E-optimal minimally connected block designs under mixed effects model

Summary Considering a mixed effects model in a minimally connected block design set-up, we obtain designs which areE-optimal, uniformly in the ratio of the variance components, for inference on varietal contrasts which constitute the fixed effects in the model. Work supported by the Centre for Management and Development Studies, Indian Institute of Management Calcutta. Work supported by the National Sciences and Engineering Research Council of Canada under grant number 7272 and partially done while visiting Indian Statistical Institute, Calcutta, India.     

Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming‐Based Approach

This paper uses semidefinite programming (SDP) to construct Bayesian optimal design for nonlinear regression models. The setup here extends the formulation of the optimal designs problem as an SDP problem from linear to nonlinear models. Gaussian quadrature formulas (GQF) are used to compute the expectation in the Bayesian design criterion, such as D‐, A‐ or E‐optimality. As an illustrative example, we demonstrate the approach using the power‐logistic model and compare results in the literature. Additionally, we investigate how the optimal design is impacted by different discretising schemes for the design space, different amounts of uncertainty in the parameter values, different choices of GQF and different prior distributions for the vector of model parameters, including normal priors with and without correlated components. Further applications to find Bayesian D‐optimal designs with two regressors for a logistic model and a two‐variable generalised linear model with a gamma distributed response are discussed, and some limitations of our approach are noted.     

Quasi-likelihood inference in a generalized linear mixed model for balanced data

For modelling the effect of crossed, fixed factors on the response variable in balanced designs with nested stratifications, a generalized linear mixed model is proposed. This model is based on a set of quasi-likelihood assumptions which imply quadratic variance functions. From these variance functions, deviances are obtained to quantify the variation per stratification. The effects of the fixed factors will be tested, an dispersion components will be estimated. The practical use of the model is illustrated by reanalysing a soldering failures problem.     

Optimal variance- and efficiency-balanced designs for one- and two-way elimination of heterogeneity

Summary In this paper, a series ofE-optimal non-binary variance balanced (block or row-column) designs and a series ofE-optimal non-binary efficiency balanced (block or row-column) designs are provided in certain broad classes of competing designs. Furthermore, their high efficiencies by the usualA- andD-optimality criteria are shown.     

Efficiencies of Rounded Optimal Approximate Designs for Small Samples

Optimal exact designs are notoriously hard to study and only a few of them are known for polynomial models. Using recently obtained optimal exact designs (I mhof , 1997), we show that the efficiency of the frequently used rounded optimal approximate designs can be sensitive if the sample size is small. For some criteria, the efficiency of the rounded optimal approximate design can vary by as much as 25% when the sample size is changed by one unit. The paper also discusses lower efficiency bounds and shows that they are sometimes the best possible bounds for the rounded optimal approximate designs.     

RANK TESTS IN 2X2 DESIGNS

Abstract. In literature numerous attempts can be found for the evaluation of two factor designs with fixed effects by means of rank tests. The aim of the present article is to show the limits of these methods and to give some new procedures for 2X2 designs. First, functionals of distribution functions shall be defined whose relations to the usual parameters of the linear model are analysed. These functionals are free of nuissance parameters under the respective hypothesis; they are estimated by special ranks of the data. The asymptotic distribution of these statistics is derived by a generalization of the Chemoff–Savage theorem for correlated random variables. The asymptotic variance depends on the parent distribution function but it can be estimated by using special rank methods. Thus, one obtains asymptotically distribution–free tests for two–factor designs with fixed effects. Some counter examples show why it is not possible to construct suitable rank tests for greater designs than the 2X2 design. The paper closes with a discussion of the drawbacks of the well known rank transform.     

Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

A note on the robustness of Box-Behnken designs to the unavailability of data

Box-Behnken designs and central composite designs are efficient designs for fitting second order polynomials to response surfaces, because they use relatively small numbers of observations to estimate the parameters. In this paper we investigate the robustness of Box-Behnken designs to the unavailability of observations, in the sense of finding t _max , the maximum number of arbitrary rows in the design matrix that can be removed and still leave all of the parameters of interest estimable. The results are compared to the known results for the central composite designs found in MacEachern, Notz, Whittinghill & Zhu (1995). The blocked Box-Behnken designs are equally as robust as those that are not blocked. Received December 1997     

Multi-level k-circulant Supersaturated Designs

In this paper we present a new method for constructing multi-level supersaturated designs with n rows, m columns and the equal occurrence property. We investigate the existence of multi-level supersaturated designs using a single generator vector and its k-cyclic permutations as rows. We find the conditions needed, in order this vector to generate a balanced supersaturated design. These conditions are simplified for the case of three level supersaturated designs. By using the proposed method three level supersaturated designs are constructed and explored. Moreover, many new, optimal and near optimal, multi-level supersaturated designs are provided as well.     

On the role of weights rounding in applications of resampling based on pseudopopulations

Resampling methods are widely studied and increasingly employed in applied research and practice. When dealing with complex sampling designs, common resampling techniques require adjusting noninteger sampling weights in order to construct the so called “pseudopopulation” in order to perform the actual resampling. The practice of rounding, however, has been empirically shown to be harmful under general designs. In this paper, we present asymptotic results concerning, in particular, the practice of rounding resampling weights to the nearest integer, an approach that is commonly adopted by virtue of its reduced computational burden, as opposed to randomization‐based alternatives. We prove that such approach leads to nonconsistent estimation of the distribution function of the survey variable; we provide empirical evidence of the practical consequences of the nonconsistency when the point estimation of the variance of complex estimators is of interest.     

Performance of empirical Bayes estimators of random coefficients in multilevel analysis: Some results for the random intercept-only model

For a multilevel model with two levels and only a random intercept, the quality of different estimators of the random intercept is examined. Analytical results are given for the marginal model interpretation where negative estimates of the variance components are allowed for. Except for four or five level-2 units, the Empirical Bayes Estimator (EBE) has a lower average Bayes risk than the Ordinary Least Squares Estimator (OLSE). The EBEs based on restricted maximum likelihood (REML) estimators of the variance components have a lower Bayes risk than the EBEs based on maximum likelihood (ML) estimators. For the hierarchical model interpretation, where estimates of the variance components are restricted being positive, Monte Carlo simulations were done. In this case the EBE has a lower average Bayes risk than the OLSE, also for four or five level-2 units. For large numbers of level-1 (30) or level-2 units (100), the performances of REML-based and ML-based EBEs are comparable. For small numbers of level-1 (10) and level-2 units (25), the REML-based EBEs have a lower Bayes risk than ML-based EBEs only for high intraclass correlations (0.5).     

Asymptotically Optimal Response‐Adaptive Designs for Allocating the Best Treatment: An Overview

Response‐adaptive designs are being used increasingly in applications, and this is especially so in early phase clinical trials. This paper reviews a particular class of response‐adaptive designs that have the property of picking the superior treatment with probability tending to one. This is a desirable property from an ethical point of view in clinical trials. The model underlying such designs is a randomly reinforced urn. This paper provides an overview of results for these designs, starting from the early paper of Durham and Yu (1990) until the recent work by Flournoy, May, Moler and Plo (2010).     

Efficiencies for optimum designs when transforming the response in nonlinear models with nonconstant variance

Often the responses from mechanistic models have to be transformed to achieve error distributions that are symmetric and have constant variance. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples given of optimum designs for particular values of λ, together with the efficiency of these designs as λ varies. Approaches to finding designs robust to variations in λ are indicated and exemplified.     

Optimality and efficiency of circular neighbor balanced designs for correlated observations

The purpose of this paper is to apply the Kunert and Martins (2000b) method for finding optimal designs into the case of dependence. Using this method we study optimality of circular neighbor balanced designs at distances 1 and 2 under the one-dimensional interference model with errors correlated according to a circular autoregressive process. We determine the efficiency of binary designs for specified values of correlation coefficient, for which these designs are not optimal.Research partially supported by the KBN Grant Number 5 P03A 041 21.     

On the speed of convergence of some urn designs for the balanced allocation of two treatments

The present paper deals with sequential urn designs for the balanced allocation of two treatments.Wei (1977, 1978b) was among the first authors to suggest an algorithm based on the probabilistic properties of the generalized Friedman urn and recently Chen (2000) has suggested the Ehrenfest design, namely a sequential procedure based on the Ehrenfest urn process. Some extensions of these algorithms are discussed: in particular we focus on a generalization of Chen's procedure, called the Ehrenfest-type urn designs, recently introduced by Baldi Antognini (2004). By analyzing some convergence properties of the Ehrenfest process, we show that for an Ehrenfest-type urn it is possible to evaluate the variance of the design at each step and analyze the convergence to balance. Furthermore, the generalized Friedman urn and the Ehrenfest-type procedures are compared in terms of speed of convergence. The Ehrenfest design proposed by Chen converges to balance faster than the other urn procedures.     

Optimal designs for generalized linear models with biased response

This paper is concerned with the search for locally optimal designs when the observations of the response variable arise from a weighted distribution in the exponential family. Locally optimal designs are derived for regression models in which the response follows a weighted version of Normal, Gamma, Inverse Gaussian, Poisson or Binomial distributions. Some conditions are given under which the optimal designs for the weighted and original (non-weighted) distributions are the same. An efficiency study is performed to find out the behavior of the D-optimal designs for the original distribution when they are used to estimate models with weighted distributions.     

Quasi-longitudinal Designs in SEM state Space Modeling

The collection of longitudinal data over the full model time scope is often an appropriate way to estimate the dynamic state space model with time-varying parameters. Nevertheless, in many situations it is possible and preferable to collect and combine data from independent groups of subjects, each covering a shorter interval than the full dynamic model. Several quasi-longitudinal designs are discussed: overlapping designs (overlapping cohort design OCD and overlapping samples design OSD) as well as nonoverlapping designs up to the exclusively cross-sectional design. The use of the structural equation modeling (SEM) program Mx and continuous time state space modeling is recommended. Finally, a number of the quasi-longitudinal designs is empirically evaluated, comparing the results with those of the full-longitudinal design.     

Algorithms for optimal design with application to multiple polynomial regression

The approximate theory of optimal linear regression design leads to specific convex extremum problems for numerical solution. A conceptual algorithm is stated, whose concrete versions lead us from steepest descent type algorithms to improved gradient methods, and finally to second order methods with excellent convergence behaviour. Applications are given to symmetric multiple polynomial models of degree three or less, where invariance structures are utilized. A final section is devoted to the construction of efficientexact designs of sizeN from the optimal approximate designs. For the multifactor cubic model and some of the most popular optimality criteria (D-, A-, andI-criteria) fairly efficient exact designs are obtained, even for small sample sizeN. AMS Subject Classification: 62K05.Abbreviated Title: Algorithms for Optimal Design.Invited paper presented at the International Conference on Mathematical Statistics,ProbaStat '94, Smolenice, Slovakia.     

Optimal extended complete block designs for dependent observations

Optimal designs under general dependence structures are usually difficult to specify theoretically or find algorithmically. However, they can sometimes be found for a specific dependence structure and a particular parameter value. In this paper, a class of generalized binary block designs with t treatments and b blocks of size k>t is considered. Each block consists of h consecutive complete blocks and, at the end, an incomplete block of size k−ht (if k > ht). For a suitable number of blocks, a universally optimal design is found for a first-order stationary autoregressive process with positive correlations. Optimal generalized binary designs and balanced block designs are also considered. Some constructions for a universally optimal design are described. A negative dependence parameter, and some other dependence structures, are also considered.     

Design and cost optimization for hierarchical data

In this paper, we consider balanced hierarchical data designs for both one‐sample and two‐sample (two‐treatment) location problems. The variances of the relevant estimates and the powers of the tests strongly depend on the data structure through the variance components at each hierarchical level. Also, the costs of a design may depend on the number of units at different hierarchy levels, and these costs may be different for the two treatments. Finally, the number of units at different levels may be restricted by several constraints. Knowledge of the variance components, the costs at each level, and the constraints allow us to find the optimal design. Solving such problems often requires advanced optimization tools and techniques, which we briefly explain in the paper. We develop new analytical tools for sample size calculations and cost optimization and apply our method to a data set on Baltic herring.