On the Determination and Construction of Optimal Block Designs in the Presence of Linear Trends

Jacroux, Mike; Majumdar, Dibyen; Shah, Kirti R.

doi:10.1080/01621459.1997.10473636

Cited by 16 publications

(12 citation statements)

References 14 publications

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“…As k À 1 is the highest possible degree to which a block design with block size k can be orthogonal, a design that is orthogonal to a (k À 1Þth order time trend will simply be referred to as a trend-free design. Jacroux et al (1995Jacroux et al ( , 1997 and Jacroux (1998) tackle the problem of identifying and constructing block designs for comparing several treatments in the presence of possibly different linear trends within the blocks. However, it is known that for certain combinations of the design parameters trend-free block designs do not exist.…”

Section: Time Trend Effectsmentioning

confidence: 99%

The Optimal Design of Blocked Experiments in Industry

Goos¹,

Tack²,

Vandebroek³

2005

Applied Optimal Designs

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Section: Time Trend Effectsmentioning

confidence: 99%

The Optimal Design of Blocked Experiments in Industry

Goos¹,

Tack²,

Vandebroek³

2005

Applied Optimal Designs

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“…The research focus is usually on combinatorial construction of exact designs orthogonal to time trend (or trend free). These are designs that satisfy that no information is lost due to the time trend (see, e.g., Jacroux et al [1997], Bailey et al [1992]).…”

Section: Trend Resistant Designsmentioning

confidence: 99%

“…For example, any experiment that consists of multiple trials performed in a time sequence may be subject to a nuisance time trend caused by the ageing of the material used for the experiment, wearing down of the experimental devices, changes in the temperature, etc. Many agricultural experiments are subject to a two-dimensional nuisance trend, resulting from the arrangement of the trials in a two-dimensional field, see, e.g., Jacroux et al [1997] and Bailey and Williams [2007]. The objective of the experimental design in such cases is to eliminate the nuisance effects, or to provide as much information as possible on the effects of interest.…”

Section: Introductionmentioning

confidence: 99%

“…There is a large amount of literature on exact designs in such models, especially on block designs (e.g., Majumdar and Notz [1983], Majumdar [1996], Jacroux [2002]), but also on trend resistant designs (e.g., Cox [1951], Cheng [1990], Atkinson and Donev [1996]) or block designs in the presence of a trend (e.g., Bradley and Yeh [1980], Jacroux et al [1997]). However, these results are usually tailored for a particular model and a system of contrasts, often with limiting assumptions (e.g., the relationship between the number of blocks and treatments, or trends modelled by polynomials of given degrees).…”

Section: Introductionmentioning

confidence: 99%

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Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects

Rosa

Harman

2016

Stat Papers

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Suppose that we intend to perform an experiment consisting of a set of independent trials.The mean value of the response of each trial is assumed to be equal to the sum of the effect of the treatment selected for the trial, and some nuisance effects, e.g., the effect of a time trend, or blocking. In this model, we examine optimal approximate designs for the estimation of a system of treatment contrasts, with respect to a wide range of optimality criteria.We show that it is necessary for any optimal design to attain the optimal treatment proportions, which may be obtained from the marginal model that excludes the nuisance effects. Moreover, we prove that for a design to be optimal, it is sufficient that it attains the optimal treatment proportions and satisfies conditions of resistance to nuisance effects.For selected natural choices of treatment contrasts and optimality criteria, we calculate the optimal treatment proportions and give an explicit form of optimal designs. In particular, we obtain optimal treatment proportions for comparison of a set of new treatments with a set of controls. The results allow us to construct a method of calculating optimal approximate designs with a small support by means of linear programming. As a consequence, we can construct efficient exact designs by a simple heuristic. 1We show that a Φ-optimal approximate design may be obtained in two steps: (i) Calculate Φ-optimal proportions of treatment replications (treatment weights). These optimal proportions depend on the choice of contrasts of interest and on the optimality criterion Φ; however, they do not depend on the nuisance effects. (ii) Subject to keeping the optimal proportions of treatment replications, distribute the treatments to nuisance conditions such that the resulting design is resistant to nuisance effects. The designs resistant to nuisance effects are an extension of the designs orthogonal to the time trend (balanced for trend, or trend-free, cf. Cox [1951], Jacroux and Ray [1990]) to a more general class of models and treatment contrasts.The approach of first finding a design in a simpler model and then assuring that the information is retained in a finer model was used, e.g., in Schwabe [1996] and Kunert [1983].Schwabe [1996] studied optimal product designs, unlike the present paper, where optimal designs with non-product structure are provided too; Kunert [1983] studied exact designs in the case of universal optimality. Universal optimality, formulated by Kiefer [1975], means optimality for estimating a maximal system of orthonormal contrasts (which is a special case of the general system of contrasts that we consider), with respect to a wide range of criteria.For selected systems of treatment contrasts and a wide class of optimality criteria Φ, we calculate Φ-optimal treatment weights and thus obtain a class of Φ-optimal designs. For instance, for the estimation of contrasts for comparing a set of new treatments with a set of controls, we provide M V -optimal designs and optimal designs with respect to Kiefer'...

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“…In the experiments performed to estimate the effects of a selected set of treatments, it is common that the responses are also affected by some other experimental conditions (the covariates -e.g., time trend, block effects, the ages or the gender of the subjects in a clinical trial). The effects of the covariates are traditionally considered to be nuisance effects in the experimental design literature (e.g., Cox [1951], Majumdar and Notz [1983], Atkinson and Donev [1996], Jacroux et al [1997], Rosa and Harman [2016]). Recently, Atkinson [2015] suggested that particularly with the growing importance of personalized medicine, the covariate effects may also be of prominent interest.…”

Section: Introductionmentioning

confidence: 99%

Optimal experimental designs for treatment contrasts in heteroscedastic models with covariates

Rosa

2020

Journal of Statistical Planning and Inference

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In clinical trials, the response of a given subject often depends on the selected treatment as well as on some covariates. We study optimal approximate designs of experiments in the models with treatment and covariate effects. We allow for the variances of the responses to depend on the chosen treatments, which introduces heteroscedasticity into the models. For estimating systems of treatment contrasts and linear functions of the covariates, we extend known results on D-optimality of product designs by providing product designs that are optimal with respect to general eigenvalue-based criteria. In particular, A-and E-optimal product designs are obtained. We then formulate a method based on linear programming for constructing optimal designs with smaller supports from the optimal product designs. The sparser designs can be more easily converted to practically applicable exact designs. The provided results and the proposed sparsification method are demonstrated on some examples.

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On the Determination and Construction of Optimal Block Designs in the Presence of Linear Trends

Cited by 16 publications

References 14 publications

The Optimal Design of Blocked Experiments in Industry

The Optimal Design of Blocked Experiments in Industry

Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects

Optimal experimental designs for treatment contrasts in heteroscedastic models with covariates

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