Improved procedures for estimation of disease prevalence using ranked set sampling

Ranked set sampling (RSS) is a sampling procedure that can be considerably more efficient than simple random sampling (SRS). When the variable of interest is binary, ranking of the sample observations can be implemented using the estimated probabilities of success obtained from a logistic regression model developed for the binary variable. The main objective of this study is to use substantial data sets to investigate the application of RSS to estimation of a proportion for a population that is different from the one that provides the logistic regression. Our results indicate that precision in estimation of a population proportion is improved through the use of logistic regression to carry out the RSS ranking and, hence, the sample size required to achieve a desired precision is reduced. Further, the choice and the distribution of covariates in the logistic regression model are not overly crucial for the performance of a balanced RSS procedure.     

Ranked set sampling: cost and optimal set size

McIntyre (1952, Australian Journal of Agricultural Research 3, 385-390) introduced ranked set sampling (RSS) as a method for improving estimation of a population mean in settings where sampling and ranking of units from the population are inexpensive when compared with actual measurement of the units. Two of the major factors in the usefulness of RSS are the set size and the relative costs of the various operations of sampling, ranking, and measurement. In this article, we consider ranking error models and cost models that enable us to assess the effect of different cost structures on the optimal set size for RSS. For reasonable cost structures, we find that the optimal RSS set sizes are generally larger than had been anticipated previously. These results will provide a useful tool for determining whether RSS is likely to lead to an improvement over simple random sampling in a given setting and, if so, what RSS set size is best to use in this case.     

General ranked set sampling with cost considerations

Nahhas, Wolfe, and Chen (2002, Biometrics58, 964-971) considered optimal set size for ranked set sampling (RSS) with fixed operational costs. This framework can be very useful in practice to determine whether RSS is beneficial and to obtain the optimal set size that minimizes the variance of the population estimator for a fixed total cost. In this article, we propose a scheme of general RSS in which more than one observation can be taken from each ranked set. This is shown to be more cost-effective in some cases when the cost of ranking is not so small. We demonstrate using the example in Nahhas, Wolfe, and Chen (2002, Biometrics58, 964-971), by taking two or more observations from one set even with the optimal set size from the RSS design can be more beneficial.     

On Estimating a Population Proportion via Ranked Set Sampling

This paper explores the use of the rank set sampling (RSS) protocol as it pertains to the estimation of a population proportion. The maximum likelihood estimator (MLE) and the sample proportion, both based on the RSS data, are discussed and their corresponding asymptotic distributions are derived. Based on these results the MLE is found to be uniformly more efficient than the sample proportion. Nevertheless, both estimators are more efficient than the simple random sample proportion. The greatest gains in efficiency are obtained at the center of the parameter space. Finally, these results remain valid in the presence of judgment error. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal adaptive designs for binary response trials

Summary. We derive the optimal allocation between two treatments in a clinical trial based on the following optimality criterion: for fixed variance of the test statistic, what allocation minimizes the expected number of treatment failures? A sequential design is described that leads asymptotically to the optimal allocation and is compared with the randomized play‐the‐winner rule, sequential Neyman allocation, and equal allocation at similar power levels. We find that the sequential procedure generally results in fewer treatment failures than the other procedures, particularly when the success probabilities of treatments are smaller.     

Pair Rank Set Sampling

Ranked set sampling (RSS) as suggested by McIntyre (1952) may be modified to introduced a new sampling method called pair rank set sampling (PRSS), which might be used in some area of application instead of the RSS to increase the efficiency of the estimators relative to the simple random sampling (SRS) method. Estimators of the population mean are considered. An example using real data is presented to illustrate computations.     

Standard errors for attributable risk for simple and complex sample designs

Adjusted attributable risk (AR) is the proportion of diseased individuals in a population that is due to an exposure. We consider estimates of adjusted AR based on odds ratios from logistic regression to adjust for confounding. Influence function methods used in survey sampling are applied to obtain simple and easily programmable expressions for estimating the variance of AR. These variance estimators can be applied to data from case-control, cross-sectional, and cohort studies with or without frequency or individual matching and for sample designs with subject samples that range from simple random samples to (sample) weighted multistage stratified cluster samples like those used in national household surveys. The variance estimation of AR is illustrated with: (i) a weighted stratified multistage clustered cross-sectional study of childhood asthma from the Third National Health and Examination Survey (NHANES III), and (ii) a frequency-matched case-control study of melanoma skin cancer.     

Recommendations for estimating mark rate of cetaceans in photo-ID research: A critique of field sampling protocols and variance estimation

Mark rate, or the proportion of the population with unique, identifiable marks, must be determined in order to estimate population size from photographic identification data. In this study we address field sampling protocols and estimation methods for robust estimation of mark rate and its uncertainty in cetacean populations. We present two alternatives for estimating the variance of mark rate: (1) a variance estimator for clusters of unequal sizes (SRCS) and (2) a hierarchical Bayesian model (SRCS-Bayes), and compare them to the simple random sampling (SRS) variance estimator. We tested these variance estimators using a simulation to see how they perform at varying mark rates, number of groups sampled, photos per group, and mean group sizes. The hierarchical Bayesian model outperformed the frequentist variance estimators, with the true mark rate of the population held in its 95% HDI 91.9% of the time (compared with coverage of 79% for the SRS method and 76.3% for the SRCS-Cochran method). The simulation results suggest that, ideally, mark rate and its precision should be quantified using hierarchical Bayesian modeling, and researchers should attempt to sample as many unique groups as possible to improve accuracy and precision.     

Multiple sensitive estimation and optimal sample size allocation in the item sum technique

For surveys of sensitive issues in life sciences, statistical procedures can be used to reduce nonresponse and social desirability response bias. Both of these phenomena provoke nonsampling errors that are difficult to deal with and can seriously flaw the validity of the analyses. The item sum technique (IST) is a very recent indirect questioning method derived from the item count technique that seeks to procure more reliable responses on quantitative items than direct questioning while preserving respondents' anonymity. This article addresses two important questions concerning the IST: (i) its implementation when two or more sensitive variables are investigated and efficient estimates of their unknown population means are required; (ii) the determination of the optimal sample size to achieve minimum variance estimates. These aspects are of great relevance for survey practitioners engaged in sensitive research and, to the best of our knowledge, were not studied so far. In this article, theoretical results for multiple estimation and optimal allocation are obtained under a generic sampling design and then particularized to simple random sampling and stratified sampling designs. Theoretical considerations are integrated with a number of simulation studies based on data from two real surveys and conducted to ascertain the efficiency gain derived from optimal allocation in different situations. One of the surveys concerns cannabis consumption among university students. Our findings highlight some methodological advances that can be obtained in life sciences IST surveys when optimal allocation is achieved.     

Non‐Parametric Selected Ranked Set Sampling

A nonparametric selected ranked set sampling is suggested. The estimator of population mean based on the new approach is compared with that using the simple random sampling (SRS), the ranked set sampling (RSS) and the median ranked set sampling (MRSS) methods. The estimator of population mean using the new approach is found to be more efficient than its counter‐parts for almost all the cases considered.     

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.     

Bayesian Estimation Using Ranked Set Sampling

Bayesian Estimation of the parameter of a distribution is considered using Ranked set sampling (RSS). It is shown that for at least one RSS plan, the Bayes estimator has smaller Bayes risk than the Bayes estimator using simple random sampling (SRS). Furthermore, for exponential family with conjugate prior, the Bayes estimator of the mean using balanced RSS dominates, in terms of its Bayes risk, the Bayes estimator of the mean using SRS. This procedure is used to estimate the average Milk yield of four hundreds and two sheep. The empirical efficiency supports the theoretical findings.     

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.     

Using Allele Frequencies and Geographic Subdivision to Reconstruct Gene Trees within a Species: Molecular Variance Parsimony

We formalize the use of allele frequency and geographic information for the construction of gene trees at the intraspecific level and extend the concept of evolutionary parsimony to molecular variance parsimony. The central principle is to consider a particular gene tree as a variable to be optimized in the estimation of a given population statistic. We propose three population statistics that are related to variance components and that are explicit functions of phylogenetic information. The methodology is applied in the context of minimum spanning trees (MSTs) and human mitochondrial DNA restriction data, but could be extended to accommodate other tree-making procedures, as well as other data types. We pursue optimal trees by heuristic optimization over a search space of more than 1.29 billion MSTs. This very large number of equally parsimonious trees underlines the lack of resolution of conventional parsimony procedures. This lack of resolution is highlighted by the observation that equally parsimonious trees yield very different estimates of population genetic diversity and genetic structure, as shown by null distributions of the population statistics, obtained by evaluation of 10,000 random MSTs. We propose a non-parametric test for the similarity between any two trees, based on the distribution of a weighted coevolutionary correlation. The ability to test for tree relatedness leads to the definition of a class of solutions instead of a single solution. Members of the class share virtually all of the critical internal structure of the tree but differ in the placement of singleton branch tips.     

Estimation of Ratio Using Rank Set Sampling

Ranked set sampling (RSS) as suggested by McIntyre (1952) and developed by Takahasi and Wakimoto (1968) is used to estimate the ratio. It is proved that by using RSS method the efficiency of the estimator relative to the simple random sampling (SRS) method has increased. Computer simulated results are given. An example using real data is presented to illustrate the computations.     

Parameters Estimation in a Simple Linear Regression Using Rank Set Sampling

Ranked set sampling (RSS) as suggested by MCINTYRE (1952) and TAKAHASI and WAKIMOTO (1968) may be used to estimate the parameters of the simple regression line. The objective is to use the RSS method to increase the efficiency of the estimators relative to the simple random sampling (SRS) method. Estimators of the slope and intercept are considered. Computer simulated results are given, and an example using real data presented to illustrate the computations.     

Estimating Genetic Variability with Restriction Endonucleases

The estimation of the amount of sequence variation in samples of homologous DNA segments is considered. The data are assumed to have been obtained by restriction endonuclease digestion of the segments, from which the numbers and frequencies of the cleavage sites in the sample are determined. An estimator, p, of the proportion of sites that are polymorphic in the sample is derived without assuming any particular population genetic model for the evolution of the population. The estimator is very close to the EWENS, SPIELMAN and HARRIS (1981) estimator that was derived with the symmetric WRIGHT-FISHER neutral mode. ENGELS (1981) has also recently proposed an estimator of the same quantity, and he arrived at his estimator without assuming a particular population genetic model. The sampling variance of p and ENGELS' estimator are derived. It is found that the sampling variance of p is lower than the sampling variance of ENGELS' estimator. Also, the sampling variance of theta, an estimate of theta (=4Nu) is obtained for the symmetric WRIGHT-FISHER neutral model with free recombination and with no recombination.     

Bias-corrected variance estimation and hypothesis testing for spatial point and marked point processes using subsampling

Summary We introduce novel regression extrapolation based methods to correct the often large bias in subsampling variance estimation as well as hypothesis testing for spatial point and marked point processes. For variance estimation, our proposed estimators are linear combinations of the usual subsampling variance estimator based on subblock sizes in a continuous interval. We show that they can achieve better rates in mean squared error than the usual subsampling variance estimator. In particular, for n×n observation windows, the optimal rate of n^?2 can be achieved if the data have a finite dependence range. For hypothesis testing, we apply the proposed regression extrapolation directly to the test statistics based on different subblock sizes, and therefore avoid the need to conduct bias correction for each element in the covariance matrix used to set up the test statistics. We assess the numerical performance of the proposed methods through simulation, and apply them to analyze a tropical forest data set.     

Efficient regression analysis with ranked-set sampling

This article is motivated by a lung cancer study where a regression model is involved and the response variable is too expensive to measure but the predictor variable can be measured easily with relatively negligible cost. This situation occurs quite often in medical studies, quantitative genetics, and ecological and environmental studies. In this article, by using the idea of ranked-set sampling (RSS), we develop sampling strategies that can reduce cost and increase efficiency of the regression analysis for the above-mentioned situation. The developed method is applied retrospectively to a lung cancer study. In the lung cancer study, the interest is to investigate the association between smoking status and three biomarkers: polyphenol DNA adducts, micronuclei, and sister chromatic exchanges. Optimal sampling schemes with different optimality criteria such as A-, D-, and integrated mean square error (IMSE)-optimality are considered in the application. With set size 10 in RSS, the improvement of the optimal schemes over simple random sampling (SRS) is great. For instance, by using the optimal scheme with IMSE-optimality, the IMSEs of the estimated regression functions for the three biomarkers are reduced to about half of those incurred by using SRS.     

Estimation of Parameters for One-Way Layout with Rank Set Sampling

Ranked set sampling (RSS) as suggested by McIntyre (1952) and independently by Takahasi and Wakimoto (1968) may be used to estimate the parameters of the one-way layout. The objective is to use the RSS method to increase the efficiency of the estimators relative to the simple random (SRS) method. Estimators of the populations (treatments) effect are considered. Computer simulated results are given, and an example using real data presented to illustrate the computations.