Adaptive cluster double sampling with post stratification with application to an epiphytic lichen community

The implementation of an adaptive cluster sampling design often becomes logistically challenging because variation in the final sampling effort introduces uncertainty in survey planning. To overcome this drawback, an inexpensive and easy to measure auxiliary variable could be used in a two-phase survey strategy, called adaptive cluster double sampling (Félix-Medina and Thompson in Biometrika 91:877–891, 2004). In this paper, a two-phase sampling strategy is proposed which combines the idea of adaptive cluster double sampling with the principle of post-stratification. In the first-phase an adaptive cluster sample is selected by means of an inexpensive auxiliary variable. Networks from the first phase sampling are then post-stratified according to their size. In the second-phase, the network structure is used to select a subsample of units by means of stratified random sampling. The proposed sampling strategy employs stratification without requiring an a priori delineation of the strata. Indeed, the strata sizes are estimated in the course of the two-phase sampling process. Therefore, it is suitable for situations where stratification is suspected to be efficient but strata cannot be easily delineated in advance. In this framework, a new type of estimator for the population mean which mimics the stratified sampling mean estimator and an estimator of the sampling variance are proposed. The results of a simulation study confirm, as expected, that the use of post-stratification leads to gain in precision for the estimator. The proposed sampling strategy is applied for targeting an epiphytic lichen community Lobarion pulmonariae in a forest area of the Northern Apennines (N-Italy), characterized by several species of conservation concern.     

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.     

Use of post-stratification in composite sampling for estimating mean

We show with the results of a study conducted in the Hamadan Province, Iran as to how the use of composite sampling for estimating mean zinc concentration in the soil can help reduce analytical costs by reducing the number of analysis required. We also introduce post-stratification methodology in the composition step to take advantage of possible spatial dispersion. We speculate that the zinc concentration value depends on the sample location, we first stratified the sample set and then composite units from different strata randomly. The results of a simulation study show that the use of this approach not only reduces the total costs but also increases the precision of the estimator.     

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.     

Complex national sampling design for long-term monitoring of protected dry grasslands in Switzerland

We describe a probabilistic sampling design of circular permanent plots for the long-term monitoring of protected dry grasslands in Switzerland. The population under study is defined by the perimeter of a national inventory. The monitoring focus is on the species composition of the protected grassland vegetation and derived conservation values. Efficient trend estimations are required for the whole country and for some predefined target groups (six biogeographical regions and eleven vegetation types). The target groups are equally important regardless of their size. Consequently, intensified sampling of the less frequent groups is essential for sample efficiency. The prior information needed to draw a targeted sample is obtained from the sampling frame and external databases. The logistics and generalized delineation of the target population may pose further problems. Thus, investments in fieldwork and travel time should be well balanced by selecting a cluster sample. Second, any access problems in the field and non-target units in the sample should be compensated for by selecting reserve plots as they otherwise may considerably reduce the effective sample size. Finally, the design has to be flexible as the sampling frame may change over time and sampling intensity might have to be adjusted to redefined budgets or requirements. Likewise, the variables and biological items of interest may change. To fulfil all these constraints and to optimally use the available prior information, we propose a multi-stage self-weighted unequal probability sampling design. The design uses modern techniques such as: balanced sampling, spreading, stratified balancing, calibration, unequal probability sampling and power allocation. This sampling design meets the numerous requirements of this study and provides a very efficient estimator.     

Spatially stratified sampling using auxiliary information for geostatistical mapping

This paper presents a method of spatial sampling based on stratification by Local Moran’s I _i calculated using auxiliary information. The sampling technique is compared to other design-based approaches including simple random sampling, systematic sampling on a regular grid, conditional Latin Hypercube sampling and stratified sampling based on auxiliary information, and is illustrated using two different spatial data sets. Each of the samples for the two data sets is interpolated using regression kriging to form a geostatistical map for their respective areas. The proposed technique is shown to be competitive in reproducing specific areas of interest with high accuracy.     

Spatial structure, sampling design and abundance estimates in sandy beach macroinfauna: some warnings and new perspectives

We discuss methodological aspects directed to quantify the across-shore population structure and abundance of sandy beach macroinfauna. The reliability of estimates derived from design-based (stratified random sampling) and model-based (geostatistics, kriging) approaches is discussed. Our analysis also addresses potential biases arising from environmentally driven designs that consider a priori fixed strata for sampling macroinfauna, as opposed to species-driven sampling designs, in which the entire range of across-shore distribution is covered. Model-based approaches showed, spatially, highly autocorrelated and persistent structures in two intertidal populations of the Uruguayan coast: the isopod Excirolana armata and the yellow clam Mesodesma mactroides. Both populations presented zonation patterns that ranged from the base of the dunes to upper levels of the subtidal. The Gaussian model consistently explained the spatial distribution of species and population components (clam recruits and adults), with a minor contribution (Е%) of unresolved, small-scale variability. The consistent structure of spatial dependence in annual data strongly suggests an across-shore-structured process covering close to 35 m. Kriging predictions through cross-validation corroborated the appropriateness of the models fitted through variographic analysis, and the derived abundance estimates were very similar (maximum difference=7%) to those obtained from linear interpolation. Monthly analysis of E. armata data showed marked variations in its zonation and an unstable spatial structure according to the Gaussian model. The clear spatial structure resulting from species-driven sampling was not observed when data was truncated to simulate an environmentally driven sampling design. In this case, the linear semivariogram indicated a spatial gradient, suggesting that sampling was not performed at the appropriate spatial scale. Further, the cross-validation procedure was not significant, and both density and total abundance were underestimated. We conclude that: (1) geostatistics provides useful additional information about population structure and aids in direct abundance estimation; thus we suggest it as a powerful tool for further applications in the study of sandy beach macroinfauna; and that (2) environmentally driven sampling strategies fail to provide conclusive results about population structure and abundance, and should be avoided in studies of sandy beach populations. This is especially true for microtidal beaches, where unpredictable swash strength precludes a priori stratification through environmental reference points. The need to use adaptive sampling designs and avoid snapshot sampling is also stressed. Methodological implications for the detection of macroecological patterns in sandy beach macroinfauna are also discussed.     

Spatial methods for plot-based sampling of wildlife populations

Classical sampling methods can be used to estimate the mean of a finite or infinite population. Block kriging also estimates the mean, but of an infinite population in a continuous spatial domain. In this paper, I consider a finite population version of block kriging (FPBK) for plot-based sampling. The data are assumed to come from a spatial stochastic process. Minimizing mean-squared-prediction errors yields best linear unbiased predictions that are a finite population version of block kriging. FPBK has versions comparable to simple random sampling and stratified sampling, and includes the general linear model. This method has been tested for several years for moose surveys in Alaska, and an example is given where results are compared to stratified random sampling. In general, assuming a spatial model gives three main advantages over classical sampling: (1) FPBK is usually more precise than simple or stratified random sampling, (2) FPBK allows small area estimation, and (3) FPBK allows nonrandom sampling designs.     

A combination line transect and capture-recapture sampling model for multiple observers in aerial surveys

We present a robust sampling methodology to estimate population size using line transect and capture-recapture procedures for aerial surveys. Aerial surveys usually underestimate population density due to animals being missed. A combination of capture-recapture and line transect sampling methods with multiple observers allows violation of the assumption that all animals on the centreline are sighted from the air. We illustrate our method with an example of inanimate objects which shows evidence of failure of the assumption that all objects on the centreline have probability 1 of being detected. A simulation study is implemented to evaluate the performance of three variations of the Lincoln-Petersen estimator: the overall estimator, the stratified estimator, and the general stratified estimator based on the combined likelihood proposed in this paper. The stratified Lincoln-Petersen estimator based on the combined likelihood is found to be generally superior to the other estimators.     

Ranked set sampling allocation models for multiple skewed variables: an application to agricultural data

The mean of a balanced ranked set sample is more efficient than the mean of a simple random sample of equal size and the precision of ranked set sampling may be increased by using an unbalanced allocation when the population distribution is highly skewed. The aim of this paper is to show the practical benefits of the unequal allocation in estimating simultaneously the means of more skewed variables through real data. In particular, the allocation rule suggested in the literature for a single skewed distribution may be easily applied when more than one skewed variable are of interest and an auxiliary variable correlated with them is available. This method can lead to substantial gains in precision for all the study variables with respect to the simple random sampling, and to the balanced ranked set sampling too.     

Triangulation based inclusion probabilities: a design-unbiased sampling approach

A probabilistic sampling approach for design-unbiased estimation of area-related quantitative characteristics of spatially dispersed population units is proposed. The developed field protocol includes a fixed number of 3 units per sampling location and is based on partial triangulations over their natural neighbors to derive the individual inclusion probabilities. The performance of the proposed design is tested in comparison to fixed area sample plots in a simulation with two forest stands. Evaluation is based on a general approach for areal sampling in which all characteristics of the resulting population of possible samples is derived analytically by means of a complete tessellation of the areal sampling frame. The example simulation shows promising results. Expected errors under this design are comparable to sample plots including a much greater number of trees per plot.     

Estimation of rare and clustered population mean using stratified adaptive cluster sampling

For many clustered populations, the prior information on an initial stratification exists but the exact pattern of the population concentration may not be predicted. Under this situation, the stratified adaptive cluster sampling (SACS) may provide more efficient estimates than the other conventional sampling designs for the estimation of rare and clustered population parameters. For practical interest, we propose a generalized ratio estimator with the single auxiliary variable under the SACS design. The expressions of approximate bias and mean squared error (MSE) for the proposed estimator are derived. Numerical studies are carried out to compare the performances of the proposed generalized estimator over the usual mean and combined ratio estimators under the conventional stratified random sampling (StRS) using a real population of redwood trees in California and generating an artificial population by the Poisson cluster process. Simulation results show that the proposed class of estimators may provide more efficient results than the other estimators considered in this article for the estimation of highly clumped population.

     

Adjusted two-stage adaptive cluster sampling

An adjusted two-stage sampling procedure is discussed for adaptive cluster sampling where some networks are large and others are small. A two-stage sample is drawn from the large networks and a single-stage sample is drawn from the rest. The simple random sampling (SRS) procedure without replacement is used at the initial stage. An estimator for the population mean along with its properties is discussed.     

A hierarchical model for spatial capture-recapture data

Estimating density is a fundamental objective of many animal population studies. Application of methods for estimating population size from ostensibly closed populations is widespread, but ineffective for estimating absolute density because most populations are subject to short-term movements or so-called temporary emigration. This phenomenon invalidates the resulting estimates because the effective sample area is unknown. A number of methods involving the adjustment of estimates based on heuristic considerations are in widespread use. In this paper, a hierarchical model of spatially indexed capture-recapture data is proposed for sampling based on area searches of spatial sample units subject to uniform sampling intensity. The hierarchical model contains explicit models for the distribution of individuals and their movements, in addition to an observation model that is conditional on the location of individuals during sampling. Bayesian analysis of the hierarchical model is achieved by the use of data augmentation, which allows for a straightforward implementation in the freely available software WinBUGS. We present results of a simulation study that was carried out to evaluate the operating characteristics of the Bayesian estimator under variable densities and movement patterns of individuals. An application of the model is presented for survey data on the flat-tailed horned lizard (Phrynosoma mcallii) in Arizona, USA.     

Design based estimation for ranked set sampling in finite populations

In this paper, we consider design-based estimation using ranked set sampling (RSS) in finite populations. We first derive the first and second-order inclusion probabilities for an RSS design and present two Horvitz–Thompson type estimators using these inclusion probabilities. We also develop an alternate Hansen–Hurwitz type estimator and investigate its properties. In particular, we show that this alternate estimator always outperforms the usual Hansen–Hurwitz type estimator in the simple random sampling with replacement design with comparable sample size. We also develop formulae for ratio estimator for all three developed estimators. The theoretical results are augmented by numerical and simulation studies as well as a case study using a well known data set. These show that RSS design can yield a substantial improvement in efficiency over the usual simple random sampling design in finite populations.     

On ranked set sampling for multiple characteristics

We consider the selection of samples in ranked set sampling when several attributes of each sample are of interest. We describe approaches that have appeared previously in the literature and present a novel method that seeks to achieve samples that are nearly balanced with respect to the ranks of all attributes. This method is shown to result in very little loss of precision compared to problems in which only a single sample attribute is of interest.     

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.     

The Relative Efficiency of Ranked Set Sampling in Ordinary Least Squares Regression

Ranked set sampling was developed for situations where measurement cost is expensive compared with unit acquisition. This paper presents results of simulations and theory examining the impact of balanced ranked set sampling on the relative efficiencies of the slope and intercept estimators of an ordinary least squares regression. Perfect ranking of either the independent or the dependent variable is assumed throughout. In contradistinction to most of the published ranked set sampling work, it is demonstrated that balanced ranked set sampling offers at most little improvement in the relative efficiencies of the slope estimator at any sample size.     

Spatial uncertainty analysis of population models

This paper describes an approach for conducting spatial uncertainty analysis of spatial population models, and illustrates the ecological consequences of spatial uncertainty for landscapes with different properties. Spatial population models typically simulate birth, death, and migration on an input map that describes habitat. Typically, only a single “reference” map is available, but we can imagine that a collection of other, slightly different, maps could be drawn to represent a particular species’ habitat. As a first approximation, our approach assumes that spatial uncertainty (i.e., the variation among values assigned to a location by such a collection of maps) is constrained by characteristics of the reference map, regardless of how the map was produced. Our approach produces lower levels of uncertainty than alternative methods used in landscape ecology because we condition our alternative landscapes on local properties of the reference map. Simulated spatial uncertainty was higher near the borders of patches. Consequently, average uncertainty was highest for reference maps with equal proportions of suitable and unsuitable habitat, and no spatial autocorrelation. We used two population viability models to evaluate the ecological consequences of spatial uncertainty for landscapes with different properties. Spatial uncertainty produced larger variation among predictions of a spatially explicit model than those of a spatially implicit model. Spatially explicit model predictions of final female population size varied most among landscapes with enough clustered habitat to allow persistence. In contrast, predictions of population growth rate varied most among landscapes with only enough clustered habitat to support a small population, i.e., near a spatially mediated extinction threshold. We conclude that spatial uncertainty has the greatest effect on persistence when the amount and arrangement of suitable habitat are such that habitat capacity is near the minimum required for persistence.