Optimal and efficient repeated measurements designs for comparing test treatments with a control

Rearranging the sequence of test treatments and control treatment within each block of an A-optimal balanced treatment block design such that certain conditions are satisfied, the resulting design is an A-optimal repeated measurements designs when blocks are regarded as units or periods. The efficiencies of designs which are obtained from universally optimal repeated measurements designs with test treatments only by changing some treatment labels into control treatment are given.     

Universally optimal designs under mixed interference models with and without block effects

The literature on neighbor designs as introduced by Rees (Biometrics 23:779–791, 1967) is mainly devoted to construction methods, providing few results on their statistical properties, such as efficiency and optimality. A review of the available literature, with special emphasis on the optimality of neighbor designs under various fixed effects interference models, is given in Filipiak and Markiewicz (Commun Stat Theory Methods 46:1127–1143, 2017). The aim of this paper is to verify whether the designs presented by Filipiak and Markiewicz (2017) as universally optimal under fixed interference models are still universally optimal under models with random interference effects. Moreover, it is shown that for a specified covariance matrix of random interference effects, a universally optimal design under mixed interference models with block effects is universally optimal over a wider class of designs. In this paper the method presented by Filipiak and Markiewicz (Metrika 65:369–386, 2007) is extended and then applied to mixed interference models without or with block effects.     

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.     

Optimal block designs for diallel crosses

Dey and Midha (Biometrika 83(2):484–489, 1996) constructed optimal block designs for complete diallel cross experiment using triangular partially balanced incomplete block designs with two associate classes. They listed optimal block designs for the lines in the range from 5 ≤ v ≤ 10. In this paper, we are also proposing additional optimal block designs for complete diallel cross experiment using two associate class partially balanced block designs for some additional values of v. Our method yields designs for proper and non-proper settings for complete diallel cross experiments. The proper and non proper designs are optimal in the sense of Kempthorne (Genetics 41:451–459, 1956) and non-proper designs are universally optimal in the sense of Kiefer (A survey of statistical design and linear models, North Holland, Amsterdam, 1975). The list of practically important designs is also given.     

On a property of orthogonal arrays and optimal blocking of fractional factorial plans

It is shown that fractional factorial plans represented by orthogonal arrays of strength three are universally optimal under a model that includes the mean, all main effects and all two-factor interactions between a specified factor and each of the other factors. Thus, such plans exhibit a kind of model robustness in being universally optimal under two different models. Procedures for obtaining universally optimal block designs for fractional factorial plans represented by orthogonal arrays are also discussed. Acknowledgements. The authors wish to thank the referees for making several useful comments on a previous version.     

The Optimality of Single-group Designs for Certain Mixed Models

In this paper optimal designs for the estimation of the fixed effects (population parameters) in a certain class of mixed models are investigated. Two classes of designs are compared: the class of single-group designs, where all individuals are observed under the same approximate design, and the class of more-group designs with the same mean number of observations per individual as before, where each individual can be observed under a different approximate design. It is shown that any design that is Φ-optimal in the class of single-group designs is also Φ-optimal in the larger class of more-group designs. The considered optimality criteria only have to satisfy mild assumptions, which is eg the case for the D-criterion and all linear criteria.     

Robustness of Williams Square change-over designs

In literature, Williams Square change-over designs balanced for first residual effects of treatments have been proved to be universally optimal. Here, an attempt has been made to show that these designs are robust against missing of last α [≤v−1; v being the number of periods in the design for v treatments] observations from an experimental unit. Received May 2000     

D‐optimal designs for a continuous predictor in longitudinal trials with discrete‐time survival endpoints

In designing an experiment with one single, continuous predictor, the questions are composed of what is the optimal number of the predictor's values, what are these values, and how many subjects should be assigned to each of these values. In this study, locally D‐optimal designs for such experiments with discrete‐time event occurrence data are studied by using a sequential construction algorithm. Using the Weibull survival function for modeling the underlying time to event function, it is shown that the optimal designs for a linear effect of the predictor have two points that coincide with the design region's boundaries, but the design weights highly depend on the predictor effect size and its direction, the survival pattern, and the number of time points. For a quadratic effect of the predictor, three or four design points are needed.     

Generalized Wordtype Pattern for Nonregular Factorial Designs with Multiple Groups of Factors

This paper introduces the generalized wordtype pattern of a nonregular fractional factorial design and considers its connection with the distance distribution. Based on the corresponding relationship between a fractional factorial design and a code, we develop a consulting design theory for fractional factorial designs with two groups of factors. It works for regular and nonregular designs and covers the previous results as special cases. As an illustration, it is further applied to the selection of optimal two-level single arrays derived from Hadamard Matrices.     

Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments

This article provides an overview of the recent literature on the design of blocked and split-plot experiments with quantitative experimental variables. A detailed literature study introduces the ongoing debate between an optimal design approach to constructing blocked and split-plot designs and approaches where the equivalence of ordinary least squares and generalized least squares estimates are envisaged. Examples where the competing design strategies lead to totally different designs are given, as well as examples in which the optimal experimental designs are orthogonally blocked or equivalent-estimation split-plot designs.     

Optimal designs for a mixed interference model

This paper generalizes Kunert and Martin’s (Ann Stat 28:1728–1742, 2000) method for finding optimal designs under a fixed interference model, to find optimal designs under a mixed interference model. The results are based on the properties of information matrices in fixed and mixed models given in Markiewicz (J Stat Plan Inference 59:127–137, 1997). The method is applied to find a design which is optimal for any given variances of random neighbor effects. Research partially supported by the KBN Grant Number 5 P03A 041 21.     

A typology of mixed methods research designs

The mixed methods paradigm is still in its adolescence, and, thus, is still relatively unknown and confusing to many researchers. In general, mixed methods research represents research that involves collecting, analyzing, and interpreting quantitative and qualitative data in a single study or in a series of studies that investigate the same underlying phenomenon. Over the last several years, a plethora of research designs have been developed. However, the number of designs that currently prevail leaves the doctoral student, the beginning researcher, and even the experienced researcher who is new to the field of mixed methods research with the challenge of selecting optimal mixed methods designs. This paper presents a three-dimensional typology of mixed methods designs that represents an attempt to rise to the challenge of creating an integrated typology of mixed methods designs. An example for each design is included as well as a notation system that fits our eight-design framework. This paper won the James E. McLean outstanding paper award.     

Minimax robust designs for field experiments

Experimental designs for field experiments are useful in planning agricultural experiments, environmental studies, etc. Optimal designs depend on the spatial correlation structures of field plots. Without knowing the correlation structures exactly in practice, we can study robust designs. Various neighborhoods of covariance matrices are introduced and discussed. Minimax robust design criteria are proposed, and useful results are derived. The generalized least squares estimator is often more efficient than the least squares estimator if the spatial correlation structure belongs to a small neighborhood of a covariance matrix. Examples are given to compare robust designs with optimal designs. The work was partially supported by research grants from the Natural Science and Engineering Research Council of Canada.     

When product type experimental design is optimal? Brief survey and new results

In the paper the problem of optimum experimental design for estimating parameters of multivariate regression functions is considered. We address the question: under what conditions one can compose the optimal design from partial designs, obtained by considering partial regressions, which depend on reduced number of variables. After reinterpreting and reviewing briefly existing results we provide some new conditions.     

Optimal Designs for Asymmetric Linear Paired Comparisons with a Profile Strength Constraint

Within the linear model framework the problem of determining optimal designs for paired comparisons of alternatives which are described by a set of discrete attributes is considered under the constraint that the alternatives in a pair are only allowed to differ with regard to a certain number of attributes. Whereas in previous treatments of this problem it was assumed that all attributes possess the same number of levels, here the general asymmetric case is discussed. We provide a characterization of optimal designs and demonstrate how this can be used to derive a solution of the design problem for many situations of interest.     

Selecting Adaptive Survey Design Strata with Partial R‐indicators

Recent survey literature shows an increasing interest in survey designs that adapt data collection to characteristics of the survey target population. Given a specified quality objective function, the designs attempt to find an optimal balance between quality and costs. Finding the optimal balance may not be straightforward as corresponding optimisation problems are often highly non‐linear and non‐convex. In this paper, we discuss how to choose strata in such designs and how to allocate these strata in a sequential design with two phases. We use partial R‐indicators to build profiles of the data units where more or less attention is required in the data collection. In allocating cases, we look at two extremes: surveys that are run only once, or infrequent, and surveys that are run continuously. We demonstrate the impact of the sample size in a simulation study and provide an application to a real survey, the Dutch Crime Victimisation Survey.     

Optimality and efficiency of incomplete split-block designs

In this paper, a balanced incomplete split-block design (BISBD) is shown to be universally optimum for the estimation of the interaction effects with respect to each of the intra-plot stratum estimation and the GLSE. Furthermore, the efficiency factor of an optimal design is investigated.     

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.     

Maximal Rank Minimum Aberration Foldover Plans for 2<Superscript><Emphasis Type="Italic">m</Emphasis>-<Emphasis Type="Italic">k</Emphasis></Superscript> Fractional Factorial Designs

Two level regular fractional factorial designs are often used in industry as screening designs to help identify early on in an experimental process those experimental or system variables which have significant effects on the process being studied. In a recent paper, Li and Lin (2003) suggested a strategy for constructing optimal follow up designs using the well known foldover technique and the minimum aberration criterion. In this paper, we extend the results of Li and Lin (2003) by giving an alternate technique for constructing optimal follow up designs using the foldover technique in conjunction with the maximal rank–minimum aberration criterion suggested in Jacroux (2003).     

Optimal Adaptive Designs with Inverse Ordinary Differential Equations

Many industrial and engineering applications are built on the basis of differential equations. In some cases, parameters of these equations are not known and are estimated from measurements leading to an inverse problem. Unlike many other papers, we suggest to construct new designs in the adaptive fashion ‘on the go’ using the A‐optimality criterion. This approach is demonstrated on determination of optimal locations of measurements and temperature sensors in several engineering applications: (1) determination of the optimal location to measure the height of a hanging wire in order to estimate the sagging parameter with minimum variance (toy example), (2) adaptive determination of optimal locations of temperature sensors in a one‐dimensional inverse heat transfer problem and (3) adaptive design in the framework of a one‐dimensional diffusion problem when the solution is found numerically using the finite difference approach. In all these problems, statistical criteria for parameter identification and optimal design of experiments are applied. Statistical simulations confirm that estimates derived from the adaptive optimal design converge to the true parameter values with minimum sum of variances when the number of measurements increases. We deliberately chose technically uncomplicated industrial problems to transparently introduce principal ideas of statistical adaptive design.