sampling

Practical considerations often motivate employing variable probability sampling designs when estimating characteristics of forest populations. Three distribution function estimators, the Horvitz-Thompson estimator, a difference estimator, and a ratio estimator, are compared following variable probability sampling in which the inclusion probabilities are proportional to an auxiliary variable, X. Relative performance of the estimators is affected by several factors, including the distribution of the inclusion probabilities, the correlation () between X and the response Y, and the position along the distribution function being estimated. Both the ratio and difference estimators are superior to the Horvitz-Thompson estimator. The difference estimator gains better precision than the ratio estimator toward the upper portion of the distribution function, but the ratio estimator is superior toward the lower end of the distribution function. The point along the distribution function at which the difference estimator becomes more precise than the ratio estimator depends on the sampling design, as well as the coefficient of variation of X and . A simple confidence interval procedure provides close to nominal coverage for intervals constructed from both the difference and ratio estimators, with the exception that coverage may be poor for the lower tail of the distribution function when using the ratio estimator.     

Correlation of primary production as measured aboard ship in Southern California Coastal waters and as estimated from satellite chlorophyll images

Chlorophyll images suitable for the regional analysis of the Southern California Bight were obtained by satellite (Nimbus-7 Coastal Zone Color Scanner) on 27 February, 6 March and 11 March 1979. An algorithm to convert satellite chlorophyll data to primary production has been developed. In addition, primary production was measured along with other biological and physical factors from ships cruising in the Southern California Bight during late February and early March 1979. There was fair agreement between shipboard measurements and satellite estimations; for example, primary production (mg carbon m^-2 d^-1) averaged 554 mg from shipboard measurement and 403 mg from satellite estimation when averaged for the entire Bight for the course of the study. The high standard error associated with the variance in productivity index used in our algorithm prevented more accurate conversion of chlorophyll concentrations into primary productivity. However, the synoptic data provided by concurrent ship and satellite measurements can be used to quantify, and perhaps reduce, sampling errors associated with ship sampling alone and to provide estimates of the accuracy of primary productivity on a regional basis.     

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.     

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.     

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.     

Estimating abundance: a non parametric mark recapture approach for open and closed systems

We present a novel, non-parametric, frequentist approach for capture-recapture data based on a ratio estimator, which offers several advantages. First, as a non-parametric model, it does not require a known underlying distribution for parameters nor the associated assumptions, eliminating the need for post-hoc corrections or additional modeling to account for heterogeneity and other violated assumptions. Second, the model explicitly deals with dependence of trials by considering trials to be dependent; therefore, cluster sampling is handled naturally and additional adjustments are not necessary. Third, it accounts for ordering, utilizing the fact that a system with a small population will have a greater frequency of recaptures “early” in the survey work compared to an identical system with a larger population. We provide mathematical proof that our estimator attains asymptotic minimum variance under open systems. We apply the model to a data set of bottlenose dolphins (Tursiops truncatus) and compare results to those from classic closed models. We show that the model has an impressive rate of convergence and demonstrate that there’s an inverse relationship between population size and the proportion of the population that need to be sampled, while achieving the same degree of accuracy for abundance estimates. The model is flexible and can apply to ecological situations as well as other situations that lend themselves to capture recapture sampling.     

Hierarchical linear mixed models in multi-stage sampling soil studies

The issue of variances of different soil variables prevailing at different sampling scales is addressed. This topic is relevant for soil science, agronomy and landscape ecology. In multi-stage sampling there are randomness components in each stage of sampling which can be taken into account by introducing random effects in analysis through the use of hierarchical linear mixed models (HLMM). Due to the nested sampling scheme, there are several hierarchical sub-models. The selection of the best model can be carried out through likelihood ratio tests (LRTs) or Wald tests, which are asymptotically equivalent under standard conditions. However, when the comparison leads to a restricted hypothesis of variance components, standard conditions are not maintained, which leads to more elaborated versions of LRTs. These versions are not disseminated among environmental scientists. The present study shows the modeling of soil data from a sampling where sites, fields within sites, transects within fields, and sampling points within transects were selected in order to take samples from different vegetation types (open and shade). For soil data, several sub-models were compared using Wald tests, classic LRTs and adjusted LRTs where the distribution of the test statistic under the null hypothesis is the Chi-square mixture of Chi-square distributions. The inclusion of random effects via HLMM and suggested by the latest version of LRT allowed us to detect effects of vegetation type on soil properties that were not detected under a classical ANOVA.     

Multiple data sources improve DNA-based mark-recapture population estimates of grizzly bears.

A fundamental challenge to estimating population size with mark-recapture methods is heterogeneous capture probabilities and subsequent bias of population estimates. Confronting this problem usually requires substantial sampling effort that can be difficult to achieve for some species, such as carnivores. We developed a methodology that uses two data sources to deal with heterogeneity and applied this to DNA mark-recapture data from grizzly bears (Ursus arctos). We improved population estimates by incorporating additional DNA "captures" of grizzly bears obtained by collecting hair from unbaited bear rub trees concurrently with baited, grid-based, hair snag sampling. We consider a Lincoln-Petersen estimator with hair snag captures as the initial session and rub tree captures as the recapture session and develop an estimator in program MARK that treats hair snag and rub tree samples as successive sessions. Using empirical data from a large-scale project in the greater Glacier National Park, Montana, USA, area and simulation modeling we evaluate these methods and compare the results to hair-snag-only estimates. Empirical results indicate that, compared with hair-snag-only data, the joint hair-snag-rub-tree methods produce similar but more precise estimates if capture and recapture rates are reasonably high for both methods. Simulation results suggest that estimators are potentially affected by correlation of capture probabilities between sample types in the presence of heterogeneity. Overall, closed population Huggins-Pledger estimators showed the highest precision and were most robust to sparse data, heterogeneity, and capture probability correlation among sampling types. Results also indicate that these estimators can be used when a segment of the population has zero capture probability for one of the methods. We propose that this general methodology may be useful for other species in which mark-recapture data are available from multiple sources.     

Regression-type estimators for adaptive two-stage sequential sampling

Adaptive two-stage sequential sampling (ATSSS) design was developed to observe more rare units and gain higher efficiency, in the sense of having a smaller variance estimator, than conventional sampling designs with equal effort for rare and spatially cluster populations. For certain rare populations, incorporating auxiliary variables into a sampling design can further improve the observation of rare units and increase efficiency. In this article, we develop regression-type estimators for ATSSS so that auxiliary variables can be incorporated into the ATSSS design when warranted. Simulation studies on two populations show that the regression-type estimators can significantly increase the efficiency of ATSSS and the detection of more rare units as compared to conventional sampling counterparts. Simulation of sampling of desert shrubs in Inner Mongolia (one of the two populations studied) showed that by incorporating a GIS auxiliary variable into ATSSS with the regression estimators resulted in a gain in efficiency over ATSSS without the auxiliary variable. Further, we found that the use of the GIS auxiliary variable in a conventional two-stage design with a regression estimator did not show a gain in efficiency.     

Self-organization of tropical seasonal rain forest in southwest China

How to measure development of ecosystems is both a theoretical and practical question in ecology. Species richness and biomass accumulation are familiar figures of merit, but they cannot be instant watched. Self-organization is a tacit character. However, methods to measure the degree of self-organization of ecosystem are problematic. To this end Lin et al. (2009) have devised indicators of energy capture and dissipation so that self-organization defined via maximum energy dissipation can be quantified easily. Here the method is used to analyze long-term data (2004-2006) of a tropical seasonal rain forest included in the ChinaFLUX program. Three years of average self-organization values were clearly separated by seasonal variation. Reflection and long wave radiation are the main two pathways of energy loss. For tropical seasonal rain forest studied, long wave radiation contributed most to energy loss, and was negatively correlated with energy capture ability (Rn/DR). The nocturnal difference between canopy and air temperatures had a strong negative correlation with the long wave radiation loss ratio. However, the long wave radiation loss ratio was slightly lower than the reflection loss ratio in rainy season, when values were very low. Precipitation and wind had significant impact on energy dissipation ability in the hot dry season, but the correlation coefficients between precipitation and wind with thermal response numbers (TRNs) were very low. The results indicated that the self-organization estimation system based on “maximum energy dissipation theory” is applicable for tropical forest.     

Including model uncertainty in estimating variances in multiple capture studies

We present bootstrap-based methods which incorporate model uncertainty in estimating variances in multiple capture studies. Each of our three methods has a specific set of properties, and we discuss when each method should be used. Our first method can be used in any multiple capture setting, but it gives an estimate of the variance conditional on the number of observed animals. Our other two methods yield estimates of the unconditional variance; they require good estimates of part or all of the specific probability model, respectively. Smoothed estimated cell probabilities are utilized by the latter method. We contrast the three methods on a real-life data set, and then conduct simulations for a simple setting. Finally, we detail the use of our methodology for specific settings and discuss adaptations for tag-return studies.