Use of best linear unbiased prediction for hot spot identification in two-way compositing

Compositing of individual samples is a cost-effective method for estimating a population mean, but at the expense of losing information about the individual sample values. The largest of these sample values (hotspot) is sometimes of particular interest. Sweep-out methods attempt to identify the hotspot and its value by quantifying a (hopefully, small) subset of individual values as well as the usual quantification of the composites. Sweep-out design is concerned with the sequential selection of individual samples for quantification on the basis of all earlier quantifications (both composite and individual). The design-goal is for the number of individual quantifications to be small (ideally, minimal). Previous sweep-out designs have applied to traditional (i.e., disjoint) compositing. This paper describes a sweep-out design suitable for two-way compositing. That is, the individual samples are arranged in a rectangular array and a composite is formed from each row and also from each column. At each step, the design employs all available measurements (composite and individual) to form the best linear unbiased predictions for the currently unquantified cells. The cell corresponding to the largest predicted value is chosen next for individual measurement. The procedure terminates when the hotspot has been identified with certainty.     

Identification of the largest individual sample value using composite sample data and certain modifications of the sweep-out method

Cost-effective hotspot identification is an important issue in hazardous waste site characterization and evaluation. Composite sampling techniques are known to be cost effective when the cost of measurement is substantially higher than the cost of sampling. Although compositing incurs no loss of information on the means, information on individual sample values is lost due to compositing. In particular, if the interest is in identifying the largest individual sample value, the composite sampling techniques are not able to do so. Under certain assumptions, it may be possible to satisfactorily predict individual sample values using the composite sample data, but it is not generally possible to identify the largest individual sample value. In this paper, we propose two methods of identifying the largest individual sample value with some additional measurement effort. Both methods are modifications of the simple sweep-out method proposed earlier. Since analytical results do not seem to be feasible, performance of the proposed methods is assessed via simulation. The simulation results show that both the proposed methods, namely the locally sequential sweep-out and the globally sequential sweep-out, are better than the simple sweep-out method.Prepared with partial support from the Statistical Analysis and Computing Branch, Environmental Statistics and Information Division, Office of Policy, Planning, and Evaluation, United States Environmental Protection Agency, Washington, DC under a Cooperative Agreement Number CR-821531. The contents have not been subjected to Agency review and therefore do not necessarily reflect the views of the Agency and no official endorsement should be inferred.     

Estimating prevalence using composites

We are interested in estimating the fraction of a population that possesses a certain trait, such as the presence of a chemical contaminant in a lake. A composite sample drawn from a population has the trait in question whenever one or more of the individual samples making up the composite has the trait. Let the true fraction of the population that is contaminated be p. Classical estimators of p, such as the MLE and the jackknife, have been shown to be biased. In this study, we introduce a new shrinking estimator which can be used when doing composite sampling. The properties of this estimator are investigated and compared with those of the MLE and the jackknife.     

Editorial: Composite sampling

The high costs of laboratory analytical procedures frequently strain environmental and public health budgets. Whether soil, water or biological tissue is being analysed, the cost of testing for chemical and pathogenic contaminants can be quite prohibitive.Composite sampling can substantially reduce analytical costs because the number of required analyses is reduced by compositing several samples into one and analysing the composited sample. By appropriate selection of the composite sample size and retesting of select individual samples, composite sampling may reveal the same information as would otherwise require many more analyses.Many of the limitations of composite sampling have been overcome by recent research, thus bringing out more widespread potential for using composite sampling to reduce costs of environmental and public health assessments while maintaining and often increasing the precision of sample-based inference.     

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.     

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     

Cost analysis of composite sampling for classification

When an environmental sampling objective is to classify all the sample units as contaminated or not, composite sampling with selective retesting can substantially reduce costs by reducing the number of units that require direct analysis. The tradeoff, however, is increased complexity that has its own hidden costs. For this reason, we propose a model for assessing the relative cost, expressed as the ratio of total expected cost with compositing to total expected cost without compositing (initial exhaustive testing). Expressions are derived for the following retesting protocols: (i) exhaustive, (ii) sequential and (iii) binary split. The effects of both false positive and false negative rates are also derived and incorporated. The derived expressions of relative cost are illustrated for a range of values for various cost components that reflect typical costs incurred with hazardous waste site monitoring. Results allow those who are designing sampling plans to evaluate if any of these compositing/retesting protocols will be cost effective for particular applications.     

Identifying extremely large values using composite sample data

Quantifying a composite sample results in a loss of information on the values of the constituent individual samples. As a consequence of this information loss, it is impossible to identify individual samples having large values, based on composite sample measurements alone. However, under certain circumstances, it is possible to identify individual samples having large values without exhaustively measuring all individual samples. In addition to composite sample measurements, a few additional measurements on carefully selected individual samples are sufficient to identify the individual samples having large values. In this paper, we present a statistical method to recover extremely large individual sample values using composite sample measurements. An application to site characterization is used to illustrate the method.The paper has been prepared with partial support from the United States Environmental Protection Agency Number CR815273. The contents have not been subject to Agency review and therefore do not necessarily reflect the views or policies of the Agency and no official endorsement should be inferred.     

Composite sampling: A novel method to accomplish observational economy in environmental studies: A monograph introduction

This monograph on composite sampling, co-authored by Patil, Gore, and Taillie provides, for the first time, a most comprehensive statistical account of composite sampling as an ingenious environmental sampling method to help accomplish observational economy in a variety of environmental and ecological studies. Sampling consists of selection, acquisition, and quantification of a part of the population. But often what is desirable is not affordable, and what is affordable is not adequate. How do we deal with this dilemma? Operationally, composite sampling recognizes the distinction between selection, acquisition, and quantification. In certain applications, it is a common experience that the costs of selection and acquisition are not very high, but the cost of quantification, or measurement, is substantially high. In such situations, one may select a sample sufficiently large to satisfy the requirement of representativeness and precision and then, by combining several sampling units into composites, reduce the cost of measurement to an affordable level. Thus composite sampling offers an approach to deal with the classical dilemma of desirable versus affordable sample sizes, when conventional statistical methods fail to resolve the problem. Composite sampling, at least under idealized conditions, incurs no loss of information for estimating the population means. But an important limitation to the method has been the loss of information on individual sample values, such as the extremely large value. In many of the situations where individual sample values are of interest or concern, composite sampling methods can be suitably modified to retrieve the information on individual sample values that may be lost due to compositing. In this monograph, we present statistical solutions to these and other issues that arise in the context of applications of composite sampling. The monograph is published in the Monograph Series: Environmental and Ecological Statistics <http://www.springer.com/series/7506>, vol. 4, The authors are Patil, Ganapati P., Gore, Sharad D., Taillie, Charles with the monograph co-ordinates,1st Edition., 2011, XIII, 275 p. 47 illus., SpringerLink <http://www.springerlink.com/content/978-1-4419-7627-7>, Hardcover, >  ISBN 978-1-4419-7627-7.     

Contributions to composite sampling

The initial use of composite sampling involved the analysis of many negative samples with relatively high laboratory cost (Dorfman sampling). We propose a method of double compositing and compare its efficiency with Dorfman sampling. The variability of composite measurement samples has environmental interest (hot spots). The precision of these estimates depends on the kurtosis of the distribution; leptokurtic distributions (₂ > 0) have increased precision as the number of field samples is increased. The opposite effect is obtained for platykurtic distributions. In the lognormal case, coverage probabilities are reasonable for < 0.5. The Poisson distribution can be associated with temporal compositing, of particular interest where radioactive measurements are taken. Sample size considerations indicate that the total sampling effort is directly proportional to the length of time sampled. If there is background radiation then increasing levels of this radiation require larger sample sizes to detect the same difference in radiation.     

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.     

溪流底栖动物定量与半定量采样法比较研究

溪流底栖动物定量和半定量采样法在个体数、物种数、物种相似性及生物指数方面的比较研究表明:(1)急流生境中,半定量样(踢网)的个体和物种数高于定量样(索伯网);静水-缓流生境中,半定量样(D形网)的个体和物种数一般高于定量样,且物种数有显著差异(z=-2.032,P<0.05).(2)同一样点半定量样(踢网加D形网)与定量样之间的物种相似性(平均为0.68)高于急流生境(0.56)和静水-缓流生境(0.45).(3)同一样点半定量样和定量样单独计算的生物指数值之间无显著差异.建议在应用溪流底栖动物开展水质生物评价时,可用半定量采样法完成野外采样.图4表1参8     

Quantifying the efficiency of soil sampling designs: A multivariate approach

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

Ranked set sample design for environmental investigations

Ranked set sampling can provide an efficient basis for estimating parameters of environmental variables, particularly when sampling costs are intrinsically high. Various ranked set estimators are considered for the population mean and contrasted in terms of their efficiencies and useful- ness, with special concern for sample design considerations. Specifically, we consider the effects of the form of the underlying random variable, optimisation of efficiency and how to allocate sampling effort for best effect (e.g. one large sample or several smaller ones of the same total size). The various prospects are explored for two important positively skew random variables (lognormal and extreme value) and explicit results are given for these cases. Whilst it turns out that the best approach is to use the largest possible single sample and the optimal ranked set best linear estimator (ranked set BLUE), we find some interesting qualitatively different conclusions for the two skew distributions     

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.     

Detecting Rare Species with Random or Subjective Sampling: a Case Study of Red-Listed Saproxylic Beetles in Boreal Sweden

Abstract:  Efficient sampling design in field studies is important for economical and statistical reasons. We compared two ways to distribute sampling effort over an area, either randomly or subjectively. We searched for red-listed saproxylic (wood-living) beetles in 30 spruce stands in boreal Sweden by sifting wood from dead trees. We randomly selected positions within each stand with a geographic positioning system and sampled the nearest dead tree (random sample). In the same stand we also sampled dead trees that, based on literature, were likely to host such species (subjective sampling). The subjective sampling (two to five samples per stand, depending on stand size) was compared with the higher, random sampling effort (fixed level of 12 samples/stand). Subjective sampling was significantly more efficient. Red-listed species were found in 36% of the subjective samples and in 16% of the random samples. Nevertheless, the larger random effort resulted in a comparable number of red-listed species per stand and in 13 detected species in total (vs. 12 species with subjective sampling). Random sampling was less efficient, but provided an unbiased alternative more suitable for statistical purposes, as needed in, for example, monitoring programs. Moreover, new species-specific knowledge can be gained through random searches.     

The use of composite sampling in contaminated sites—a case study

Simulated composite sampling was carried out using data from a contaminated site. The values obtained by composite sampling were compared with the results obtained using discrete (individual) samples. It is appropriate to use a modified investigation level (MIL) when using composite samples. The MIL is lower than the standard investigation level, (IL). Various MILs were considered in this study. Too low an MIL will indicate that some composite samples require further investigation, when none of the discrete samples comprising the composite would have exceeded the IL. Too high an MIL will result in some discrete samples that exceed the IL being missed. A suggested MIL is IL/ where n is the number of discrete samples in the composite sample. This MIL was found to give few false negatives but many fewer false positives than the IL/n rule. Although this MIL was effective on the test data it could be site specific. Some local areas of high concentration may be missed with composite samples if a lower investigation level is used. These however do not make a large contribution to the health risk because they will have a contaminant level only slightly higher than the IL, and the neighboring samples must have a low concentration of the contaminant. The increased risk due this cause may be more than offset by the higher sampling density made possible through the economies of composite sampling When composite sampling is used as the first phase of an adaptive cluster-sampling scheme, it must be augmented by additional samples to delineate the contaminated area to be cleaned up. Composite sampling can also be effectively used in a clean up unit technique, where a clean up unit is represented by one or more composite samples. Suggestions are given for when composite sampling can be used effectively.     

Distribution-free ranked-set sample procedures for umbrella alternatives in the m-sample setting

In settings where measurements are costly and/or difficult to obtain but ranking of the potential sample data is relatively easy and reliable, the use of statistical methods based on a ranked-set sampling approach can lead to substantial improvement over analogous methods associated with simple random samples. Previous nonparametric work in this area has been concentrated almost exclusively on the one- and two-sample location problems. In this paper we develop ranked-set sample procedures for the m-sample location setting where the treatment effect parameters follow a restricted umbrella pattern. Distribution-free testing procedures are developed for both the case where the peak of the umbrella is known and for the case where it is unknown. Small sample and asymptotic null distribution properties are provided for the peak-known test statistic.     

Post-war levels of persistent organic pollutants (POPs) in air from Serbia determined by active and passive sampling methods

For the duration of the war accident in former Yugoslavia, several industrial and military targets were burnt and damaged, resulting in a significant release of persistent organic pollutants. Locations heavily targeted in the attacks were later defined by UNEP as four “hot spots”: Kragujevac, Novi Sad, Pancevo and Bor. We analyzed concentration levels of pollutants collected in 2004 and 2005 in air samples from the city of Kragujevac, Serbia, following the war accident of 1999. Pollutants included polycyclic aromatic hydrocarbons (PAHs), hexachlorocyclohexane (HCH), dichloro-diphenyl-trichloroethane (DDT), dichloro-diphenyl-dichloroethylene (DDE), dichloro-diphenyl-dichloroethane (DDD) and polychlorinated biphenyls (PCBs). We present results obtained during air sampling campaign conducted in July 2004 by the active sampling method; and during September 2004–June 2005 by the passive sampling method. Our findings show the occurrence of residual quantities of DDT, HCH, PCBs and PAHs in air samples. High levels of PCBs are probably due to the destruction of transformers during the war accident.