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     

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.     

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.     

Design-based ranked set sampling using auxiliary variables

A ranked set sampling protocol is proposed when an auxiliary variable is available in addition to the target variable in sample surveys. The protocol may be practically carried out without additional sampling effort or costs. Under the suggested sampling scheme, the estimators usually adopted in surveys with auxiliary information - such as the ratio estimator or the regression estimator - display surprising theoretical properties as well as high performance in practice.     

Randomly selected order statistics in ranked set sampling: A less expensive comparable alternative to simple random sampling

Rank-based sampling designs are powerful alternatives to simple random sampling (SRS) and often provide large improvements in the precision of estimators. In many environmental, ecological, agricultural, industrial and/or medical applications the interest lies in sampling designs that are cheaper than SRS and provide comparable estimates. In this paper, we propose a new variation of ranked set sampling (RSS) for estimating the population mean based on the random selection technique to measure a smaller number of observations than RSS design. We study the properties of the population mean estimator using the proposed design and provide conditions under which the mean estimator performs better than SRS and some existing rank-based sampling designs. Theoretical results are augmented with some numerical studies and a real-life example, where we also study the performance of our proposed design under perfect and imperfect ranking situations.     

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 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.     

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.     

Nonparametric mean estimation using partially ordered sets

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

On spline-based approaches to spatial linear regression for geostatistical data

For spatial linear regression, the traditional approach to capture spatial dependence is to use a parametric linear mixed-effects model. Spline surfaces can be used as an alternative to capture spatial variability, giving rise to a semiparametric method that does not require the specification of a parametric covariance structure. The spline component in such a semiparametric method, however, impacts the estimation of the regression coefficients. In this paper, we investigate such an impact in spatial linear regression with spline-based spatial effects. Statistical properties of the regression coefficient estimators are established under the model assumptions of the traditional spatial linear regression. Further, we examine the empirical properties of the regression coefficient estimators under spatial confounding via a simulation study. A data example in precision agriculture research regarding soybean yield in relation to field conditions is presented for illustration.     

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.     

Ranked set sampling: its essence and some new applications

Ranked set sampling is a simple idea of great use. It was proposed half a century ago. The last 15 years or so have witnessed considerable development in the research and applications of ranked set sampling. In this paper, we give an overview on ranked set sampling. We review several variants of ranked set sampling developed since the original idea was proposed. We discuss the essence and the theoretical foundation of ranked set sampling. We present some novel applications of ranked set sampling in areas such as clinical trials, genetic quantitative trait loci mappings and others. By doing so, we wish to provide the reader with a philosophical view on ranked set sampling and shed some lights on a broader range of its applications.     

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.     

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.     

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.     

Asymptotics in adaptive cluster sampling

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

Adaptive cluster sampling with clusters selected without replacement and stopping rule

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

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.     

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 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.