Two-stage estimation of ungulate abundance in Mediterranean areas using pellet group count

A design-based strategy for estimating wildlife ungulate abundance in a Mediterranean protected area (Maremma Regional Park) is considered. The estimation is based on pellet group count (clearance count technique) in a set of plots, whose size and number is established on the basis of practical considerations and available resources. The sampling scheme involves a preliminary stratification and subsequent two-stage sampling. In the first stage, large strata (defined through habitat features) are partitioned into spatial units and a sample of units is selected by means of a sampling scheme ensuring inclusion probabilities proportional to unit size, but avoiding the selection of contiguous units. Then, the abundances of the selected units are estimated in a second stage, in which plots are located using a random scheme ensuring an even coverage of the units. In small strata, only the second stage is performed. Unbiased estimators of abundance and conservative estimators of their variances are derived for each strata and for the whole study area. The proposed strategy has been applied since the Summer of 2006 and the estimation results reveal substantial improvement with respect to the previous results obtained by means of an alternative strategy.     

Estimation of small woodlot and tree row attributes in large-scale forest inventories

Forest surveys performed over a large scale (e.g. national inventories) involve several phases of sampling. The first phase is usually performed by means of a systematic search of the study region, in which the region is partitioned into regular polygons of the same size and points are randomly or systematically selected, one per polygon. In most cases, first-phase points are selected and recognized in orthophotos or very high resolution satellite images available for the whole study area. Disregarding the subsequent phases, the first phase of sampling can be effectively adopted to select small woodlots and tree rows, in the sense that a unit is selected when at least one first-phase point falls within it. On the basis of such a scheme of sampling, approximately unbiased estimators of abundance, coverage and other physical attributes readily measurable from orthophotos (e.g. tree-row length) are proposed, together with estimators of the corresponding variances. A simulation study is performed in order to check the performance of the estimators under several distributions of units over the study area (random, clustered, spatially trended).     

Sampling from partially rank-ordered sets

In this paper we introduce a new sampling design. The proposed design is similar to a ranked set sampling (RSS) design with a clear difference that rankers are allowed to declare any two or more units are tied in ranks whenever the units can not be ranked with high confidence. These units are replaced in judgment subsets. The fully measured units are then selected from these partially ordered judgment subsets. Based on this sampling scheme, we develop unbiased estimators for the population mean and variance. We show that the proposed sampling procedure has some advantages over standard ranked set sampling.     

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.     

Network inclusion probabilities and Horvitz-Thompson estimation for adaptive simple Latin square sampling

Consider a survey of a plant or animal species in which abundance or presence/absence will be recorded. Further assume that the presence of the plant or animal is rare and tends to cluster. A sampling design will be implemented to determine which units to sample within the study region. Adaptive cluster sampling designs Thompson (1990) are sampling designs that are implemented by first selecting a sample of units according to some conventional probability sampling design. Then, whenever a specified criterion is satisfied upon measuring the variable of interest, additional units are adaptively sampled in neighborhoods of those units satisfying the criterion. The success of these adaptive designs depends on the probabilities of finding the rare clustered events, called networks. This research uses combinatorial generating functions to calculate network inclusion probabilities associated with a simple Latin square sample. It will be shown that, in general, adaptive simple Latin square sampling when compared to adaptive simple random sampling will (i) yield higher network inclusion probabilities and (ii) provide Horvitz-Thompson estimators with smaller variability.     

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.     

Two neighborhood-free plot designs for adaptive sampling of forests

Adaptive cluster sampling (ACS) has the potential of being superior for sampling rare and geographically clustered populations. However, setting up an efficient ACS design is challenging. In this study, two adaptive plot designs are proposed as alternatives: one for fixed-area plot sampling and the other for relascope sampling (also known as variable radius plot sampling). Neither includes a neighborhood search which makes them much easier to execute. They do, however, include a conditional plot expansion: at a sample point where a predefined condition is satisfied, sampling is extended to a predefined larger cluster-plot or a larger relascope plot. Design-unbiased estimators of population total and its variance are derived for each proposed design, and they are applied to ten artificial and one real tree position maps to estimate density (number of trees per ha) and basal area (the cross-sectional area of a tree stem at breast height) per hectare. The performances—in terms of relative standard error (SE%)—of the proposed designs and their non-adaptive alternatives are compared. The adaptive plot designs were superior for the clustered populations in all cases of equal sample sizes and in some cases of equal area of sample plots. However, the improvement depends on: (1) the plot size factor; (2) the critical value (the minimum number of trees triggering an expansion); (3) the subplot distance for the adapted cluster-plots, and (4) the spatial arrangement of the sampled population. For some spatial arrangements, the improvement is relatively small. The adaptive designs may be particularly attractive for sampling in rare and compactly clustered populations with critical value of 1, subplot distance equal to the diameter of initial circular plots, or plot size factor of 2.5 for an initial basal area factor of 2.     

Markov Chain Monte Carlo on optimal adaptive sampling selections

Under a Bayesian population model with a given prior distribution, the optimal sampling strategy with a fixed sample size n is an n-phase adaptive one. That is, the selection of the next sampling units should sequentially depend on the information obtained from the previously selected units, including the observed values of interest. Such an optimal strategy is in general not executable in practice due to its intensive computation. In many survey sampling situations, an important problem is that one would like to select a set of units in addition to a certain number of sampling units which have been observed. If the optimal strategy is an adaptive one, the selection of the additional units should take both the labels and the observed values of the already selected units into account. Hence, a simpler optimal two-phase adaptive sampling strategy under a Bayesian population model is proposed in this article for practical interest. A Markov chain Monte Carlo method is used to approximate the posterior joint distribution of the unobserved population units after the first phase sampling, for the optimal selection of the second phase sample. This approximation method is found to be successful to select the optimal second-phase sample. Finally, this optimal strategy is applied to a set of data from a study of geothermal CO₂ emissions in Yellowstone National Park as a practical illustrative example.     

A framework for adapting survey design through time for wildlife population assessment

Sampling strategies for monitoring the status and trends in wildlife populations are often determined before the first survey is undertaken. However, there may be little information about the distribution of the population and so the sample design may be inefficient. Through time, as data are collected, more information about the distribution of animals in the survey region is obtained but it can be difficult to incorporate this information in the survey design. This paper introduces a framework for monitoring motile wildlife populations within which the design of future surveys can be adapted using data from past surveys whilst ensuring consistency in design-based estimates of status and trends through time. In each survey, part of the sample is selected from the previous survey sample using simple random sampling. The rest is selected with inclusion probability proportional to predicted abundance. Abundance is predicted using a model constructed from previous survey data and covariates for the whole survey region. Unbiased design-based estimators of status and trends and their variances are derived from two-phase sampling theory. Simulations over the short and long-term indicate that in general more precise estimates of status and trends are obtained using this mixed strategy than a strategy in which all of the sample is retained or all selected with probability proportional to predicted abundance. Furthermore the mixed strategy is robust to poor predictions of abundance. Estimates of status are more precise than those obtained from a rotating panel design.     

Asymptotics in adaptive cluster sampling

In this article we consider asymptotic properties of the Horvitz-Thompson and Hansen-Hurwitz types of estimators under the adaptive cluster sampling variants obtained by selecting the initial sample by simple random sampling without replacement and by unequal probability sampling with replacement. We develop an asymptotic framework, which basically assumes that the number of units in the initial sample, as well as the number of units and networks in the population tend to infinity, but that the network sizes are bounded. Using this framework we prove that under each of the two variants of adaptive sampling above mentioned, both the Horvitz-Thompson and Hansen-Hurwitz types of estimators are design-consistent and asymptotically normally distributed. In addition we show that the ordinary estimators of their variances are also design-consistent estimators.     

Adaptive cluster sampling with clusters selected without replacement and stopping rule

Adaptive cluster sampling (ACS) has received much attention in recent years since it yields more precise estimates than conventional sampling designs when applied to rare and clustered populations. These results, however, are impacted by the availability of some prior knowledge about the spatial distribution and the absolute abundance of the population under study. This prior information helps the researcher to select a suitable critical value that triggers the adaptive search, the neighborhood definition and the initial sample size. A bad setting of the ACS design would worsen the performance of the adaptive estimators. In particular, one of the greatest weaknesses in ACS is the inability to control the final sampling effort if, for example, the critical value is set too low. To overcome this drawback one can introduce ACS with clusters selected without replacement where one can fix in advance the number of distinct clusters to be selected or ACS with a stopping rule which stops the adaptive sampling when a predetermined sample size limit is reached or when a given stopping rule is verified. However, the stopping rule breaks down the theoretical basis for the unbiasedness of the ACS estimators introducing an unknown amount of bias in the estimates. The current study improves the performance of ACS when applied to patchy and clustered but not rare populations and/or less clustered populations. This is done by combining the stopping rule with ACS without replacement of clusters so as to further limit the sampling effort in form of traveling expenses by avoiding repeat observations and by reducing the final sample size. The performance of the proposed design is investigated using simulated and real data.     

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.     

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.     

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.     

Nonparametric mean estimation using partially ordered sets

In ranked-set sampling (RSS), the ranker must give a complete ranking of the units in each set. In this paper, we consider a modification of RSS that allows the ranker to declare ties. Our sampling method is simply to break the ties at random so that we obtain a standard ranked-set sample, but also to record the tie structure for use in estimation. We propose several different nonparametric mean estimators that incorporate the tie information, and we show that the best of these estimators is substantially more efficient than estimators that ignore the ties. As part of our comparison of estimators, we develop new results about models for ties in rankings. We also show that there are settings where, to achieve more efficient estimation, ties should be declared not just when the ranker is actually unsure about how units rank, but also when the ranker is sure about the ranking, but believes that the units are close.     

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.     

Halton iterative partitioning: spatially balanced sampling via partitioning

A new spatially balanced sampling design for environmental surveys is introduced, called Halton iterative partitioning (HIP). The design draws sample locations that are well spread over the study area. Spatially balanced designs are known to be efficient when surveying natural resources because nearby locations tend to be similar. The HIP design uses structural properties of the Halton sequence to partition a resource into nested boxes. Sample locations are then drawn from specific boxes in the partition to ensure spatial diversity. The method is conceptually simple and computationally efficient, draws spatially balanced samples in two or more dimensions and uses standard design-based estimators. Furthermore, HIP samples have an implicit ordering that can be used to define spatially balanced over-samples. This feature is particularly useful when sampling natural resources because we can dynamically add spatially balanced units from the over-sample to the sample as non-target or inaccessible units are discovered. We use several populations to show that HIP sampling draws spatially balanced samples and gives precise estimates of population totals.     

Estimation of the CDF of a finite population in the presence of acalibration sample

We compare the performance of a number of estimators of the cumulative distribution function (CDF) for the following scenario: imperfect measurements are taken on an initial sample from afinite population and perfect measurements are obtained on a small calibration subset of the initial sample. The estimators we considered include two naive estimators using perfect and imperfect measurements; the ratio, difference and regression estimators for a two-phasesample; a minimum MSE estimator; Stefanski and Bay's SIMEX estimator (1996); and two proposed estimators. The proposed estimators take the form of a weighted average of perfect and imperfect measurements. They are constructed by minimizing variance among the class of weighted averages subject to an unbiasedness constraint. They differ in the manner of estimating the weight parameters. The first one uses direct sample estimates. The second one tunes the unknown parameters to an underlying normal distribution. We compare the root mean square error (RMSE) of the proposed estimator against other potential competitors through computer simulations. Our simulations show that our second estimator has the smallest RMSE among thenine compared and that the reduction in RMSE is substantial when the calibration sample is small and the error is medium or large.     

Comparison of a two-stage sampling design and its composite sample alternative: An application to soil studies

The objective of a long-term soil survey is to determine the mean concentrations of several chemical parameters for the pre-defined soil layers and to compare them with the corresponding values in the past. A two-stage random sampling procedure is used to achieve this goal. In the first step, n subplots are selected from N subplots by simple random sampling without replacement; in the second step, m sampling sites are chosen within each of the n selected subplots. Thus n · m soil samples are collected for each soil layer. The idea of the composite sample design comes from the challenge of reducing very expensive laboratory analyses: m laboratory samples from one subplot and one soil layer are physically mixed to form a composite sample. From each of the n selected subplots, one composite sample per soil layer is analyzed in the laboratory, thus n per soil layer in total. In this paper we show that the cost is reduced by the factor m — 1 when instead of the two-stage sampling its composite sample alternative is used; however, the variance of the composite sample mean is increased. In the case of positive intraclass correlation the increase is less than 12.5%; in the case of negative intraclass correlation the increase depends on the properties of the variable as well. For the univariate case we derive the optimal number of subplots and sampling sites. A case study is discussed at the end.     

Comparison between Hansen-Hurwitz and Horvitz-Thompson estimators for adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design. This sampling design has been shown to be a useful sampling method for parameter estimation of a clustered and scattered population (Roesch, 1993; Smith et al., 1995; Thompson and Seber, 1996). Two estimators, the modified Hansen-Hurwitz (HH) and Horvitz-Thompson (HT) estimators, are available to estimate the mean or total of a population. Empirical results from previous researches indicate that the modified HT estimator has smaller variance than the modified HH estimator. We analytically compare the properties of these two estimators. Some results are obtained in favor of the modified HT estimator so that practitioners are strongly recommended to use the HT estimator despite easiness of computations for the HH estimator.