A geometric characterization of optimal designs for regression models with correlated observations

We consider the problem of optimal design of experiments for random effects models, especially population models, where a small number of correlated observations can be taken on each individual, while the observations corresponding to different individuals are assumed to be uncorrelated. We focus on c-optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of Elfving (1952) from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients. The theory is demonstrated by finding optimal designs for a linear model with correlated observations and a nonlinear random effects population model, which is commonly used in pharmacokinetics.     

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.     

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.     

Confidence intervals for the mean of a population containing many zero values under unequal‐probability sampling

In many applications, a finite population contains a large proportion of zero values that make the population distribution severely skewed. An unequal‐probability sampling plan compounds the problem, and as a result the normal approximation to the distribution of various estimators has poor precision. The central‐limit‐theorem‐based confidence intervals for the population mean are hence unsatisfactory. Complex designs also make it hard to pin down useful likelihood functions, hence a direct likelihood approach is not an option. In this paper, we propose a pseudo‐likelihood approach. The proposed pseudo‐log‐likelihood function is an unbiased estimator of the log‐likelihood function when the entire population is sampled. Simulations have been carried out. When the inclusion probabilities are related to the unit values, the pseudo‐likelihood intervals are superior to existing methods in terms of the coverage probability, the balance of non‐coverage rates on the lower and upper sides, and the interval length. An application with a data set from the Canadian Labour Force Survey‐2000 also shows that the pseudo‐likelihood method performs more appropriately than other methods. The Canadian Journal of Statistics 38: 582–597; 2010 © 2010 Statistical Society of Canada     

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.     

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.     

Orthogonal three-level parallel flats designs for user-specified resolution

Orthogonal factorial and fractional factorial designs are very popular in many experimental studies, particularly the two-level and three-level designs used in screening experiments. When an experimenter is able to specify the set of possibly nonnegligible factorial effects, it is sometimes possible to obtain an orthogonal design belonging to the class of parallel flats designs, that has a smaller run-size than a suitable design from the class of classical fractional factorial designs belonging to the class of single flat designs. Sri-vastava and Li (1996) proved a fundamental theorem of orthogonal s-level, s being a prime, designs of parallel flats type for the user-specified resolution. They also tabulated a series of orthogonal designs for the two-level case. No orthogonal designs for three-level case are available in their paper. In this paper, we present a simple proof for the theorem given in Srivastava and Li (1996) for the three-level case. We also give a dual form of the theorem, which is more useful for developing an algorithm for construction of orthogonal designs. Some classes of three-level orthogonal designs with practical run-size are given in the paper.     

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.     

Non-linear design problem in a chemical kinetic model with non-constant error variance

For a locally optimum non-linear design problem for a chemical kinetic model, we investigate the influence of the dispersion structure of the random observation errors on the design and its efficiency. We find that there are two kinds of design determined by the model parameters and the error variance function: “interior” designs, and “boundary” designs that depend also on the design range. We give an exact criterion for determining which kind of design will arise and we illustrate the qualitative difference between the two kinds of design in terms of the design locus and the equivalence theorem. We tabulate quantitative details of the designs for a range of parameter values.     

Minimization of model inadequacy when observations subject to shift in means and variances

In this article optimality of experimental design for fitting a lower-order polynomial to a higher order response function for the situation in which observations may be subject to shift in means as well as in variances is considered. It is found that Karson, Manson and Hader‘s (1969) optimum designs provide pro-tection, in some sense, against model inadequacies even when observations are subject to shift in means and variances.     

An Optimal Multivariate Stratified Sampling Design Using Dynamic Programming

Numerous optimization problems arise in survey designs. The problem of obtaining an optimal (or near optimal) sampling design can be formulated and solved as a mathematical programming problem. In multivariate stratified sample surveys usually it is not possible to use the individual optimum allocations for sample sizes to various strata for one reason or another. In such situations some criterion is needed to work out an allocation which is optimum for all characteristics in some sense. Such an allocation may be called an optimum compromise allocation. This paper examines the problem of determining an optimum compromise allocation in multivariate stratified random sampling, when the population means of several characteristics are to be estimated. Formulating the problem of allocation as an all integer nonlinear programming problem, the paper develops a solution procedure using a dynamic programming technique. The compromise allocation discussed is optimal in the sense that it minimizes a weighted sum of the sampling variances of the estimates of the population means of various characteristics under study. A numerical example illustrates the solution procedure and shows how it compares with Cochran's average allocation and proportional allocation.     

A New Proof of Murthy's Estimator which Applies to Sequential Sampling

Murthy's estimator has been used for constructing an unbiased estimator of a population total or mean from a sample of fixed size when there is unequal probability sampling without replacement. Traditionally, the estimator is derived by constructing an unordered version of Raj's ordered unbiased estimator. This paper presents an elementary proof of Murthy's estimator which applies the Rao–Blackwell theorem to a very simple estimator. This proof includes any sequential sampling scheme, thus extending the usefulness of Murthy's estimator. We demonstrate this extension by deriving unbiased estimators for inverse sampling.     

Optimal decomposition of probabilistic networks by simulated annealing

This paper investigates the applicability of a Monte Carlo technique known as simulated annealing to achieve optimum or sub-optimum decompositions of probabilistic networks under bounded resources. High-quality decompositions are essential for performing efficient inference in probabilistic networks. Optimum decomposition of probabilistic networks is known to be NP-hard (Wen, 1990). The paper proves that cost-function changes can be computed locally, which is essential to the efficiency of the annealing algorithm. Pragmatic control schedules which reduce the running time of the annealing algorithm are presented and evaluated. Apart from the conventional temperature parameter, these schedules involve the radius of the search space as a new control parameter. The evaluation suggests that the inclusion of this new parameter is important for the success of the annealing algorithm for the present problem.     

On the Convergence of the Horvitz-Thompson Estimator

We investigate the degree to which the Horvitz-Thompson estimator approximates the population mean in survey sampling. Specifically, conditions for (mean square) consistency are given, along with rates of convergence. The conditions involve only first and second-order inclusion probabilities, and for various sampling designs they are easy to check.     

Note on umvsu-estimation under randomized response model

Considering a class ofs randomized response trials for eliciting sensitive information from a sample survey and a class of ordered sampling designs, a uniformly minimum variance unbiased estimator of population variance (of the sensitive character) has been obtained. This note indicates that a theorem (theorem 3.9) of Cassel, Sarndal and Wretman (1977) and the results in the present note can be extended to estimation of any symmetric function of population values in the field of direct response surveys and randomized response surveys respectively.     

Optimal designs for statistical analysis with Zernike polynomials

The Zernike polynomials arise in several applications such as optical metrology or image analysis on a circular domain. In the present paper, we determine optimal designs for regression models which are represented by expansions in terms of Zernike polynomials. We consider two estimation methods for the coefficients in these models and determine the corresponding optimal designs. The first one is the classical least squares method and Φ _p -optimal designs in the sense of Kiefer [Kiefer, J., 1974, General equivalence theory for optimum designs (approximate theory). Annals of Statistics, 2 849–879.] are derived, which minimize an appropriate functional of the covariance matrix of the least squares estimator. It is demonstrated that optimal designs with respect to Kiefer's Φ _p -criteria (p>?∞) are essentially unique and concentrate observations on certain circles in the experimental domain. E-optimal designs have the same structure but it is shown in several examples that these optimal designs are not necessarily uniquely determined. The second method is based on the direct estimation of the Fourier coefficients in the expansion of the expected response in terms of Zernike polynomials and optimal designs minimizing the trace of the covariance matrix of the corresponding estimator are determined. The designs are also compared with the uniform designs on a grid, which is commonly used in this context.     

Influence analysis on crossover design experiment in bioequivalence studies

Crossover designs are commonly used in bioequivalence studies. However, the results can be affected by some outlying observations, which may lead to the wrong decision on bioequivalence. Therefore, it is essential to investigate the influence of aberrant observations. Chow and Tse in 1990 discussed this issue by considering the methods based on the likelihood distance and estimates distance. Perturbation theory provides a useful tool for the sensitivity analysis on statistical models. Hence, in this paper, we develop the influence functions via the perturbation scheme proposed by Hampel as an alternative approach on the influence analysis for a crossover design experiment. Moreover, the comparisons between the proposed approach and the method proposed by Chow and Tse are investigated. Two real data examples are provided to illustrate the results of these approaches. Our proposed influence functions show excellent performance on the identification of outlier/influential observations and are suitable for use with small sample size crossover designs commonly used in bioequivalence studies. Copyright © 2013 John Wiley & Sons, Ltd.     

Locally optimal designs for the simple death process

We demonstrate the existence and uniqueness of optimal observation schedules for discrete observation of the simple death process. Optimality is framed in the sense of observing the process at times that maximise the Fisher information for the death rate. We examine the relationship of our designs with those obtained by other authors for the simple birth process. A number of interesting properties are uncovered. Practical considerations in the application of these designs in an experimental setting are investigated, and we examine the performance of some approximately optimal designs that are usually simpler to implement. We will show that our optimal designs are highly robust to mis-specification of the death rate.     

D-optimal statistical designs with restricted string property

This paper applies Frechet derivatives to derive asymp-totically D-optimal statistical designs where the designmatrix is a (O,l)-matrix having exactly one run [of length at most k(, the number of parameters) of l's in each row. These asymptotic results have been utilized in dealing with the more intractable design problem with a finite number of observations. The problemof E-optimality has also been considered.     

Tolerance‐based error reduction in multi‐agent bioassays

Some degree of error is inevitable in multi‐agent bioassays regardless of design or measurement technology. Estimation error can be reduced post facto by exploiting the matrix partial ordering of the bioassay survival estimates. The standard method for this is order‐restricted regression (ORR). If the joint action of the bioassay agents admits a tolerance‐based interpretation, additional structure beyond matrix partial ordering is available, leading to a new method of error reduction. This tolerance‐based error reduction (TBER) procedure almost always outperforms ORR. Like ORR, TBER applies to complete factorial bioassay designs and, using weighting, to incomplete designs.