Joint Inclusion Probabilities Research Articles

Sampling Designs on Finite Populations with Spreading Control Parameters

We present new sampling methods in finite population that allow to control the joint inclusion probabilities of units and especially the spreading of sampled units in the population. They are based on the use of renewal chains and multivariate discrete distributions to generate the difference of population ranks between two successive selected units. With a Bernoulli sampling design, these differences follow a geometric distribution, and with a simple random sampling design they follow a negative hypergeometric distribution. We propose to use other distributions and introduce a large class of sampling designs with and without fixed sample size. The choice of the rank-difference distribution allows us to control units joint inclusion probabilities with a relatively simple method and closed form formula. Joint inclusion probabilities of neighboring units can be chosen to be larger, or smaller, compared to those of Bernoulli or simple random sampling, thus allowing to more or less spread the sample on the population. This can be useful when neighboring units have similar characteristics or, on the contrary, are very different. A set of simulations illustrates the qualities of this method.

Applying the Hájek Approach in Formula‐Based Variance Estimation

AbstractThe variance formula derived for a two‐stage sampling design without replacement employs the joint inclusion probabilities in the first‐stage selection of clusters. One of the difficulties encountered in data analysis is the lack of information about such joint inclusion probabilities. One way to solve this issue is by applying Hájek's approximation of the joint probabilities in variance estimation. To assess the Hájek approach, several estimators of Hájek'scanddare proposed. The application is illustrated with simulation and real data. A Monte Carlo simulation is employed to compare the results of joint inclusion probabilities yielded from the probability‐proportional‐to‐size sampling methods with the results from Hájek's approximation. Empirically estimated variances from the jackknife procedure are also compared with the formula‐based variances with incorporated Hájek's approximation.

Empirical Likelihood Inference for the Rao‐Hartley‐Cochran Sampling Design

AbstractThe Hartley‐Rao‐Cochran sampling design is an unequal probability sampling design which can be used to select samples from finite populations. We propose to adjust the empirical likelihood approach for the Hartley‐Rao‐Cochran sampling design. The approach proposed intrinsically incorporates sampling weights, auxiliary information and allows for large sampling fractions. It can be used to construct confidence intervals. In a simulation study, we show that the coverage may be better for the empirical likelihood confidence interval than for standard confidence intervals based on variance estimates. The approach proposed is simple to implement and less computer intensive than bootstrap. The confidence interval proposed does not rely on re‐sampling, linearization, variance estimation, design‐effects or joint inclusion probabilities.

Bayesian Variable Selection Under Collinearity

In this article, we highlight some interesting facts about Bayesian variable selection methods for linear regression models in settings where the design matrix exhibits strong collinearity. We first demonstrate via real data analysis and simulation studies that summaries of the posterior distribution based on marginal and joint distributions may give conflicting results for assessing the importance of strongly correlated covariates. The natural question is which one should be used in practice. The simulation studies suggest that posterior inclusion probabilities and Bayes factors that evaluate the importance of correlated covariates jointly are more appropriate, and some priors may be more adversely affected in such a setting. To obtain a better understanding behind the phenomenon, we study some toy examples with Zellner’s g-prior. The results show that strong collinearity may lead to a multimodal posterior distribution over models, in which joint summaries are more appropriate than marginal summaries. Thus, we recommend a routine examination of the correlation matrix and calculation of the joint inclusion probabilities for correlated covariates, in addition to marginal inclusion probabilities, for assessing the importance of covariates in Bayesian variable selection.

Empirical Likelihood Confidence Intervals for Complex Sampling Designs

Summary We define an empirical likelihood approach which gives consistent design-based confidence intervals which can be calculated without the need of variance estimates, designeffects, resampling, joint inclusion probabilities and linearization, even when the point estimator is not linear. It can be used to construct confidence intervals for a large class of sampling designs and estimators which are solutions of estimating equations. It can be used for means, regressions coefficients, quantiles, totals or counts even when the population size is unknown. It can be used with large sampling fractions and naturally includes calibration constraints. It can be viewed as an extension of the empirical likelihood approach to complex survey data. This approach is computationally simpler than the pseudoempirical likelihood and the bootstrap approaches. The simulation study shows that the confidence interval proposed may give better coverages than the confidence intervals based on linearization, bootstrap and pseudoempirical likelihood. Our simulation study shows that, under complex sampling designs, standard confidence intervals based on normality may have poor coverages, because point estimators may not follow a normal sampling distribution and their variance estimators may be biased.

Approximation of rejective sampling inclusion probabilities and application to high order correlations

This paper is devoted to rejective sampling. We provide an expansion of joint inclusion probabilities of any order in terms of the inclusion probabilities of order one, extending previous results by Hajek (1964) and Hajek (1981) and making the remainder term more precise. Following Hajek (1981), the proof is based on Edgeworth expansions. The main result is applied to derive bounds on higher order correlations, which are needed for the consistency and asymptotic normality of several complex estimators.

Conditional Logistic Regression With Survey Data

When data consist of clusters of potentially correlated observations then conditional logistic regression can be used to estimate the association between a binary outcome and covariates conditionally on the cluster effects. Surveys can use multistage sampling with potentially differential probabilities of sampling individuals from the same conditioning cluster (e.g., family). We show that conditional logistic regression of survey data using standard inflation (weighted) estimation (i.e., observations are weighted by the inverse of their inclusion probabilities) can result in biased estimators when the sample sizes of the observations sampled from the conditioning clusters are small. We propose using methods based on weighted pseudo-pairwise likelihoods that combine the conditional logistic likelihoods for all pairs of observations consisting of a positive and a negative outcome within a conditioning cluster and weights the pairwise likelihoods by the inverse of their joint inclusion probabilities within the cluster. Design-based variance estimators for regression coefficient estimators are provided. Limited simulations demonstrate that the proposed methods produce approximately unbiased regression coefficients and variance estimates, but can be considerably less efficient than maximum likelihood estimation when the sampling is uninformative. The proposed methods are illustrated with an analysis of data from the Hispanic Health and Nutrition Examination Survey.

Systematic sampling is a minimum support design

In order to select a sample in a finite population of N units with given inclusion probabilities, it is possible to define a sampling design on at most N samples that have a positive probability of being selected. Designs defined on minimal sets of samples are called minimum support designs. It is shown that, for any vector of inclusion probabilities, systematic sampling always provides a minimum support design. This property makes it possible to extensively compute the sampling design and the joint inclusion probabilities. Random systematic sampling can be viewed as the random choice of a minimum support design. However, even if the population is randomly sorted, a simple example shows that some joint inclusion probabilities can be equal to zero. Another way of randomly selecting a minimum support design is proposed, in such a way that all the samples have a positive probability of being selected, and all the joint inclusion probabilities are positive.

Variance approximation under balanced sampling

A balanced sampling design has the interesting property that Horvitz–Thompson estimators of totals for a set of balancing variables are equal to the totals we want to estimate, therefore the variance of Horvitz–Thompson estimators of variables of interest are reduced in function of their correlations with the balancing variables. Since it is hard to derive an analytic expression for the joint inclusion probabilities, we derive a general approximation of variance based on a residual technique. This approximation is useful even in the particular case of unequal probability sampling with fixed sample size. Finally, a set of numerical studies with an original methodology allows to validate this approximation.

A Simple Variance Estimator for Unequal Probability Sampling without Replacement

Survey sampling textbooks often refer to the Sen–Yates–Grundy variance estimator for use with without-replacement unequal probability designs. This estimator is rarely implemented because of the complexity of determining joint inclusion probabilities. In practice, the variance is usually estimated by simpler variance estimators such as the Hansen–Hurwitz with replacement variance estimator; which often leads to overestimation of the variance for large sampling fractions that are common in business surveys. We will consider an alternative estimator: the Hájek (1964) variance estimator that depends on the first-order inclusion probabilities only and is usually more accurate than the Hansen–Hurwitz estimator. We review this estimator and show its practical value. We propose a simple alternative expression; which is as simple as the Hansen–Hurwitz estimator. We also show how the Hájek estimator can be easily implemented with standard statistical packages.

Estimating Variance Components by Using Survey Data

Summary Inflation-type weighted estimators for variance components can be badly biased. Modified weighted estimators suggested in the literature are also badly biased for certain sampling designs. We propose new estimators for variance components, some of which are approximately unbiased regardless of the sampling design. These estimators require knowledge of the joint inclusion probabilities of the observations. The small sample properties of the estimators are studied via simulation for the simple one-way random-effects model. An application is given by using data from the US Hispanic Health and Nutrition Examination Survey.

A Multiyear Lattice Sampling Design for Maryland-Wide Fish Abundance Estimation

A latticed, multiyear, multistaged sampling design developed for the Maryland Biological Stream Survey (MBSS) enabled the Maryland Department of Natural Resources to make design-based, Maryland-wide estimates of fish abundance in wadable nontidal streams. Because budget and labor constraints prohibited completing the field work required for statewide estimates within a single year, a 3-year sampling program was designed. A sampling lattice was constructed with the 3 years of sampling (1995, 1996, 1997) as the columns and three regions within the state of Maryland (eastern, central, and western) as the rows. Each one of the three regions consisted of six river basins, and the primary sampling units (PSUs) were defined as the combinations of these basins with the 3 years for a total of 54 PSUs. Two basins from each region were sampled in each year so that all basins were sampled once. In addition, one basin within each region was sampled in a second year under a restricted randomized design to complete the first stage of sampling. This additional sample insured that all joint inclusion probabilities of primary sampling units were nonzero. In all 3 years, the second stage consisted of simple random samples of reaches proportionally selected within strata defined by Strahler order. However, the MBSS's latticed sampling design combined with estimators based on the Horvitz-Thompson method would allow the calculation of statewide totals using different sampling designs within PSUs. Two species of fish, creek chub and largemouth bass, are used to illustrate the approach.

An elimination procedure for unequal probability sampling without replacement

SUMMARY An exact without replacement sampling procedure with fixed sample size and unequal inclusion probabilities is presented. The proposed method eliminates a unit of the population at each of the N - n steps of the algorithm in order finally to retain n units. A simple expression is given for the joint inclusion probabilities.

Some Remarks on Unequal Probability Sampling Designs without Replacement

A set of demands is presented which should be satisfied by a good unequal-probability sampling method without replacement. First it is shown that some of these demands are contradictory. ln particular, it is shown that a sequential algorithm that ensures strictly positive joint inclusion probabilities does not exist; not does there exist a sequential procedure that yields a result which is not dependent on the order of units in the data file. Next, a way is discussed to build approximations of the joint-inclusion probabilities for sampling design implemented by means of a sequential algorithm and preceded by a random sort of the data file. An original approximation is proposed which resorts to a method of adjustment to marginal totals. Finally, several approximations are compared to systematic sampling, and to Sunter's method.

Optimal controlled sampling designs

Optimal controlled sampling designs that minimize the probability of selecting a nonpreferred sample are obtained. These plans satisfy either of the following two requirements: (i) match specified joint inclusion probabilities, π ij associated with an uncontrolled sampling plan having certain desirable properties; (ii) match specified first-order inclusion probabilities and provide stable, nonnegative unbiased variance estimators. Limitations of other approaches using methods of experimental designs are demonstrated

A method of inclusion probability proportional to size selection

The purpose of this article is to propose a procedure for selecting an inclusion probability proportional to size sample n. The simplicity in sample selection and in computing joint inclusion probabilities is achieved by using the combinatorial properties of balanced incomplete block designs. The proposed method gives the same second order inclusion probabilities as that of Midzuno-Sen's procedure.

On the Robustness of LISREL (Maximum Likelihood Estimation) Against Small Sample Size and Non-normality.

This book is a comprehensive and readable guide to the large body of literature on unequal probability sampling. It may be recommended strongly to students and practitioners of sampling theory. After a brief introduction, the authors list in their order of appearance in the literature (1949-79) 50 procedures for probability proportional to sampling without replacement. They classify these by manner of selection, joint inclusion probabilities, and the appropriate type of estimator. In Chapter 2 the procedures are described in detail and given more refined classifications. Their strengths and limitations are compared in Chapters 3 and 4, with useful summary tables rounding out the discussion. Chapter 5 gives a brief look at the algebra of unequal probability multistage sampling. In Chapter 6 (to this reviewer the most interesting) the authors discuss recent work on robust sampling estimation, particularly Brewer (1979). The approach taken is conventional in that the performance criteria for design-estimator strategies include ease of implementation, designunbiasedness or asymptotic design-unbiasedness of the estimator, and availability of a design-unbiased variance estimator. Although a regression model with size as the independent variable is used to compare expected mean-squared errors of the strategies, the influence of the model on the inference itself is downplayed. Nevertheless, the authors often invoke for estimators what they called the ratio estimator property.' They do not define it in full generality, but it seems mathematically equivalent to unbiasedness under the regression model. Also, the large-sample variance estimator that they suggest when the design-based one is difficult to compute is one that estimates expected variance under the model. The authors have attempted to argue that the conventional approach is appropriate for enumerative as opposed to analytic inference. Some effort has been made at a theoretical justification of the performance criteria they consider. However, the authors are most convincing when they argue for unequal probability sampling, on practical grounds, as an alternative to fine stratification. There are infrequent misprints in the text, and Chapter 5 could have referred to Stuart (1963) and Durbin (1953). All in all, however, the book may be hailed as a fitting summary of 30 years of sampling research.