Ranked set sampling for replicated sampling designs

In practical ecological sampling studies, a certain design (such as plot sampling or line-intercept sampling) is usually replicated more than once. For each replication, the Horvitz-Thompson estimation of the objective parameter is considered. Finally, an overall estimator is achieved by averaging the single Horvitz-Thompson estimators. Because the design replications are drawn independently and under the same conditions, the overall estimator is simply the sample mean of the Horvitz-Thompson estimators under simple random sampling. This procedure may be wisely improved by using ranked set sampling. Hence, we propose the replicated protocol under ranked set sampling, which gives rise to a more accurate estimation than the replicated protocol under simple random sampling.     

Inverse Adaptive Cluster Sampling

Consider a population in which the variable of interest tends to be at or near zero for many of the population units but a subgroup exhibits values distinctly different from zero. Such a population can be described as rare in the sense that the proportion of elements having nonzero values is very small. Obtaining an estimate of a population parameter such as the mean or total that is nonzero is difficult under classical fixed sample-size designs since there is a reasonable probability that a fixed sample size will yield all zeroes. We consider inverse sampling designs that use stopping rules based on the number of rare units observed in the sample. We look at two stopping rules in detail and derive unbiased estimators of the population total. The estimators do not rely on knowing what proportion of the population exhibit the rare trait but instead use an estimated value. Hence, the estimators are similar to those developed for poststratification sampling designs. We also incorporate adaptive cluster sampling into the sampling design to allow for the case where the rare elements tend to cluster within the population in some manner. The formulas for the variances of the estimators do not allow direct analytic comparison of the efficiency of the various designs and stopping rules, so we provide the results of a small simulation study to obtain some insight into the differences among the stopping rules and sampling approaches. The results indicate that a modified stopping rule that incorporates an adaptive sampling component and utilizes an initial random sample of fixed size is the best in the sense of having the smallest variance.     

Improved nonparametric estimation of the optimal diagnostic cut‐off point associated with the Youden index under different sampling schemes

A diagnostic cut‐off point of a biomarker measurement is needed for classifying a random subject to be either diseased or healthy. However, the cut‐off point is usually unknown and needs to be estimated by some optimization criteria. One important criterion is the Youden index, which has been widely adopted in practice. The Youden index, which is defined as the maximum of (sensitivity + specificity ?1), directly measures the largest total diagnostic accuracy a biomarker can achieve. Therefore, it is desirable to estimate the optimal cut‐off point associated with the Youden index. Sometimes, taking the actual measurements of a biomarker is very difficult and expensive, while ranking them without the actual measurement can be relatively easy. In such cases, ranked set sampling can give more precise estimation than simple random sampling, as ranked set samples are more likely to span the full range of the population. In this study, kernel density estimation is utilized to numerically solve for an estimate of the optimal cut‐off point. The asymptotic distributions of the kernel estimators based on two sampling schemes are derived analytically and we prove that the estimators based on ranked set sampling are relatively more efficient than that of simple random sampling and both estimators are asymptotically unbiased. Furthermore, the asymptotic confidence intervals are derived. Intensive simulations are carried out to compare the proposed method using ranked set sampling with simple random sampling, with the proposed method outperforming simple random sampling in all cases. A real data set is analyzed for illustrating the proposed method.     

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Summary Nested case–control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.     

On Use of Distinct Respondents in Randomized Response Surveys

It is well known for direct response surveys (DR), where the responses are obtained from the respondents directly, that the sample mean, based on distinct units of a simple random sample selected with replacement (SRSWR) method, is more efficient than the sample mean based on all the units including repetition. In this paper, it is shown that a linear unbiased estimator based on distinct units is inadmissible for estimating a finite population mean when the sample is selected by an arbitrary with replacement (WR) sampling scheme and the responses are obtained independently by some RR technique. Efficiencies for a few linear unbiased estimators are compared under SRSWR sampling.     

Additional Circular Systematic Sampling Methods

Several systematic sampling methods have been used to estimate the population mean when size of the population is a multiple of sample size. Among these, only few methods have been extended and used to estimate mean of the population when its size is not a multiple of sample size. In this paper, new methods called balanced circular systematic sampling and centered circular systematic sampling are introduced by extending balanced systematic sampling method and centered systematic sampling method respectively. These methods are compared with circular systematic sampling using average variance of corrected sample means for populations exhibiting approximate linear and parabolic trends. The suggested methods are found suitable to estimate the population mean.     

Ranked set sampling with unequal samples

A ranked set sampling procedure with unequal samples (RSSU) is proposed and used to estimate the population mean. This estimator is then compared with the estimators based on the ranked set sampling (RSS) and median ranked set sampling (MRSS) procedures. It is shown that the relative precisions of the estimator based on RSSU are higher than those of the estimators based on RSS and MRSS. An example of estimating the mean diameter at breast height of longleaf-pine trees on the Wade Tract in Thomas County, Georgia, is presented.     

Comparison of density and basal area estimation of mountain natural forests based on distance-based sampling methods in Zhejiang,China

ObjectiveThe sustainable development of forest ecology and forest management practices is inseparable from the support of forest surveys. Different sampling methods have an unavoidable impact on the collection of natural community characteristic information. An appropriate method reduces the cost of the investigation to the maximum degree under the premise of ensuring accuracy. Distance-based sampling methods are widely used because of their excellent performance in estimating forest population characteristics. The purpose of this study is to compare and find an efficient sampling method of natural broad-leaved forests in mountainous areas of Zhejiang, China, which is of great significance to large-scale field survey practice in similar areas.MethodOur study used census survey data from fixed monitoring sample plots of natural broad-leaved forest in Wuyanling National Nature Reserve, Zhejiang, China as an example and simulated different distance-based sampling methods, including n-tree distance (NTD), point-centered quarter (PCQ), and T-square (Ts), combined with several estimators to estimate the stand density and basal area. The results were compared with the actual mean values of the 100% survey.ResultWe found that different sampling methods and estimators significantly influenced the results. NTD1 overestimated both the stand density and basal area, while NTD2 performed the best, with the lowest RMSE. Secondary performance was obtained with Ts3, Ts5, and Ts6, with small RMSEs of density and basal area. The RMSEs of the PCQ and Ts sampling methods based on a single distance were all large.ConclusionThe NTD sampling method with the NTD2 estimator is recommended to estimate the stand density and basal area for field investigation of natural forests in the Zhejiang mountainous area.     

Nonparametric Lower Bounds for Species Richness and Shared Species Richness under Sampling without Replacement

Summary A number of species richness estimators have been developed under the model that individuals (or sampling units) are sampled with replacement. However, if sampling is done without replacement so that no sampled unit can be repeatedly observed, then the traditional estimators for sampling with replacement tend to overestimate richness for relatively high-sampling fractions (ratio of sample size to the total number of sampling units) and do not converge to the true species richness when the sampling fraction approaches one. Based on abundance data or replicated incidence data, we propose a nonparametric lower bound for species richness in a single community and also a lower bound for the number of species shared by multiple communities. Our proposed lower bounds are derived under very general sampling models. They are universally valid for all types of species abundance distributions and species detection probabilities. For abundance data, individuals' detectabilities are allowed to be heterogeneous among species. For replicated incidence data, the selected sampling units (e.g., quadrats) need not be fully censused and species can be spatially aggregated. All bounds converge correctly to the true parameters when the sampling fraction approaches one. Real data sets are used for illustration. We also test the proposed bounds by using subsamples generated from large real surveys or censuses, and their performance is compared with that of some previous estimators.     

Design‐based inference on Bernstein type estimators for continuous populations

The estimation of the values of a variable at any point of a study area is performed using Bernstein polynomials when the sampling scheme is implemented by selecting a point in each polygon of a regular grid overimposed onto the area. The evaluation of the precision of the resulting estimates is investigated under a completely design‐based framework. Moreover, as the main contribution to the mean squared error of the Bernstein‐type estimator is due to the bias, also a pseudo‐jackknife estimator is proposed. The performance of both estimators is investigated theoretically and by means of a simulation study. An application to a soil survey performed in Berkshire Downs in Oxfordshire (UK) is considered.     

Bayesian inference for disease prevalence using negative binomial group testing

Group testing, also known as pooled testing, and inverse sampling are both widely used methods of data collection when the goal is to estimate a small proportion. Taking a Bayesian approach, we consider the new problem of estimating disease prevalence from group testing when inverse (negative binomial) sampling is used. Using different distributions to incorporate prior knowledge of disease incidence and different loss functions, we derive closed form expressions for posterior distributions and resulting point and credible interval estimators. We then evaluate our new estimators, on Bayesian and classical grounds, and apply our methods to a West Nile Virus data set.     

Optimizing performance of nonparametric species richness estimators under constrained sampling

Understanding the functional relationship between the sample size and the performance of species richness estimators is necessary to optimize limited sampling resources against estimation error. Nonparametric estimators such as Chao and Jackknife demonstrate strong performances, but consensus is lacking as to which estimator performs better under constrained sampling. We explore a method to improve the estimators under such scenario. The method we propose involves randomly splitting species‐abundance data from a single sample into two equally sized samples, and using an appropriate incidence‐based estimator to estimate richness. To test this method, we assume a lognormal species‐abundance distribution (SAD) with varying coefficients of variation (CV), generate samples using MCMC simulations, and use the expected mean‐squared error as the performance criterion of the estimators. We test this method for Chao, Jackknife, ICE, and ACE estimators. Between abundance‐based estimators with the single sample, and incidence‐based estimators with the split‐in‐two samples, Chao2 performed the best when CV < 0.65, and incidence‐based Jackknife performed the best when CV > 0.65, given that the ratio of sample size to observed species richness is greater than a critical value given by a power function of CV with respect to abundance of the sampled population. The proposed method increases the performance of the estimators substantially and is more effective when more rare species are in an assemblage. We also show that the splitting method works qualitatively similarly well when the SADs are log series, geometric series, and negative binomial. We demonstrate an application of the proposed method by estimating richness of zooplankton communities in samples of ballast water. The proposed splitting method is an alternative to sampling a large number of individuals to increase the accuracy of richness estimations; therefore, it is appropriate for a wide range of resource‐limited sampling scenarios in ecology.     

Models and estimators linking individual-based and sample-based rarefaction,extrapolation and comparison of assemblages

Aims In ecology and conservation biology, the number of species counted in a biodiversity study is a key metric but is usually a biased underestimate of total species richness because many rare species are not detected. Moreover, comparing species richness among sites or samples is a statistical challenge because the observed number of species is sensitive to the number of individuals counted or the area sampled. For individual-based data, we treat a single, empirical sample of species abundances from an investigator-defined species assemblage or community as a reference point for two estimation objectives under two sampling models: estimating the expected number of species (and its unconditional variance) in a random sample of (i) a smaller number of individuals (multinomial model) or a smaller area sampled (Poisson model) and (ii) a larger number of individuals or a larger area sampled. For sample-based incidence (presence–absence) data, under a Bernoulli product model, we treat a single set of species incidence frequencies as the reference point to estimate richness for smaller and larger numbers of sampling units.Methods The first objective is a problem in interpolation that we address with classical rarefaction (multinomial model) and Coleman rarefaction (Poisson model) for individual-based data and with sample-based rarefaction (Bernoulli product model) for incidence frequencies. The second is a problem in extrapolation that we address with sampling-theoretic predictors for the number of species in a larger sample (multinomial model), a larger area (Poisson model) or a larger number of sampling units (Bernoulli product model), based on an estimate of asymptotic species richness. Although published methods exist for many of these objectives, we bring them together here with some new estimators under a unified statistical and notational framework. This novel integration of mathematically distinct approaches allowed us to link interpolated (rarefaction) curves and extrapolated curves to plot a unified species accumulation curve for empirical examples. We provide new, unconditional variance estimators for classical, individual-based rarefaction and for Coleman rarefaction, long missing from the toolkit of biodiversity measurement. We illustrate these methods with datasets for tropical beetles, tropical trees and tropical ants.Important findings Surprisingly, for all datasets we examined, the interpolation (rarefaction) curve and the extrapolation curve meet smoothly at the reference sample, yielding a single curve. Moreover, curves representing 95% confidence intervals for interpolated and extrapolated richness estimates also meet smoothly, allowing rigorous statistical comparison of samples not only for rarefaction but also for extrapolated richness values. The confidence intervals widen as the extrapolation moves further beyond the reference sample, but the method gives reasonable results for extrapolations up to about double or triple the original abundance or area of the reference sample. We found that the multinomial and Poisson models produced indistinguishable results, in units of estimated species, for all estimators and datasets. For sample-based abundance data, which allows the comparison of all three models, the Bernoulli product model generally yields lower richness estimates for rarefied data than either the multinomial or the Poisson models because of the ubiquity of non-random spatial distributions in nature.     

Estimating the encounter rate variance in distance sampling

Summary .  The dominant source of variance in line transect sampling is usually the encounter rate variance. Systematic survey designs are often used to reduce the true variability among different realizations of the design, but estimating the variance is difficult and estimators typically approximate the variance by treating the design as a simple random sample of lines. We explore the properties of different encounter rate variance estimators under random and systematic designs. We show that a design-based variance estimator improves upon the model-based estimator of Buckland et al. (2001, Introduction to Distance Sampling. Oxford: Oxford University Press, p. 79) when transects are positioned at random. However, if populations exhibit strong spatial trends, both estimators can have substantial positive bias under systematic designs. We show that poststratification is effective in reducing this bias.     

基于空间结构调查的林分密度估计

利用林分空间结构抽样调查技术,采用测量抽样点到其第k株最近相邻木的距离(距离法)进行密度估计,分别选取k=4和k=6两种距离调查方法进行分析,并对Prodan、Persson、Thompson 3种不同密度估计方法的预估能力进行检验.结果表明：不同预估方法的预估能力受林木水平分布格局影响.Prodan法在均匀分布的林分中有较强的预估能力,随着分布格局聚集性增加会产生越来越大的偏差;Persson计算法在均匀及随机分布的林分中产生正偏差,但随着分布格局聚集性增加产生的相对误差接近0,预估能力增强;Thompson计算法对随机或接近随机分布的林分有较强的预估能力,而在均匀分布和聚集分布的格局中分别产生正偏差和负偏差.抽样点为49个时,选择6株木与4株木预估能力无显著差异,因此,密度估计可与选取4株相邻木的空间结构参数调查整合在一起.     

Strategies to estimate national forest carbon stocks from inventory data: the 1990 New Zealand baseline

An estimate of live tree carbon stored in New Zealand forests at 1990 was made to partially satisfy New Zealand's international obligations under the Framework Convention for Climate Change. A national database was compiled of 4956 forest inventory plots measured as recently as possible to 1990. Plot biomass estimates were obtained by applying species allometric relationships derived from harvested stands. Forest areas and classes were taken from a 1987 national map of vegetation cover. Regularly spaced grids, based on an initial 1 km × 1 km grid, were overlaid on the total forest area and plots were tested for bias against site characteristics at the grid points. As grid point density and sample size increased, bias was minimal in regional sampling intensity and in total annual precipitation. Differences in mean elevation and annual temperature remained stable as grid point density increased, and showed little correlation with stem biomass. This sampling method gave a measure of precision not available from previous estimates. An efficient sample size to estimate the mean within a 5% level of precision (at 95% probability) required a sample of 574 plots selected from a 4‐km grid. This strategy generated a mean estimate for the 1990 New Zealand forest carbon biomass of 179.3 ± 4.9 Mg ha^?1 (± SE), totalling 919.1 ± 25.1 Mt for the 5.1 million ha mapped forest area. The mean was 6–10% lower than previous estimates, and was within the range reported for other countries. Within forest classes, mean carbon biomass ranged from 105 Mg ha^?1 in pure podocarp forest to 215 Mg ha^?1 in mixed lowland podocarp–broadleaved–beech forest. Of the major taxa groups throughout the forest estate, beech (Nothofagus) contributed 60% of the national forest carbon biomass reservoir, 26.7% was in other hardwoods, 13.2% in conifers, and 0.1% in other taxa (e.g. tree ferns).     

Associated LAH Numbers,Factorial Series Distributions and Unbiased Estimation of a Size Parameter

A finite population consists of kN individuals of N different categories with k individuals each. It is required to estimate the unknown parameter N, the number of different classes in the population. A sequential sampling scheme is considered in which individuals are sampled until a preassigned number of repetitions of already observed categories occur in the sample. Corresponding fixed sample size schemes were considered by Charalambides (1981). The sequential sampling scheme has the advantage of always allowing unbiased estimation of the size parameter N. It is shown that relative to Charalambides' fixed sample size scheme only minor adjustments are required to account for the sequential scheme. In particular, MVU estimators of parametric functions are expressible in terms of the C-numbers introduced by Charalambides.     

Median Ranked Set Sampling with Size Biased Probability of Selection

Median ranked set sampling may be combined with size biased probability of selection. A two-phase sample is assumed. In the first phase, units are selected with probability proportional to their size. In the second phase, units are selected using median ranked set sampling to increase the efficiency of the estimators relative to simple random sampling. There is also an increase in the efficiency relative to ranked set sampling (for some probability distribution functions). There will be a loss in efficiency depending on the amount of errors in ranking the units, the median ranked set sampling can be used to reduce the errors in ranking the units selected from the population. Estimators of the population mean and the population size are considered. The median ranked set sampling with probability proportion to size and with errors in ranking is considered and compared with ranked set sampling with errors in ranking. Computer simulation results for some probability distributions are also given.     

On the Use of Jack‐Knife Techniques in Systematic Sampling

We have proposed two general classes of ratio and product type estimators to estimate an unknown population parameter of a response variable y under systematic sampling strategy. Jack‐Knife technique is employed to make the classes almost/exactly unbiased and sampling variance of the proposed estimators are derived to the first order of approximation. The merits of the proposed estimators over other estimators are discussed in this paper.