mODa 6 — Advances in Model-Oriented Design and Analysis

Frontmatter

Trend-free Repeated Measurement Designs

Abstract

The existence and non-existence of trend-free repeated measurement designs are investigated. Two families of efficient/optimal repeated measurement designs which are very popular among experimenters are shown to be trend-free or trend-free with respect to treatments.

K. Afsarinejad

Minimax Optimal Designs for Nonparametric Regression — A Further Optimality Property of the Uniform Distribution

Abstract

In the common nonparametric regression model y _{i n} =g(t _{i n} ) + σ(t _{i n} ) ε _i i = 1,…,n with i.i.d. noise and nonrepeatable design pointsti we consider the problem of choosing an optimal design for the estimation of the mean functiong. A minimax approach is adopted which searches for designs minimizing the maximum of the asymptotic integrated mean squared error, where the maximum is taken over an appropriately bounded class of functions(g σ). The minimax designs are found explicitly, and for certain special cases the optimality of the uniform distribution can be established.

S. Biedermann, H. Dette

Optimization of Monitoring Networks for Estimation of the Semivariance Function

Abstract

The optimal adjustment of an existing monitoring network for estimation of the semivariance function by means of optimal design of experiments is discussed. The difference between neglecting and including correlation between point pairs, from which the semivariance function is estimated, is visualized for a simple adjustment of a monitoring network. A branch-and-bound algorithm is applied to calculate an exact optimal configuration of monitoring sites (design). For a case study it is shown that the optimal design is robust against misspecified parameter values and model choice.

E. P. J. Boer, E. M. T. Hendrix, D. A. M. K. Rasch

A-optimal Chemical Balance Weighing Designs with Diagonal Covariance Matrix of Errors

Abstract

The paper deals with the problem of estimating the weights of objects using an A-optimal chemical balance weighing design with the covariance matrix of errors o²G, where G is an n x n known positive definite diagonal matrix. A lower bound of tr(X’G^-1X)^-1is obtained and a necessary and sufficient condition for this lower bound to be attained is given. The incidence matrices of balanced incomplete block designs have been used to construct A-optimal chemical balance weighing designs.

B. Ceranka, K. Katulska

Replications with Gröbner Bases

Abstract

We present an extension of the Gröbner basis method for experimental design introduced in Pistone and Wynn (1996) to designs with replicates. This extension is presented in an abstract regression analysis framework, based on direct computations with functions and inner products. Explicit examples are presented to illustrate our approach.

A. M. Cohen, A. Di Bucchianico, E. Riccomagno

Extracting Information from the Variance Function: Optimal Design

Abstract

Regression models with the variance function depending on unknown parameters appear in a number of practical problems (variogram fitting and mixed effect models are popular examples). We found that estimation of parameters entering both response and variance functions can be combined in a rather simple way. The proposed estimator belongs to the class of iterated estimators and is numerically very close to the fixed point method, which takes the form of the reweighted least squares method in our setting. The proposed estimators lead to design problems with additive information matrices and therefore can be treated within traditional convex design theory.

D. Downing, V. V. Fedorov, S. Leonov

Model Validity Range in Multicentre Clinical Trials

Abstract

Analyses of multicentre trials consider the estimated treatment effect differences of the individual centres and combine them into an estimate of the overall treatment effect. There has been much debate in the literature concerning the best way to combine these treatment effect differences. We emphasize that first of all one should define the combined response to treatment (CRT), the object that has to be estimated from the results of a multicentre clinical trial. Having the defined target in mind, the least squares estimators of the CRT under three possible models of increasing complexity for multicentre data are derived. We compare these estimators in terms of their mean squared errors. It is shown that the choice of CRT determines not only the best estimator, but also the allocation of patients among the centres that minimizes the MSE.

V. Dragalin, V. V. Fedorov

Two Models of Nonadaptive Group Testing for Designing Screening Experiments

Abstract

We discuss two non-standard models of nonadaptive combinatorial search which develop the conventional disjunct search model of Du and Hwang (1993) for a small number of defective elements contained in a finite ground set or a population. The first model called asearch of defective supersets (complexes)was suggested in D’yachkovetal. (2000c,d). The second model which can be called asearch of defective subsets in the presence of inhibitorswas introduced for the case of an adaptive search by Farachetal. (1997) and De Bonis and Vaccaro (1998). For these models, we study the constructive search methods based on the known constructions for the disjunct model from Kautz and Singleton (1964) and from D’yachkovetal. (2000a,b).

A. G. D’yachkov, A. J. Macula, D. C. Torney, P. A. Vilenkin

Optimal Designs for a Continuation-ratio Model

Abstract

Optimal designs for a trinomial response under the continuation-ratio model are discussed. Examples of locally D-optimal and Bayesian optimal designs for some simple prior distributions are given together with closed form expressions for designs which are approximately optimal.

S. K. Fan, K. Chaloner

Bayesian Interpolation Schemes for Monitoring Systems

Abstract

Environmental monitoring gives rise to several statistical problems of optimal design. We study the question of interpolation from a Bayesian point of view. Although the considered models are motivated by applications we focus on a theoretical concept rather than the presentation of case studies. We derive Bayesian estimates and risks for variables of interest in monitoring systems under conjugate priors for multivariate normal distributions. Numerical approaches such as MCMC algorithms are discussed in addition.

K. Felsenstein

Optimality of the Wald SPRT for Processes with Continuous Time Parameter

Abstract

The paper deals with the Wald sequential test of two simple hypotheses for processes with continuous time parameter. The observation is a likelihood ratio process, which is a right-continuous local martingale having left-side limits. The Wald sequential test is proved to be optimal in the sense that it minimizes the Kullback-Leibler information under both hypotheses among all tests having no larger error probabilities. The proof is based on the explicit solution of the related optimal stopping problem.

L. I. Galtchouk

Efficient Paired Comparison Designs for Utility Elicitation

Abstract

In applications data are often available from the comparison of two alternatives rather than the direct valuation of a single object on its own. Experiments have to be designed for these paired comparisons in a different way than for standard situations. In this note we deal with a problem arising from organizational psychology and present an application of design considerations to the estimation of utility functions.

H. Großmann, U. Graßhoff, H. Holling, R. Schwabe

Optimal Design for the Testing of Anti-malarial Drugs

Abstract

In dose-response experiments involving anti-malarial drugs the number of maturing parasites is recorded but the number of parasites originally present in the blood sample is unknown. This situation, commonly referred to as Wadley’s problem, can be modelled by means of a particular Poisson distribution. In this paper designs for which the parameters of interest or functions of those parameters are estimated as precisely as possible are developed. In particular locallyD- D ₃ -and c-optimal designs are constructed and compared and the D₃-optimal design is used to provide a bench mark for appraising existing procedures.

L. M. Haines, G. P. Y. Clarke, E. Gouws, W. F. Rosenberger

Optimal Adaptive Designs for Delayed Response Models: Exponential Case

Abstract

We propose a delayed response model for a Bernoulli 2-armed bandit. Patients arrive according to a Poisson process and their response times are exponential. We develop optimal solutions, and compare to previously suggested designs.

J. Hardwick, R. Oehmke, Q. F. Stout

Non-D-optimality of the Simplex Centroid Design for Regression Models Homogeneous of Degree p

Abstract

Models with regression functions homogeneous of degree p are suitable in some situations to describe the data derived from mixture experiments. The uniformly weighted simplex centroid design, supported by the barycenters corresponding to the regression functions may be an intuitive design. The aim of the investigation is to show the non-D-optimality of the design.

R.-D. Hilgers

New Upper Bounds for Maximum-Entropy Sampling

Abstract

We develop new upper bounds for the constrained maximum-entropy sampling problem. Our partition bounds are based on Fischer’s inequality. Further new bounds combine the use of Fischer’s inequality with previously developed bounds. We demonstrate this in detail by using partitioning to strengthen spectral bounds. Computations suggest that these bounds may be useful in finding optimal solutions by branch-and-bound.

A. Hoffman, J. Lee, J. Williams

Residuals

Abstract:

We consider different definitions for residuals and their distributions with a view to their use in bootstrapping. It turns out that order statistics play an important role in finding good residuals.

H. Läuter

Asymptotically Optimal Sequential Discrimination between Markov Chains

Abstract

An asymptotic lower bound is derived involving a second additive term of order \( {\text{ as }}\alpha \to 0 \) for the mean length of a sequential strategy s for discrimination between two statistical models for Markov chains. The parameter a is the maximal error probability of s. A sequential strategy is outlined attaining (or almost attaining) this asymptotic bound uniformly over the distributions of models including those from the indifference zone.

M. B. Malyutov, I. I. Tsitovich

Optimum Experimental Designs for a Modified Inverse Linear Model

Abstract:

In this paper some optimal designs for a modified inverse linear model are studied. D-optimal designs are obtained from analytical results. In order to study some of the characteristics of this modelc- LandDL-optimal designs are examined empirically. Finally, the behavior of the resulting designs is compared.

I. Martínez, I. Ortiz, C. Rodríguez

Permutation Tests for Effects in Unbalanced Repeated Measures Factorial Designs

Abstract

We deal with permutation testing for balanced and unbalanced repeated measures designs and we consider a replicated homoscedastic (balanced or unbalanced) factorial design with fixed effects (Milliken, 1984) as the basic experimental plan. It is worth noting that the new permutation approach, presented here, is exact and, being conditional to the sufficient statistic represented by the data matrix it does not require normality of error terms in the linear model for responses. A comparative simulation study has been performed in order to evaluate the power of such exact separate tests.

D. Mazzaro, F. Pesarin, L. Salmaso

The Influence of the Design on the Breakdown Point of ℓ1-type M-estimators

Abstract

Mizera and Müller (1999) showed that the breakdown of M-estimators with bounded score function in the general linear model depends via a quantityM(Xr) only on the designXand the variation exponent r of the objective function. In the case r = 0, e.g. for the Cauchy M-estimator, the quantityM(Xr) is the maximum number of design points at which the unknown parameter vector is not identifiable. We study the case r = 1, the case of the so-called ti-type M-estimators¡ªincluding the Li-estimator itself, Huber’s M-estimator, and others¡ªwhere the quantityM(X) =M(X, 1) has the potential of detecting the presence of leverage points in the design. We give a numerical algorithm for computingM(X)and calculate it for several examples, showing howM (X)can be used as a diagnostic tool for detecting leverage points and how the design influences the breakdown point of ti-type M-estimators.

I. Mizera, Ch. H. Müller

Analytical Properties of Locally D-optimal Designs for Rational Models

Abstract

The paper is devoted to studying locally D-optimal designs for rational regression models. He et al. (1997) have shown that the problem considered is equivalent to that of D-optimal designs for polynomial regression with polynomial variance function. Both problems were studied in a number of papers with the help of two usual approaches: numerical construction of optimal designs and constructing the designs in a closed analytical form. It appears that the latter form is available only in some special cases. Here another approach is developed. Points of optimal designs are studied as functions of the parameters. It turns out that these functions can be expanded in Taylor series. An example of such expansion for a model with six parameters is given.

V. B. Melas

Understanding Aliasing Using Gröbner Bases

Abstract

The now well-established Gröbner basis method in experimental design (see the authors’ monograph “Algebraic Statistics”) had the understanding of aliasing as a key motivation. The basic method asks: given an experimental design, what is estimable, or more generally what is the alias structure? The paper addresses the following related question: given a set of conditions which the design is known to satisfy, what can we say about the alias structure? Some classical and non-classical construction methods are included.

G. Pistone, E. Riccomagno, H. P. Wynn

Average D-optimum Design for Randomly Varying Experimental Conditions

Abstract

We consider a regression model η(ϑ, ξ) for which the experimental conditions ξk used at the experimentation stage may randomly fluctuate around the valuesζkspecified at the design stage. The problem consists in choosingN values ei, ζ1, ⋯, ζN to estimate the parameters /9 from observatiSons yk,k = 1,⋯,N,with yk = n(t9,.k) +ekandIrkan i.i.d. sequence. We assume that the ξk’s are observed, so that the accuracy of the estimation can be characterized by the usual information matrix. We design the experiment by maximizing the expected value of its determinant. We give the expression of this expected determinant in the case where theξk’sare stochastically independent. Deterministic exchange algorithms can then be used to determine the optimal choice of the (ζk’s. Two examples are presented.

L. Pronzato

Constrained Bayesian Optimal Designs for Phase I Clinical Trials: Continuous Dose Space

Abstract:

We derive constrained Bayesian D-optimal designs for efficient estimation in phase I clinical trials with binary responses on a continuous dose space. The constraint is based on ethical considerations that patients cannot be assigned to highly toxic doses. We find that a naive restriction on the dose space, requiring patients to be assigned below the mean of a quantile, is about as effective as the more computationally intensive constrained designs.

W. F. Rosenberger, L. M. Haines, I. Perevozskaya

Trend-Robust and Budget Constrained Optimum Designs

Abstract

Performing experiments in a time sequence may induce time dependence in the observed results. In practice, run orders that are optimally balanced for time trends are of little use because of economical reasons. This paper proposes an optimality criterion that strikes a balance between cost-efficiency and trend-resistance. The presented design algorithm serves as a proper tool for the construction of cost-efficient run orders with an optimal protection against time order dependence. The algorithm is intended to provide the experimenter with a general method for solving a wide range of practical problems. A real-life industrial case illustrates practical utility.

L. Tack, M. Vandebroek

Minimax Designs for Logistic Regression in a Compact Interval

Abstract

Minimax designs for logistic regression with one design variable are considered where the design space is a general interval [a, b] on the straight real line. This corresponds to a weighted linear regression model for a specific weight function. Torsney and López-Fidalgo (1995) computed maximum variance (MV-) optimal designs for simple linear regression for a general interval. López-Fidalgo et al. (1998) gave MV-optimal designs for symmetric weight functions and symmetric intervals, [-b, b]. Computations for general intervals for logistic regression are much more complex. Using similar approaches to these two papers and the equivalence theorem, explicit minimax optimal designs are given for different regions of points (a, b) in the semi-plane b>a for the logistic model. This complements Dette and Sahm (1998) who give a method for computing MV-optimal designs without any restriction on the design space. From this, Standardized Maximum Variance (SMV-) optimal designs immediately follow. The same approach could be used for probit, double exponential, double reciprocal and some other popular models in the biomedical sciences.

B. Torsney, J. López-Fidalgo

Sensor Motion Planning with Design Criteria in Output Space

Abstract

Motion planning of pointwise scanning sensors while estimating unknown parameters in models described by partial differential equations is addressed. In contrast to the common approach based on parameter-space criteria, close attention is paid to criteria in output space, which are of interest if the purpose of parameter estimation is to accurately predict system outputs. Some performance indices are proposed to quantify the prediction accuracy. Then the approach is to convert the problem to a state-constrained optimal-control one in which both the control forces of the sensors and the initial sensor positions are optimized. A method of successive linearizations is then employed for numerical solution.

D. Uciński

Quality Improvement of Signal-Dependent Systems

Abstract

This paper presents a model-based approach to quality improvement of signal-dependent systems. Mean and variance models for such systems are given, including the case of performance characteristic’s variance depending on time. It is shown that the additional information obtained from measurable external variables, called signals, can improve process quality considerably.

I. N. Vuchkov, L. N. Boyadjieva

Recursive Algorithm for Digital Diffusion Networks and Applications to Image Processing

Abstract

This work aims to develop a class of recursive algorithms for a digital diffusion network, which is an approximation of an analog diffusion machine. Our main effort is to prove the convergence of a continuous-time interpolation of the discrete-time algorithm to that of the analog diffusion network via weak convergence methods. The parallel processing feature of the network makes it attractive for solving large-scale optimization problems, and applicable not only to image estimation problems but also to optimal design, optimal control, and related fields. As an application, we consider image estimation problems and present simulation results.

G. Yin, P. A. Kelly, M. H. Dowell

Backmatter