Partially replicated two‐level fractional factorial designs

Liao, Chen‐Tuo; Chai, Feng-Shun

doi:10.2307/3316025

Cited by 10 publications

(7 citation statements)

References 19 publications

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“…Following Dykstra (1959), several researchers have considered partially replicated two‐level designs, although optimality is not considered. More recently, Liao and Chai (2004) and Dasgupta and Jacroux (2010) evaluated their partially replicated designs by using the standard D ‐criterion. However, in all these cases, the choice of the number of pure error degrees of freedom is made informally and separately from optimality considerations.…”

Section: Design Criteria For Inferential Proceduresmentioning

confidence: 99%

Optimum Design of Experiments for Statistical Inference

Gilmour

Trinca

2012

Journal of the Royal Statistical Society Series C: Applied Statistics

113

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One attractive feature of optimum design criteria, such as D-and A-optimality, is that they are directly related to statistically interpretable properties of the designs that are obtained, such as minimizing the volume of a joint confidence region for the parameters. However, the assumed relationships with inferential procedures are valid only if the variance of experimental units is assumed to be known. If the variance is estimated, then the properties of the inferences depend also on the number of degrees of freedom that are available for estimating the error variance. Modified optimality criteria are defined, which correctly reflect the utility of designs with respect to some common types of inference. For fractional factorial and response surface experiments, the designs that are obtained are quite different from those which are optimal under the standard criteria, with many more replicate points required to estimate error. The optimality of these designs assumes that inference is the only purpose of running the experiment, but in practice interpretation of the point estimates of parameters and checking for lack of fit of the treatment model assumed are also usually important. Thus, a compromise between the new criteria and others is likely to be more relevant to many practical situations. Compound criteria are developed, which take account of multiple objectives, and are applied to fractional factorial and response surface experiments. The resulting designs are more similar to standard designs but still have sufficient residual degrees of freedom to allow effective inferences to be carried out. The new procedures developed are applied to three experiments from the food industry to see how the designs used could have been improved and to several illustrative examples. The design optimization is implemented through a simple exchange algorithm.We shall refer to this as the DP.α/ criterion. In this paper, we shall use α = 0:05 for illustration and refer to the criterion simply as DP, but the required confidence level should be considered carefully for each experiment. Despite the above quotation, Kiefer (1959) did not suggest this additional step, since he did not separate lack of fit from pure error.Similarly, D S -optimality is intended to minimize the volume of a joint confidence region for a subset of p 2 of the parameters by minimizing |.M −1 / 22 |, where M = X X and .M −1 / 22 is the portion of its inverse corresponding to the subset of the parameters of interest. To take account of pure error estimation correctly, the .DP/ S criterion is to minimizeThis criterion should be used, for example, when a major objective of the experiment is to compare the first-order model with the second-order model. Then the higher order terms will form the subset and minimizing the volume of a confidence region for them will be equivalent to maximizing the power of a test for their existence. Note that if the parameters of interest are the treatment parameters and the nuisance parameter(s) is or are the intercept or the int...

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Section: Design Criteria For Inferential Proceduresmentioning

confidence: 99%

Optimum Design of Experiments for Statistical Inference

Gilmour

Trinca

2012

Journal of the Royal Statistical Society Series C: Applied Statistics

113

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“…Thus an f -PFD is determined by the pair of matrixes (A, C). Specifically, if there are exactly two identical c i among the f coset indicator vectors, these parallel-flats designs with partial replication are designated f -PFDRs by Liao and Chai (2004). Thus an f -PFDR contains f flats, one of which is replicated twice.…”

Section: Parallel-flats Designs With Partial Replicationmentioning

confidence: 99%

“…Some scattered results have been given by Dykstra (1959), Snee (1985), and Pigeon and McAllister (1989). Most recently, Liao and Chai (2004) recognized the partially replicated two-level factorial design presented by Pigeon and McAllister (1989) as one composed of three different fractions, belonging to a family of regular 2 n−k designs with the same defining relations, and a duplicate of one of the three fractions. In other words, it is a parallel-flats design (PFD) with two identical flats.…”

Section: Introductionmentioning

confidence: 99%

Design and Analysis of Two-Level Factorial Experiments With Partial Replication

Liao

Chai

2009

Technometrics

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In a two-level factorial experiment, we consider construction of parallel-flats designs with two identical parallel flats that allow estimation of a set of specified possibly active effects and the pure error variance. A set of sufficient conditions is presented for the designs to be D-optimal for the specified effects, assuming that the other effects are negligible, over the class of competing parallel-flats designs. In addition, an algorithm is developed to generate the D-optimal designs with a choice of flexible degrees of freedom for the pure error variance. Because the proposed partially replicated designs are highly efficient in estimating the possibly active effects and provide a replication-based estimate of the error variance, they provide a practical compromise between the power in identifying truly active effects and the number of runs in experiments. This property is verified through a simulation study.

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“…Therefore, an essential consideration is to request the augmented design as efficient as possible in estimating all the main effects with respect to a variance‐based optimality criterion. The reader is referred to Liao & Chai (), Butler & Ramos (), Liao & Chai (), Dasgupta, Jacroux & SahaRay (), Chatzopoulos, Kolyva‐Machera & Chatterjee () and Tsai, Liao & Chai () for this design issue. Specifically, Liao & Chai () explored the partially replicated designs in a class of parallel‐flats designs, such that they are D‐optimal for a set of user‐specified possibly active effects.…”

Section: Introductionmentioning

confidence: 99%

Selection of partial replication on two‐level orthogonal arrays

Tsai

Liao

2014

Can J Statistics

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Due to the lack of a replication‐based estimate for the experimental error variance, unreplicated orthogonal main‐effect plans (OMEPs) usually appear to be unsatisfactory in screening active factors, when there are certain interactions likely to be non‐negligible. In this study, partially replicated designs constructed through two‐level orthogonal arrays are recommended to compensate for this deficiency. A compound criterion called the extended minimum aberration (EMA) is proposed for choosing the twice‐replicated fractions. According to the EMA criterion, sufficient conditions for constructing the minimal two‐level OMEPs with partial replication are proposed, and a series of EMA designs with practical run‐sizes is provided for practical applications. The Canadian Journal of Statistics 42: 168–183; 2014 © 2014 Statistical Society of Canada

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Partially replicated two‐level fractional factorial designs

Cited by 10 publications

References 19 publications

Optimum Design of Experiments for Statistical Inference

Optimum Design of Experiments for Statistical Inference

Design and Analysis of Two-Level Factorial Experiments With Partial Replication

Selection of partial replication on two‐level orthogonal arrays

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