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.     

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.     

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.     

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.     

Variance decomposition in two-stage plot sampling: theoretical and empirical results

The statistical properties of two-stage plot sampling estimators of abundance are considered. In the first stage, some spatial units are selected over the whole study area according to a suitable sampling design, while in the second stage, the selected units are surveyed with floating plot sampling to estimate the abundance within. Some insights into the accuracy of the resulting estimators are obtained by splitting the sample variance into the first and second-stage components, while performance is empirically checked by means of a simulation study. Simulation results show that, in most situations, a relevant amount of the overall variance is due to the second stage sampling.     

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.     

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.     

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.     

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.     

Design unbiased estimation in line intersect sampling using segmented transects

In many applications of line intersect sampling, transects consist of multiple, connected segments in a prescribed configuration. The relationship between the transect configuration and the selection probability of a population element is illustrated and a consistent sampling protocol, applicable to populations composed of arbitrarily shaped elements, is proposed. It is shown that this protocol obviates the arbitrary practice of treating multiple intersections of a single particle as independent probabilistic events and preserves the design-unbiasedness of Kaisers (1983, Biometrics 39, 965–976) conditional and unconditional estimators, suitably generalized to segmented transect designs. The relative efficiency and utility of segmented transect designs are also discussed from a fixed population perspective.     

Sample based estimation of landscape metrics; accuracy of line intersect sampling for estimating edge density and Shannon’s diversity index

A recent trend is to estimate landscape metrics using sample data and cost-efficiency is one important reason for this development. In this study, line intersect sampling (LIS) was used as an alternative to wall-to-wall mapping for estimating Shannon’s diversity index and edge length and density. Monte Carlo simulation was applied to study the statistical performance of the estimators. All combinations of two sampling designs (random and systematic distribution of transects), four sample sizes, five transect configurations (straight line, L, Y, triangle, and quadrat), two transect orientations (fixed and random), and three configuration lengths were tested, each with a large number of simulations. Reference was 50 photos of size 1 km², already manually delineated in vector format by photo interpreters using GIS environment. The performance was compared by root mean square error (RMSE) and bias. The best combination for all three metrics was found to be the systematic design and as response design the straight line configuration with random orientation of transects, with little difference between the fixed and random orientation of transects. The rate of decrease of RMSE for increasing sample size and line length was studied with a mixed linear model. It was found that the RMSE decreased to a larger degree with the systematic design than the random one, especially with increasing sample size. Due to the nonlinearity in the definition of Shannon diversity estimator its estimator has a small and negative bias, decreasing with sample size and line length. Finally, a time study was conducted, measuring the time for registration of line intersections and their lengths on non-delineated aerial photos. The time study showed that long sampling lines were more cost-efficient than short ones for photo-interpretation.     

The application of generalized linear mixed models to multi-level sampling for insect population monitoring

Iwao's quadratic regression or Taylor's Power Law (TPL) are commonly used to model the variance as a function of the mean for sample counts of insect populations which exhibit spatial aggregation. The modeled variance and distribution of the mean are typically used in pest management programs to decide if the population is above the action threshold in any management unit (MU) (e.g., orchard, forest compartment). For nested or multi-level sampling the usual two-stage modeling procedure first obtains the sample variance for each MU and sampling level using ANOVA and then fits a regression of variance on the mean for each level using either Iwao or TPL variance models. Here this approach is compared to the single-stage procedure of fitting a generalized linear mixed model (GLMM) directly to the count data with both approaches demonstrated using 2-level sampling. GLMMs and additive GLMMs (AGLMMs) with conditional Poisson variance function as well as the extension to the negative binomial are described. Generalization to more than two sampling levels is outlined. Formulae for calculating optimal relative sample sizes (ORSS) and the operating characteristic curve for the control decision are given for each model. The ORSS are independent of the mean in the case of the AGLMMs. The application described is estimation of the variance of the mean number of leaves per shoot occupied by immature stages of a defoliator of eucalypts, the Tasmanian Eucalyptus leaf beetle, based on a sample of trees within plots from each forest compartment. Historical population monitoring data were fitted using the above approaches.     

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 bias under model selection for distance sampling detection functions

Many simulation studies have examined the properties of distance sampling estimators of wildlife population size. When assumptions hold, if distances are generated from a detection model and fitted using the same model, they are known to perform well. However, in practice, the true model is unknown. Therefore, standard practice includes model selection, typically using model comparison tools like Akaike Information Criterion. Here we examine the performance of standard distance sampling estimators under model selection. We compare line and point transect estimators with distances simulated from two detection functions, hazard-rate and exponential power series (EPS), over a range of sample sizes. To mimic the real-world context where the true model may not be part of the candidate set, EPS models were not included as candidates, except for the half-normal parameterization. We found median bias depended on sample size (being asymptotically unbiased) and on the form of the true detection function: negative bias (up to 15% for line transects and 30% for point transects) when the shoulder of maximum detectability was narrow, and positive bias (up to 10% for line transects and 15% for point transects) when it was wide. Generating unbiased simulations requires careful choice of detection function or very large datasets. Practitioners should collect data that result in detection functions with a shoulder similar to a half-normal and use the monotonicity constraint. Narrow-shouldered detection functions can be avoided through good field procedures and those with wide shoulder are unlikely to occur, due to heterogeneity in detectability.     

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.     

Combining ranking information in judgment post stratified and ranked set sampling designs

Judgment post stratified (JPS) and ranked set sampling (RSS) designs rely on the ability of a ranker to assign ranks to potential observations on available experimental units. In many settings, there are often more than one rankers available and each of these rankers provide judgment ranks. This paper proposes two sampling schemes, one for JPS and the other for RSS, to combine the judgment ranks of these rankers to produce a strength of agreement measure for each fully measured unit. This strength measure is used to draw inference for the population mean and cumulative distribution function. The paper shows that the estimators constructed based on this strength measure provide a substantial improvement over the same estimators based on judgment ranking information of a single best ranker.     

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.     

The application of adaptive cluster sampling for rare subtidal macroalgae

Adaptive cluster sampling (ACS) is a targeting sampling method that provides unbiased abundance estimators for populations of rare species that may be inadequately sampled with simple random sampling (SRS). ACS has been used successfully to estimate abundances of rockfish and sardine larvae from shipboard surveys. In this study, we describe the application of ACS for subtidal macroalgae. Using SCUBA, we measured abundances of Codium mamillosum, C. pomoides, and Halimeda cuneata at three islands and two levels of wave exposure. The three species were relatively patchy and could be sampled with ACS at one site per dive. Their distributions differed among islands and with exposure to wave energy, with H. cuneata found at only one island. ACS is a useful tool for understanding the spatial distribution and abundance of populations of rare benthic species, but, as was the case in this study, may not be as efficient as sampling with SRS with comparable replication.     

Estimator bias and efficiency for adaptive cluster sampling with order statistics and a stopping rule

Practical problems facing adaptive cluster sampling with order statistics (acsord) are explored using Monte Carlo simulation for three simulated fish populations and two known waterfowl populations. First, properties of an unbiased Hansen-Hurwitz (HH) estimator and a biased alternative Horvitz-Thompson (HT) estimator are evaluated. An increase in the level of population aggregation or the initial sample size increases the efficiencies of the two acsord estimators. For less aggregated fish populations, the efficiencies decrease as the order statistic parameter r (the number of units about which adaptive sampling is carried out) increases; for the highly aggregated fish and waterfowl populations, they increase with r. Acsord is almost always more efficient than simple random sampling for the highly aggregated populations. Positive bias is observed for the HT estimator, with the maximum bias usually occurring at small values of r. Secondly, a stopping rule at the Sth iteration of adaptive sampling beyond the initial sampling unit was applied to the acsord design to limit the otherwise open-ended sampling effort. The stopping rule induces relatively high positive bias to the HH estimator if the level of the population aggregation is high, the stopping level S is small, and r is large. The bias of HT is not very sensitive to the stopping rule and its bias is often reduced by the stopping rule at smaller values of r. For more aggregated populations, the stopping rule often reduces the efficiencies of the estimators compared to the non-stopping-rule scheme, but acsord still remains more efficient than simple random sampling. Despite its bias and lack of theoretical grounding, the HT estimator is usually more efficient than the HH estimator. In the stopping rule case, the HT estimator is preferable, because its bias is less sensitive to the stopping level.     

Cross-combined composite sampling designs for identification of elevated regions

Analyzing soils for contaminants can be costly. Generally, discrete samples are gathered from within a study area, analyzed by a laboratory and the results are used in a site-specific statistical analysis. Because of the heterogeneities that exist in soil samples within study areas, a large amount of variability and skewness may be present in the sample population. This necessitates collecting a large number of samples to obtain reliable inference on the mean contaminant concentration and to understand the spatial patterns for future remediation. Composite, or Incremental, sampling is a commonly applied method for gathering multiple discrete samples and physically combining them, such that each combination of discrete samples requires a single laboratory analysis, which reduces cost and can improve the estimates of the mean concentration. While incremental sampling can reduce cost and improve mean estimates, current implementations do not readily facilitate the characterization of spatial patterns or the detection of elevated constituent regions within study areas. The methods we present in this work provide efficient estimation and inference for the mean contaminant concentration over the entire spatial area and enable the identification of high contaminant regions within the area of interest. We develop sample design methodologies that explicitly define the characteristics of these designs (such as sample grid layout) and quantify the number of incremental samples that must be obtained under a design criteria to control false positive and false negative (Type I and II) decision errors. We present the sample design theory and specifications as well as results on simulated and real data.