Pareto Sampling versus Sampford and Conditional Poisson Sampling

Abstract.  Pareto sampling was introduced by Rosén in the late 1990s. It is a simple method to get a fixed size π ps sample though with inclusion probabilities only approximately as desired. Sampford sampling, introduced by Sampford in 1967, gives the desired inclusion probabilities but it may take time to generate a sample. Using probability functions and Laplace approximations, we show that from a probabilistic point of view these two designs are very close to each other and asymptotically identical. A Sampford sample can rapidly be generated in all situations by letting a Pareto sample pass an acceptance–rejection filter. A new very efficient method to generate conditional Poisson ( CP ) samples appears as a byproduct. Further, it is shown how the inclusion probabilities of all orders for the Pareto design can be calculated from those of the CP design. A new explicit very accurate approximation of the second-order inclusion probabilities, valid for several designs, is presented and applied to get single sum type variance estimates of the Horvitz–Thompson estimator.     

Empirical Processes in Survey Sampling with (Conditional) Poisson Designs

It is the main purpose of this paper to study the asymptotics of certain variants of the empirical process in the context of survey data. Precisely, Functional Central Limit Theorems are established under usual conditions when the sample is drawn from a Poisson or a rejective sampling design. The framework we develop encompasses sampling designs with non‐uniform first order inclusion probabilities, which can be chosen so as to optimize estimation accuracy. Applications to Hadamard differentiable functionals are considered.     

On Sampling with Prescribed Second‐order Inclusion Probabilities

Abstract. Methods to perform fixed size sampling with prescribed second‐order inclusion probabilities are presented. The focus is on a conditional Poisson design of order 2, a CP(2) design. It is an exponential design of quadratic type and it is carefully studied. In particular, methods to find the suitable values of the parameters and methods to sample are described. Small examples illustrate.     

Fitting time series models for longitudinal survey data under informative sampling

The purpose of this paper is to account for informative sampling in fitting time series models, and in particular an autoregressive model of order one, for longitudinal survey data. The idea behind the proposed approach is to extract the model holding for the sample data as a function of the model in the population and the first-order inclusion probabilities, and then fit the sample model using maximum-likelihood, pseudo-maximum-likelihood and estimating equations methods. A new test for sampling ignorability is proposed based on the Kullback–Leibler information measure. Also, we investigate the issue of the sensitivity of the sample model to incorrect specification of the conditional expectations of the sample inclusion probabilities. The simulation study carried out shows that the sample-likelihood-based method produces better estimators than the pseudo-maximum-likelihood method, and that sensitivity to departures from the assumed model is low. Also, we find that both the conventional t-statistic and the Kullback–Leibler information statistic for testing of sampling ignorability perform well under both informative and noninformative sampling designs.     

A Class of Sampling Two Units with Probability Proportional to Size

A class of sampling two units without replacement with inclusion probability proportional to size is proposed in this article. Many different well known probability proportional to size sampling designs are special cases from this class. The first and second inclusion probabilities of this class satisfy important properties and provide a non-negative variance estimator of the Horvitz and Thompson estimator for the population total. Suitable choice for the first and second inclusion probabilities from this class can be used to reduce the variance estimator of the Horvitz and Thompson estimator. Comparisons between different proportional to size sampling designs through real data and artificial examples are given. Examples show that the minimum variance of the Horvitz and Thompson estimator obtained from the proposed design is not attainable for the most cases at any of the well known designs.     

Optimal sample coordination using controlled selection

Sample coordination maximizes or minimizes the overlap of two or more samples selected from overlapping populations. It can be applied to designs with simultaneous or sequential selection of samples. We propose a method for sample coordination in the former case. We consider the case where units are to be selected with maximum overlap using two designs with given unit inclusion probabilities. The degree of coordination is measured by the expected sample overlap, which is bounded above by a theoretical bound, called the absolute upper bound, and which depends on the unit inclusion probabilities. If the expected overlap equals the absolute upper bound, the sample coordination is maximal. Most of the methods given in the literature consider fixed marginal sampling designs, but in many cases, the absolute upper bound is not achieved. We propose to construct optimal sampling designs for given unit inclusion probabilities in order to realize maximal coordination. Our method is based on some theoretical conditions on joint selection probability of two samples and on the controlled selection method with linear programming implementation. The method can also be applied to minimize the sample overlap.     

Conditional and Restricted Pareto Sampling: Two New Methods for Unequal Probability Sampling

Abstract. Two new unequal probability sampling methods are introduced: conditional and restricted Pareto sampling. The advantage of conditional Pareto sampling compared with standard Pareto sampling, introduced by Rosén (J. Statist. Plann. Inference, 62, 1997, 135, 159), is that the factual inclusion probabilities better agree with the desired ones. Restricted Pareto sampling, preferably conditioned or adjusted, is able to handle cases where there are several restrictions on the sample and is an alternative to the recent cube method for balanced sampling introduced by Deville and Tillé (Biometrika, 91, 2004, 893). The new sampling designs have high entropy and the involved random numbers can be seen as permanent random numbers.     

Randomized nomination sampling for finite populations

We propose a randomized minima–maxima nomination (RMMN) sampling design for use in finite populations. We derive the first- and second-order inclusion probabilities for both with and without replacement variations of the design. The inclusion probabilities for the without replacement variation are derived using a non-homogeneous Markov process. The design is simple to implement and results in simple and easy to calculate estimators and variances. It generalizes maxima nomination sampling for use in finite populations and includes some other sampling designs as special cases. We provide some optimality results and show that, in the context of finite population sampling, maxima nomination sampling is not generally the optimum design to follow. We also show, through numerical examples and a case study, that the proposed design can result in significant improvements in efficiency compared to simple random sampling without replacement designs for a wide choice of population types. Finally, we describe a bootstrap method for choosing values of the design parameters.     

On the efficiency of adaptive designs for flexible interim decisions in clinical trials

It is shown that the optimal group sequential designs considered in Tsiatis and Mehta [2003. On the inefficiency of the adaptive design for monitoring clinical trials. Biometrika 90, 367–378] are special cases of the more general flexible designs which allow for a valid inference after adapting a predetermined way to spend the rejection and acceptance probabilities. An unforeseen safety issue in a clinical trial, for example, could make a change of the preplanned number of interim analyses and their sample sizes appropriate. We derive flexible designs which have equivalent rejection and acceptance regions if no adaptation is performed, but at the same time allow for an adaptation of the spending functions, and have a conditional optimality property.     

An efficient non-rejective implementation of the πps sampling designs

Poisson sampling is a method for unequal probabilities sampling with random sample size. There exist several implementations of the Poisson sampling design, with fixed sample size, which almost all are rejective methods, that is, the sample is not always accepted. Thus, the existing methods can be time-consuming or even infeasible in some situations. In this paper, a fast and non-rejective method, which is efficient even for large populations, is proposed and studied. The method is a new design for selecting a sample of fixed size with unequal inclusion probabilities. For the population of large size, the proposed design is very close to the strict πps sampling which is similar to the conditional Poisson (CP) sampling design, but the implementation of the design is much more efficient than the CP sampling. And the inclusion probabilities can be calculated recursively.     

Simulation-assisted saddlepoint approximation

A general saddlepoint/Monte Carlo method to approximate (conditional) multivariate probabilities is presented. This method requires a tractable joint moment generating function (m.g.f.), but does not require a tractable distribution or density. The method is easy to program and has a third-order accuracy with respect to increasing sample size in contrast to standard asymptotic approximations which are typically only accurate to the first order.

The method is most easily described in the context of a continuous regular exponential family. Here, inferences can be formulated as probabilities with respect to the joint density of the sufficient statistics or the conditional density of some sufficient statistics given the others. Analytical expressions for these densities are not generally available, and it is often not possible to simulate exactly from the conditional distributions to obtain a direct Monte Carlo approximation of the required integral. A solution to the first of these problems is to replace the intractable density by a highly accurate saddlepoint approximation. The second problem can be addressed via importance sampling, that is, an indirect Monte Carlo approximation involving simulation from a crude approximation to the true density. Asymptotic normality of the sufficient statistics suggests an obvious candidate for an importance distribution.

The more general problem considers the computation of a joint probability for a subvector of random T, given its complementary subvector, when its distribution is intractable, but its joint m.g.f. is computable. For such settings, the distribution may be tilted, maintaining T as the sufficient statistic. Within this tilted family, the computation of such multivariate probabilities proceeds as described for the exponential family setting.     

Accounting for uncertainty in the historical response rate of the standard treatment in single‐arm two‐stage designs based on Bayesian power functions

In phase II single‐arm studies, the response rate of the experimental treatment is typically compared with a fixed target value that should ideally represent the true response rate for the standard of care therapy. Generally, this target value is estimated through previous data, but the inherent variability in the historical response rate is not taken into account. In this paper, we present a Bayesian procedure to construct single‐arm two‐stage designs that allows to incorporate uncertainty in the response rate of the standard treatment. In both stages, the sample size determination criterion is based on the concepts of conditional and predictive Bayesian power functions. Different kinds of prior distributions, which play different roles in the designs, are introduced, and some guidelines for their elicitation are described. Finally, some numerical results about the performance of the designs are provided and a real data example is illustrated. Copyright © 2016 John Wiley & Sons, Ltd.     

Conditional estimation using prior information in 2‐stage group sequential designs assuming asymptotic normality when the trial terminated early

Two‐stage designs are widely used to determine whether a clinical trial should be terminated early. In such trials, a maximum likelihood estimate is often adopted to describe the difference in efficacy between the experimental and reference treatments; however, this method is known to display conditional bias. To reduce such bias, a conditional mean‐adjusted estimator (CMAE) has been proposed, although the remaining bias may be nonnegligible when a trial is stopped for efficacy at the interim analysis. We propose a new estimator for adjusting the conditional bias of the treatment effect by extending the idea of the CMAE. This estimator is calculated by weighting the maximum likelihood estimate obtained at the interim analysis and the effect size prespecified when calculating the sample size. We evaluate the performance of the proposed estimator through analytical and simulation studies in various settings in which a trial is stopped for efficacy or futility at the interim analysis. We find that the conditional bias of the proposed estimator is smaller than that of the CMAE when the information time at the interim analysis is small. In addition, the mean‐squared error of the proposed estimator is also smaller than that of the CMAE. In conclusion, we recommend the use of the proposed estimator for trials that are terminated early for efficacy or futility.     

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

The 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.     

On Robust and Efficient Designs for Risk Estimation in Epidemiological Studies

Abstract.  We consider the design problem for the estimation of several scalar measures suggested in the epidemiological literature for comparing the success rate in two samples. The designs considered so far in the literature are local in the sense that they depend on the unknown probabilities of success in the two groups and are not necessarily robust with respect to their misspecification. A maximin approach is proposed to obtain efficient and robust designs for the estimation of the relative risk, attributable risk and odds ratio, whenever a range for the success rates can be specified by the experimenter. It is demonstrated that the designs obtained by this method are usually more efficient than the commonly used uniform design, which allocates equal sample sizes to the two groups.     

Scalable Bayes under Informative Sampling

Bayesian hierarchical formulations are utilized by the U.S. Bureau of Labor Statistics (BLS) with respondent‐level data for missing item imputation because these formulations are readily parameterized to capture correlation structures. BLS collects survey data under informative sampling designs that assign probabilities of inclusion to be correlated with the response on which sampling‐weighted pseudo posterior distributions are estimated for asymptotically unbiased inference about population model parameters. Computation is expensive and does not support BLS production schedules. We propose a new method to scale the computation that divides the data into smaller subsets, estimates a sampling‐weighted pseudo posterior distribution, in parallel, for every subset and combines the pseudo posterior parameter samples from all the subsets through their mean in the Wasserstein space of order 2. We construct conditions on a class of sampling designs where posterior consistency of the proposed method is achieved. We demonstrate on both synthetic data and in application to the Current Employment Statistics survey that our method produces results of similar accuracy as the usual approach while offering substantially faster computation.     

Optimal designs for contingent response models with application to toxicity–efficacy studies

We describe a general family of contingent response models. These models have ternary outcomes constructed from two Bernoulli outcomes, where one outcome is only observed if the other outcome is positive. This family is represented in a canonical form which yields general results for its Fisher information. A bivariate extreme value distribution illustrates the model and optimal design results. To provide a motivating context, we call the two binary events that compose the contingent responses toxicity and efficacy. Efficacy or lack thereof is assumed only to be observable in the absence of toxicity, resulting in the ternary response (toxicity, efficacy without toxicity, neither efficacy nor toxicity). The rate of toxicity, and the rate of efficacy conditional on no toxicity, are assumed to increase with dose. While optimal designs for contingent response models are numerically found, limiting optimal designs can be expressed in closed forms. In particular, in the family of four parameter bivariate location-scale models we study, as the marginal probability functions of toxicity and no efficacy diverge, limiting D optimal designs are shown to consist of a mixture of the D optimal designs for each failure (toxicity and no efficacy) univariately. Limiting designs are also obtained for the case of equal scale parameters.     

Sampling design proportional to order statistic of auxiliary variable

The sampling designs dependent on sample moments of auxiliary variables are well known. Lahiri (Bull Int Stat Inst 33:133–140, 1951) considered a sampling design proportionate to a sample mean of an auxiliary variable. Sing and Srivastava (Biometrika 67(1):205–209, 1980) proposed the sampling design proportionate to a sample variance while Wywiał (J Indian Stat Assoc 37:73–87, 1999) a sampling design proportionate to a sample generalized variance of auxiliary variables. Some other sampling designs dependent on moments of an auxiliary variable were considered e.g. in Wywiał (Some contributions to multivariate methods in, survey sampling. Katowice University of Economics, Katowice, 2003a); Stat Transit 4(5):779–798, 2000) where accuracy of some sampling strategies were compared, too.These sampling designs cannot be useful in the case when there are some censored observations of the auxiliary variable. Moreover, they can be much too sensitive to outliers observations. In these cases the sampling design proportionate to the order statistic of an auxiliary variable can be more useful. That is why such an unequal probability sampling design is proposed here. Its particular cases as well as its conditional version are considered, too. The sampling scheme implementing this sampling design is proposed. The inclusion probabilities of the first and second orders were evaluated. The well known Horvitz–Thompson estimator is taken into account. A ratio estimator dependent on an order statistic is constructed. It is similar to the well known ratio estimator based on the population and sample means. Moreover, it is an unbiased estimator of the population mean when the sample is drawn according to the proposed sampling design dependent on the appropriate order statistic.     

Testing multiple primary endpoints in clinical trials with sample size adaptation

In this paper, we propose a design that uses a short‐term endpoint for accelerated approval at interim analysis and a long‐term endpoint for full approval at final analysis with sample size adaptation based on the long‐term endpoint. Two sample size adaptation rules are compared: an adaptation rule to maintain the conditional power at a prespecified level and a step function type adaptation rule to better address the bias issue. Three testing procedures are proposed: alpha splitting between the two endpoints; alpha exhaustive between the endpoints; and alpha exhaustive with improved critical value based on correlation. Family‐wise error rate is proved to be strongly controlled for the two endpoints, sample size adaptation, and two analysis time points with the proposed designs. We show that using alpha exhaustive designs greatly improve the power when both endpoints are effective, and the power difference between the two adaptation rules is minimal. The proposed design can be extended to more general settings. Copyright © 2015 John Wiley & Sons, Ltd.     

A BETA-BINOMIAL MODEL FOR ESTIMATING THE SIZE OF A HETEROGENEOUS POPULATION

This paper compares the properties of various estimators for a beta‐binomial model for estimating the size of a heterogeneous population. It is found that maximum likelihood and conditional maximum likelihood estimators perform well for a large population with a large capture proportion. The jackknife and the sample coverage estimators are biased for low capture probabilities. The performance of the martingale estimator is satisfactory, but it requires full capture histories. The Gibbs sampler and Metropolis‐Hastings algorithm provide reasonable posterior estimates for informative priors.