Inference under pivotal sampling: Properties,variance estimation,and application to tesselation for spatial sampling

Unequal probability sampling is commonly used for sample selection. In the context of spatial sampling, the variables of interest often present a positive spatial correlation, so that it is intuitively relevant to select spatially balanced samples. In this article, we study the properties of pivotal sampling and propose an application to tesselation for spatial sampling. We also propose a simple conservative variance estimator. We show that the proposed sampling design is spatially well balanced, with good statistical properties and is computationally very efficient.     

复杂抽样推断方法体系的比较研究

对复杂样本进行推断通常有两种体系,一种是传统的基于随机化理论的统计推断,另一种是基于模型的统计推断。传统的抽样理论以随机化理论为基础,将总体取值视为固定,随机性仅体现在样本的选取上,对总体的推断依赖于抽样设计。该方法在大样本情况下具有稳健估计量,但在小样本、数据缺失等情况下失效。基于模型的抽样推断认为总体是超总体模型中抽取的一个随机样本,对总体的推断取决于模型的建立,但在不可忽略抽样设计下估计量是有偏估计。在对这两类推断方法分析的基础上,提出抽样设计辅助的模型推断,并指出该方法在复杂抽样中具有重要的应用价值。     

CONTROLLED SAMPLING WITH VARYING PROBABILITIES WITH AND WITHOUT REPLACEMENT

Goodman and Kish (1950) introduced the problem of controlled selection in the sense of decreasing the selection probability of nonpreferred combinations, such as, for example, combinations which present organisational difficulties involving additional cost, etc. The authors (Avadhani and Sukhatme (1965), Sukhatme and Avadhani (1965)) have evolved certain techniques of controlled selection which eliminate altogether some non-preferred combinations and reduce the probability of selection of the remaining combinations, if any, to the minimum possible extent without deviating from the fundamental principles of random sampling. However, it is often felt convenient in practice to draw sampling units one after another from the population rather than combinations of units. But, at present no technique by which one can select the units one after another and at the same time reduce the probability of selection of non-preferred units is available in literature.
In this paper we have suggested a solution to this problem which not only minimizes the selection probability of non-preferred units (and of samples containing predominantly large numbers of nonpreferred units) but also provides more efficient estimators than the the usual probability proportional to size (P.P.S) sampling scheme.     

Optimal PPS Sampling with Vanishing Auxiliary Variables – with Applications in Microscopy

Recently, non‐uniform sampling has been suggested in microscopy to increase efficiency. More precisely, proportional to size (PPS) sampling has been introduced, where the probability of sampling a unit in the population is proportional to the value of an auxiliary variable. In the microscopy application, the sampling units are fields of view, and the auxiliary variables are easily observed approximations to the variables of interest. Unfortunately, often some auxiliary variables vanish, that is, are zero‐valued. Consequently, part of the population is inaccessible in PPS sampling. We propose a modification of the design based on a stratification idea, for which an optimal solution can be found, using a model‐assisted approach. The new optimal design also applies to the case where ‘vanish’ refers to missing auxiliary variables and has independent interest in sampling theory. We verify robustness of the new approach by numerical results, and we use real data to illustrate the applicability.     

Variables acceptance reliability sampling plan for repairable items

Traditionally, acceptance reliability sampling plans have been developed for non-repairable items. However, the functions required for items become more and more complex and, accordingly, the items are composed of several components and tend to be repairable. In this paper, we consider variables acceptance reliability sampling plan for repairable items. We develop a variables acceptance sampling plan based on the failure and repair data observed during the testing period. It is shown that the developed sampling plan improves the reliability characteristic of the population and that the lifetimes of items before and after the reliability sampling test are stochastically ordered.     

A review of optimal designs in survey sampling

Results in five areas of survey sampling dealing with the choice of the sampling design are reviewed. In Section 2, the results and discussions surrounding the purposive selection methods suggested by linear regression superpopulation models are reviewed. In Section 3, similar models to those in the previous section are considered; however, random sampling designs are considered and attention is focused on the optimal choice of π_j. Then in Section 4, systematic sampling methods obtained under autocorrelated superpopulation models are reviewed. The next section examines minimax sampling designs. The work in the final section is based solely on the randomization. In Section 6 methods of sample selection which yield inclusion probabilities π_j = n/N and π_ij = n(n - 1)/N(N - 1), but for which there are fewer than _NC_n possible samples, are mentioned briefly.     

Multistage sampling for latent variable models

I consider the design of multistage sampling schemes for epidemiologic studies involving latent variable models, with surrogate measurements of the latent variables on a subset of subjects. Such models arise in various situations: when detailed exposure measurements are combined with variables that can be used to assign exposures to unmeasured subjects; when biomarkers are obtained to assess an unobserved pathophysiologic process; or when additional information is to be obtained on confounding or modifying variables. In such situations, it may be possible to stratify the subsample on data available for all subjects in the main study, such as outcomes, exposure predictors, or geographic locations. Three circumstances where analytic calculations of the optimal design are possible are considered: (i) when all variables are binary; (ii) when all are normally distributed; and (iii) when the latent variable and its measurement are normally distributed, but the outcome is binary. In each of these cases, it is often possible to considerably improve the cost efficiency of the design by appropriate selection of the sampling fractions. More complex situations arise when the data are spatially distributed: the spatial correlation can be exploited to improve exposure assignment for unmeasured locations using available measurements on neighboring locations; some approaches for informative selection of the measurement sample using location and/or exposure predictor data are considered.     

Weighted re‐randomization tests for minimization with unbalanced allocation

Re‐randomization test has been considered as a robust alternative to the traditional population model‐based methods for analyzing randomized clinical trials. This is especially so when the clinical trials are randomized according to minimization, which is a popular covariate‐adaptive randomization method for ensuring balance among prognostic factors. Among various re‐randomization tests, fixed‐entry‐order re‐randomization is advocated as an effective strategy when a temporal trend is suspected. Yet when the minimization is applied to trials with unequal allocation, fixed‐entry‐order re‐randomization test is biased and thus compromised in power. We find that the bias is due to non‐uniform re‐allocation probabilities incurred by the re‐randomization in this case. We therefore propose a weighted fixed‐entry‐order re‐randomization test to overcome the bias. The performance of the new test was investigated in simulation studies that mimic the settings of a real clinical trial. The weighted re‐randomization test was found to work well in the scenarios investigated including the presence of a strong temporal trend. Copyright © 2013 John Wiley & Sons, Ltd.     

Reliability sampling plan for repairable items following general failure process and its statistical analysis

Acceptance sampling plan has become an essential tool in the statistical quality control. Traditionally, most acceptance sampling plans have been developed for non-repairable items. Recently in Cha [Variables acceptance reliability sampling plan for repairable items. Statistics. 2015;49:1141–1156], variables acceptance reliability sampling plan for repairable items has been developed assuming that the failure process follows the non-homogeneous Poisson process (NHPP). In this paper, we assume that the failure process follows a new counting process which is a generalized version of the NHPP. Furthermore, it is shown that the developed sampling plan improves the reliability characteristic of the population of items which have passed the testing procedure. An illustrative example is also provided.     

Improved variance estimation for balanced samples drawn via the cube method

The cube method proposed by Deville and Tillé (2004) enables the selection of balanced samples: that is, samples such that the Horvitz-Thompson estimators of auxiliary variables match the known totals of those variables. As an exact balanced sampling design often does not exist, the cube method generally proceeds in two steps: a “flight phase” in which exact balance is maintained, and a “landing phase” in which the final sample is selected while respecting the balance conditions as closely as possible. Deville and Tillé (2005) derive a variance approximation for balanced sampling that takes account of the flight phase only, whereas the landing phase can prove to add non-negligible variance. This paper uses a martingale difference representation of the cube method to construct an efficient simulation-based method for calculating approximate second-order inclusion probabilities. The approximation enables nearly unbiased variance estimation, where the bias is primarily due to the limited number of simulations. In a Monte Carlo study, the proposed method has significantly less bias than the standard variance estimator, leading to improved confidence interval coverage.     

How to Select Representative Samples

We give a formal definition of a representative sample, but roughly speaking, it is a scaled‐down version of the population, capturing its characteristics. New methods for selecting representative probability samples in the presence of auxiliary variables are introduced. Representative samples are needed for multipurpose surveys, when several target variables are of interest. Such samples also enable estimation of parameters in subspaces and improved estimation of target variable distributions. We describe how two recently proposed sampling designs can be used to produce representative samples. Both designs use distance between population units when producing a sample. We propose a distance function that can calculate distances between units in general auxiliary spaces. We also propose a variance estimator for the commonly used Horvitz–Thompson estimator. Real data as well as illustrative examples show that representative samples are obtained and that the variance of the Horvitz–Thompson estimator is reduced compared with simple random sampling.     

大数据背景下网络调查样本的建模推断问题研究——以广义Boosted模型的倾向得分推断为例

随着大数据和网络的不断发展,网络调查越来越广泛,大部分网络调查样本属于非概率样本,难以采用传统的抽样推断理论进行推断,如何解决网络调查样本的推断问题是大数据背景下网络调查发展的迫切需求。本文首次从建模的角度提出了解决该问题的基本思路：一是入样概率的建模推断,可以考虑构建基于机器学习与变量选择的倾向得分模型来估计入样概率推断总体;二是目标变量的建模推断,可以考虑直接对目标变量建立参数、非参数或半参数超总体模型进行估计;三是入样概率与目标变量的双重建模推断,可以考虑进行倾向得分模型与超总体模型的加权估计与混合推断。最后,以基于广义Boosted模型的入样概率建模推断为例演示了具体解决方法。     

Spatially correlated Poisson sampling

A new method for sampling from a finite population that is spread in one, two or more dimensions is presented. Weights are used to create strong negative correlations between the inclusion indicators of nearby units. The method can be used to produce unequal probability samples that are well spread over the population in every dimension, without any spatial stratification. Since the method is very general there are numerous possible applications, especially in sampling of natural resources where spatially balanced sampling has proven to be efficient. Two examples show that the method gives better estimates than other commonly used designs.     

On Prediction for Correlated Domains in Longitudinal Surveys

In the survey sampling estimation or prediction of both population’s and subopulation’s (domain’s) characteristics is one of the key issues. In the case of the estimation or prediction of domain’s characteristics one of the problems is looking for additional sources of information that can be used to increase the accuracy of estimators or predictors. One of these sources may be spatial and temporal autocorrelation. Due to the mean squared error (MSE) estimation, the standard assumption is that random variables are independent for population elements from different domains. If the assumption is taken into account, spatial correlation may be assumed only inside domains. In the paper, we assume some special case of the linear mixed model with two random components that obey assumptions of the first-order spatial autoregressive model SAR(1) (but inside groups of domains instead of domains) and first-order temporal autoregressive model AR(1). Based on the model, the empirical best linear unbiased predictor will be proposed together with an estimator of its MSE taking the spatial correlation between domains into account.     

A ratio estimator of the total of a subpopulation

If the total of an auxiliary variable is known for an entire population but is unknown for some subpopulation, the usual estimator of the total of the primary variable for the subpopulation is the ratio estimator that uses the auxiliary total for the entire population. This article proposes a ratio estimator that uses an estimator of the auxiliary total over the subpopulation as suggested by Kish (1967, p. 438). Under some conditions, it is shown that the latter estimator is unbiased and has smaller variance than the former estimator in large simple random samples.     

The pseudo‐GEE approach to the analysis of longitudinal surveys

Longitudinal surveys have emerged in recent years as an important data collection tool for population studies where the primary interest is to examine population changes over time at the individual level. Longitudinal data are often analyzed through the generalized estimating equations (GEE) approach. The vast majority of existing literature on the GEE method; however, is developed under non‐survey settings and are inappropriate for data collected through complex sampling designs. In this paper the authors develop a pseudo‐GEE approach for the analysis of survey data. They show that survey weights must and can be appropriately accounted in the GEE method under a joint randomization framework. The consistency of the resulting pseudo‐GEE estimators is established under the proposed framework. Linearization variance estimators are developed for the pseudo‐GEE estimators when the finite population sampling fractions are small or negligible, a scenario often held for large‐scale surveys. Finite sample performances of the proposed estimators are investigated through an extensive simulation study using data from the National Longitudinal Survey of Children and Youth. The results show that the pseudo‐GEE estimators and the linearization variance estimators perform well under several sampling designs and for both continuous and binary responses. The Canadian Journal of Statistics 38: 540–554; 2010 © 2010 Statistical Society of Canada     

Estimating the population median by nomination sampling

In nomination sampling, the largest values from several independent random samples (nominees) are rank ordered, and an estimate of the population median is formed by interpolating between 2 of these order statistics. The resulting estimate compares favorably to the sample median of a simple random sample from the same population. When historical data sets retain only extreme values, nomination sampling may offer the only practical way to estimate the population median. The approach may also be useful when potential survey respondents will only participate if they can actively influence the selection of cases for analysis.     

规模以下工业抽样调查中代表性样本的一种探索设计:平衡抽样设计

规下工业抽样调查是社会经济统计调查的重要组成部分,为国民经济核算提供基础数据,而样本代表性直接决定统计推断结果。对企业目录库抽取平衡样本,能够使得样本结构与总体结构相似。平衡样本是指满足如下条件的样本:辅助变量的汉森赫维茨估计等于总体总量真值。平衡抽样设计需要包含丰富辅助信息的完善抽样框,政府统计数据能够为此提供足够的支撑。基于2009年工业企业数据库的实证分析表明,平衡抽样设计对总体总量的估计相对误差很小,特别是估计的均值与总体真值非常接近,近似无偏;与简单随机抽样比较,平衡抽样设计更加有效。     

Model‐Based Non‐parametric Variance Estimation for Systematic Sampling

Abstract. Systematic sampling is frequently used in surveys, because of its ease of implementation and its design efficiency. An important drawback of systematic sampling, however, is that no direct estimator of the design variance is available. We describe a new estimator of the model‐based expectation of the design variance, under a non‐parametric model for the population. The non‐parametric model is sufficiently flexible that it can be expected to hold at least approximately in many situations with continuous auxiliary variables observed at the population level. We prove the model consistency of the estimator for both the anticipated variance and the design variance under a non‐parametric model with a univariate covariate. The broad applicability of the approach is demonstrated on a dataset from a forestry survey.