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.     

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.     

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.     

Variance estimation in the EMAP strategy for sampling discrete ecological resources

The United States Environmental Protection Agency's Environmental Monitoring and Assessment Program (EMAP) is designed to describe status, trends and spatial pattern of indicators of condition of the nation's ecological resources. The proposed sampling design for EMAP is based on a triangular systematic grid and employs both variable probability and double sampling. The Horvitz-Thompson estimator provides the foundation of the design-based estimation strategy used in EMAP. However, special features of EMAP designed to accommodate the complexity of sampling environmental resources on a national scale require modifications of standard variance estimation procedures as well as development of new techniques. An overview of variance estimation methods proposed for application to EMAP's sampling strategy for discrete resources is presented.     

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.     

Rao-Blackwell versions of the Horvitz-Thompson and Hansen-Hurwitz in adaptive cluster sampling

Thompson (1990) introduced the adaptive cluster sampling design and developed two unbiased estimators, the modified Horvitz-Thompson (HT) and Hansen-Hurwitz (HH) estimators, for this sampling design and noticed that these estimators are not a function of the minimal sufficient statistics. He applied the Rao-Blackwell theorem to improve them. Despite having smaller variances, these latter estimators have not received attention because a suitable method or algorithm for computing them was not available. In this paper we obtain closed forms of the Rao-Blackwell versions which can easily be computed. We also show that the variance reduction for the HH estimator is greater than that for the HT estimator using Rao-Blackwell versions. When the condition for extra samples is 0$$ " align="middle" border="0"> , one can expect some Rao-Blackwell improvement in the HH estimator but not in the HT estimator. Two examples are given.     

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.     

Empirical likelihood confidence intervals for adaptive cluster sampling

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

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.     

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.     

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.     

Spatially-balanced sampling versus unbalanced stratified sampling for assessing forest change: evidences in favour of spatial balance

Large-scale remote sensing-based inventories of forest cover are usually carried out by combining unsupervised classifications of satellite pixels into forest/non forest classes (map data) with subsequent time-consuming visual on-screen imagery classification of a probabilistic sample of pixels taken as the ground truth (reference data). In this paper the estimation of forest change from a sample of reference data is approached by: (i) exploiting map data to construct strata in which changes are occurred, and then adopting the stratified sampling joined with the HT estimator with most sampling effort devoted to strata where changes are occurred irrespective of their size, as suggested in most remote sensing literature regarding land change assessments; (ii) adopting a spatial scheme ensuring spatially balanced samples, as suggested in most recent statistical literature regarding spatial surveys, and exploiting the map data in the difference estimator. The results of a comparison performed on an artificial population of reference data generated from a real population of map data recorded in Sardinia (Italy) discourage the use of unbalanced stratified samples that achieve the worst precision. The best results are obtained by means of spatially balanced samples or stratification with nearly proportional allocation to strata.     

A new design for sampling with adaptive sample plots

Adaptive cluster sampling (ACS) is a sampling technique for sampling rare and geographically clustered populations. Aiming to enhance the practicability of ACS while maintaining some of its major characteristics, an adaptive sample plot design is introduced in this study which facilitates field work compared to “standard” ACS. The plot design is based on a conditional plot expansion: a larger plot (by a pre-defined plot size factor) is installed at a sample point instead of the smaller initial plot if a pre-defined condition is fulfilled. This study provides insight to the statistical performance of the proposed adaptive plot design. A design-unbiased estimator is presented and used on six artificial and one real tree position maps to estimate density (number of objects per ha). The performance in terms of coefficient of variation is compared to the non-adaptive alternative without a conditional expansion of plot size. The adaptive plot design was superior in all cases but the improvement depends on (1) the structure of the sampled population, (2) the plot size factor and (3) the critical value (the minimum number of objects triggering an expansion). For some spatial arrangements the improvement is relatively small. The adaptive design may be particularly attractive for sampling in rare and compactly clustered populations with an appropriately chosen plot size factor.     

Restricted adaptive cluster sampling

Adaptive cluster sampling can be a useful design for sampling rare and patchy populations. With this design the initial sample size is fixed but the size of the final sample (and total sampling effort) cannot be predicted prior to sampling. For some populations the final sample size can be quite variable depending on the level of patchiness. Restricted adaptive cluster sampling is a proposed modification where a limit is placed on the sample size prior to sampling and quadrats are selected sequentially for the initial sample size. As a result there is less variation in the final sample size and the total sampling effort can be predicted with some certainty, which is impor- tant for many ecological studies. Estimates of density are biased with the restricted design but under some circumstances the bias can be estimated well by bootstrapping. © Rapid Science 1998     

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.     

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.     

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.     

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.     

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.