Estimation under generalized sampling of cell proportions for contingency tables subject to marginal constraints

Under simple random (multinomial) sampling the problem of estimating cell proportions for a contingency table subject to marginal constraints has been well explored. We briefly review methods that have been considered; then we develop a general method, for more complicated sampling, which reflects the variance structure of the estimated cell proportions. For stratified and cluster sampling we compare our method against earlier methods for the 2×2 table and find it potentially advantageous.     

Weighted empirical likelihood inference for multiple samples

We propose a weighted empirical likelihood approach to inference with multiple samples, including stratified sampling, the estimation of a common mean using several independent and non-homogeneous samples and inference on a particular population using other related samples. The weighting scheme and the basic result are motivated and established under stratified sampling. We show that the proposed method can ideally be applied to the common mean problem and problems with related samples. The proposed weighted approach not only provides a unified framework for inference with multiple samples, including two-sample problems, but also facilitates asymptotic derivations and computational methods. A bootstrap procedure is also proposed in conjunction with the weighted approach to provide better coverage probabilities for the weighted empirical likelihood ratio confidence intervals. Simulation studies show that the weighted empirical likelihood confidence intervals perform better than existing ones.     

A Bayesian test of independence in a two-way contingency table using surrogate sampling

We consider a Bayesian approach to the study of independence in a two-way contingency table which has been obtained from a two-stage cluster sampling design. If a procedure based on single-stage simple random sampling (rather than the appropriate cluster sampling) is used to test for independence, the p-value may be too small, resulting in a conclusion that the null hypothesis is false when it is, in fact, true. For many large complex surveys the Rao–Scott corrections to the standard chi-squared (or likelihood ratio) statistic provide appropriate inference. For smaller surveys, though, the Rao–Scott corrections may not be accurate, partly because the chi-squared test is inaccurate. In this paper, we use a hierarchical Bayesian model to convert the observed cluster samples to simple random samples. This provides surrogate samples which can be used to derive the distribution of the Bayes factor. We demonstrate the utility of our procedure using an example and also provide a simulation study which establishes our methodology as a viable alternative to the Rao–Scott approximations for relatively small two-stage cluster samples. We also show the additional insight gained by displaying the distribution of the Bayes factor rather than simply relying on a summary of the distribution.     

A note on the determination of sample sizes for hypergeometric distributions

Sample size determination for testing the hypothesis of equality of two proportions against an alternative with specified type I and type II error probabilities is considered for two finite populations. When two finite populations involved are quite different in sizes, the equal size assumption may not be appropriate. In this paper, we impose a balanced sampling condition to determine the necessary samples taken without replacement from the finite populations. It is found that our solution requires smaller samples as compared to those using binomial distributions. Furthermore, our solution is consistent with the sampling with replacement or when population size is large. Finally, three examples are given to show the application of the derived sample size formula.     

住户调查中代表性样本的一种探索获取方法——平衡抽样设计

住户调查是我国社会经济统计调查体系的重要组成部分,样本代表性直接决定统计数据质量。多阶段抽样中初级单元的方差对估计的影响是主要的,因此本文结合2010年全国第六次人口普查分县数据,采用平衡抽样设计获取初级单元的代表性样本-平衡样本。对代表性样本的事后评估结果表明,样本结构与总体结构吻合,目标估计的误差很小,说明了本文平衡设计的有效性。     

Confidence limits for estimates of totals from stratified samples, with application to medicare Part B overpayment audits

Superpopulation models are proposed that should be appropriate for modelling sample-based audits of Medicare payments and other overpayment situations. Simulations are used to estimate the coverage probabilities of confidence intervals formed using the standard Stratified Expansion and Combined Ratio estimators of the total. Despite severe departures from the usual model of normal deviations, these methods have actual coverage probabilities reasonably close to the nominal level specified by the US government's sampling guidelines. An exception occurs when all claims from a single sampling unit are either completely allowed, or completely denied, and for this situation an alternative is explored. A balanced sampling design is also examined, but shown to make no improvement over ordinary stratified samples used in conjunction with ratio estimates.     

Variance estimation of DEE and regression estimator when a first-phase cluster sample is restratified

We consider a variance estimation when a stratified single stage cluster sample is selected in the first phase and a stratified simple random element sample is selected in the second phase. We propose explicit formulas of (asymptotically), we propose explicit formulas of (asymptotically) unbiased variance estimators for the double expansion estimator and regression estimator. We perform a small simulation study to investigate the performance of the proposed variance estimators. In our simulation study, the proposed variance estimator showed better or comparable performance to the Jackknife variance estimator. We also extend the results to a two-phase sampling design in which a stratified pps with replacement cluster sample is selected in the first phase.     

A Comparison Of Two Adaptive Sampling Designs

Stratified sampling is a technique commonly used for ecological surveys. In this study there appears to be little gain in using a stratified design with adaptive cluster sampling. Two-phase adaptive sampling is preferable to adaptive cluster sampling. Even though two-phase adaptive sampling can give biased estimates, it is found that two-phase adaptive sampling has a lower MSE than adaptive cluster sampling for most populations.     

An exact test for trend among binomial proportions based on a modified Baumgartner-Weiß-Schindler statistic

The Cochran-Armitage test is the most frequently used test for trend among binomial proportions. This test can be performed based on the asymptotic normality of its test statistic or based on an exact null distribution. As an alternative, a recently introduced modification of the Baumgartner-Weiß-Schindler statistic, a novel nonparametric statistic, can be used. Simulation results indicate that the exact test based on this modification is preferable to the Cochran-Armitage test. This exact test is less conservative and more powerful than the exact Cochran-Armitage test. The power comparison to the asymptotic Cochran-Armitage test does not show a clear winner, but the difference in power is usually small. The exact test based on the modification is recommended here because, in contrast to the asymptotic Cochran-Armitage test, it guarantees a type I error rate less than or equal to the significance level. Moreover, an exact test is often more appropriate than an asymptotic test because randomization rather than random sampling is the norm, for example in biomedical research. The methods are illustrated with an example data set.     

Group Testing,the Pooled Hypergeometric Distribution,and Estimating the Number of Defectives in Small Populations

The testing of combined bacteriological samples – or “group testing” – was introduced to reduce the cost of identifying defective individuals in populations containing small proportions of defectives. It may also be applied to plants, animals, or food samples to estimate proportions infected, or to accept or reject populations. Given the proportion defective in the population, the number of positive combined samples is approximately binomial when the population is large: we find the exact distribution when groups include the same number of samples. We derive some properties of this distribution, and consider maximum-likelihood and Bayesian estimation of the number defective.     

辅以排序集样本的分层抽样方法研究

分层抽样中,样本在各层中的不同获取方式会对估计量的精度和试验费用产生一定的影响,而已有的理论方法大多不能在提高精度的同时降低调查费用。为此,将排序抽样与分层抽样方法相结合,提出了辅以排序集样本的分层抽样方案,并得到了总体均值的估计量以及这一估计量的良好性质。这些结果表明,与单一的分层随机抽样相比,这种抽样设计的估计量具有更高的精度,同时也节约了各层抽样调查的费用。     

Bayesian analysis of binary survey data

First a comprehensive treatment of the hierarchical-conjugate Bayesian predictive approach to binary survey data is presented, encompassing simple random, stratified, cluster, and two-stage sampling, as well as two-stage sampling within strata. For the case of two-stage sampling within strata when there is more than one variable of stratification, analysis using an unsaturated logit linear model on the prior means is proposed. This allows there to be cells containing no sampled clusters. Formulas for posterior predictive means, variances, and covariances of numbers of successes in unsampled portions of clusters are presented in terms of posterior expectations of certain functions of hyperparameters; these may be evaluated by existing methods. The technique is illustrated using a small subset of Canada Youth & AIDS Study data. A sample of students within each of various selected school boards was chosen and interviewed via questionnaire. The boards were stratified/poststratified in two dimensions, but some of the resulting cells contained no data. The additive logit linear model on the prior means produced estimates and posterior variances for boards in all cells. Data showed the additive model to be plausible.     

Asymptotic results for fitting marginal hazards models from stratified case-cohort studies with multiple disease outcomes

In stratified case-cohort designs, samplings of case-cohort samples are conducted via a stratified random sampling based on covariate information available on the entire cohort members. In this paper, we extended the work of Kang & Cai (2009) to a generalized stratified case-cohort study design for failure time data with multiple disease outcomes. Under this study design, we developed weighted estimating procedures for model parameters in marginal multiplicative intensity models and for the cumulative baseline hazard function. The asymptotic properties of the estimators are studied using martingales, modern empirical process theory, and results for finite population sampling.     

Confidence Intervals for the Median of a Finite Population Based on the Sign Test

This article presents methods for constructing confidence intervals for the median of a finite population under simple random sampling without replacement, stratified random sampling, and cluster sampling. The confidence intervals, as well as point estimates and test statistics, are derived from sign estimating functions which are based on the well-known sign test. Therefore, a unified approach for inference about the median of a finite population is given.     

MODEL-BASED VARIANCE ESTIMATION IN SURVEYS WITH STRATIFIED CLUSTERED DESIGN

A model-based method for estimating the sampling variances of estimators of (sub-)population means, proportions, quantiles, and regression parameters in surveys with stratified clustered design is described and applied to a survey of US secondary education. The method is compared with the jackknife by a simulation study. The model-based estimators of the sampling variances have much smaller mean squared errors than their jackknife counterparts. In addition, they can be improved by incorporating information about the unknown parameters (variances) from external sources. A regression-based smoothing method for estimating the sampling variances of the estimators for a large number of subpopulation means is proposed. Such smoothing may be invaluable when subpopulations are represented in the sample by only few subjects.     

Confidence regions for two proportions from independent negative binomial distributions

The negative binomial distribution offers an alternative view to the binomial distribution for modeling count data. This alternative view is particularly useful when the probability of success is very small, because, unlike the fixed sampling scheme of the binomial distribution, the inverse sampling approach allows one to collect enough data in order to adequately estimate the proportion of success. However, despite work that has been done on the joint estimation of two binomial proportions from independent samples, there is little, if any, similar work for negative binomial proportions. In this paper, we construct and investigate three confidence regions for two negative binomial proportions based on three statistics: the Wald (W), score (S) and likelihood ratio (LR) statistics. For large-to-moderate sample sizes, this paper finds that all three regions have good coverage properties, with comparable average areas for large sample sizes but with the S method producing the smaller regions for moderate sample sizes. In the small sample case, the LR method has good coverage properties, but often at the expense of comparatively larger areas. Finally, we apply these three regions to some real data for the joint estimation of liver damage rates in patients taking one of two drugs.     

A unified sequential test procedure for simultaneous testing the equality of several binomial proportions to a specified standard

In this article, we propose a unified sequentially rejective test procedure for testing simultaneously the equality of several independent binomial proportions to a specified standard. The proposed test procedure is general enough to include some well-known multiple testing procedures such as the Ordinary Bonferroni procedure, Hochberg procedure and Rom procedure. It involves multiple tests of significance based on the simple binomial tests (exact or approximate) which can be easily found in many elementary standard statistics textbooks. Unlike the traditional Chi-square test of the overall hypothesis, the procedure can identify the subset of the binomial proportions, which are different from the prespecified standard with the control of the familywise type I error rate. Moreover, the power computation of the procedure is provided and the procedure is illustrated by two real examples from an ecological study and a carcinogenicity study.     

Confidence limits for stress–strength reliability involving Weibull models

The problem of interval estimation of the stress–strength reliability involving two independent Weibull distributions is considered. An interval estimation procedure based on the generalized variable (GV) approach is given when the shape parameters are unknown and arbitrary. The coverage probabilities of the GV approach are evaluated by Monte Carlo simulation. Simulation studies show that the proposed generalized variable approach is very satisfactory even for small samples. For the case of equal shape parameter, it is shown that the generalized confidence limits are exact. Some available asymptotic methods for the case of equal shape parameter are described and their coverage probabilities are evaluated using Monte Carlo simulation. Simulation studies indicate that no asymptotic approach based on the likelihood method is satisfactory even for large samples. Applicability of the GV approach for censored samples is also discussed. The results are illustrated using an example.     

Properties of the beta regression model for small area estimation of proportions and application to estimation of poverty rates

Abstract

Linear mixed effects models have been popular in small area estimation problems for modeling survey data when the sample size in one or more areas is too small for reliable inference. However, when the data are restricted to a bounded interval, the linear model may be inappropriate, particularly if the data are near the boundary. Nonlinear sampling models are becoming increasingly popular for small area estimation problems when the normal model is inadequate. This paper studies the use of a beta distribution as an alternative to the normal distribution as a sampling model for survey estimates of proportions which take values in (0, 1). Inference for small area proportions based on the posterior distribution of a beta regression model ensures that point estimates and credible intervals take values in (0, 1). Properties of a hierarchical Bayesian small area model with a beta sampling distribution and logistic link function are presented and compared to those of the linear mixed effect model. Propriety of the posterior distribution using certain noninformative priors is shown, and behavior of the posterior mean as a function of the sampling variance and the model variance is described. An example using 2010 Small Area Income and Poverty Estimates (SAIPE) data is given, and a numerical example studying small sample properties of the model is presented.     

Large sample results for tests of association ii. multinomial and stratified sampling

This paper deals with the asymptotics of a class of tests for association in 2-way contingency tables based on square forms in cell frequencies, given the total number of observations (multinomial sampling) or one set of marginal totals (stratified sampling). The case when both row and column marginal totals are fixed (hypergeometric sampling) was studied in Kulinskaya (1994), The class of tests under consideration includes a number of classical measures for association, Its two subclasses are the tests based on statistics using centralized cell frequencies (asymptotically distributed as weighted sums of central chi-squares) and those using the non-centralized cell frequencies (asymptotically normal). The parameters of asymptotic distributions depend on the sampling model and on true marginal probabilities. Maximum efficiency for asymptotically normal statistics is achieved under hypergeometric sampling, If the cell frequencies or the statistic as a whole are centralized using marginal proportions as estimates for marginal probabilities, the asymptotic distribution does not differ much between models and it is equivalent to that under hypergeometric sampling. These findings give an extra justification for the use of permutation tests for association (which are based on hypergeometric sampling). As an application, several well known measures of association are analysed.