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

Abstract

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.

Selection of partial replication on two‐level orthogonal arrays

“…A design is said to be D‐optimal for over a specific class of designs, if its information matrix achieves the maximal determinant among all the competing designs. Note that the D‐optimality criterion is also used by Liao & Chai () and Tsai, Liao & Chai () for selecting partially replicated design points with respect to a user‐specified , which consists of not only the constant term and all the main effects, but also some two‐factor or higher order interactions. Under the assumption that the full‐effect model is the true model, the expectation of , the best linear unbiased estimator (BLUE) for , is given by where is the alias or bias matrix.…”

“…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.…”

Two-level Orthogonal and Saturated Designs for Estimating Main Effects and Certain Important Interactions

“…Interestingly, their simulation study showed that the 48-run designs are plentiful in which each two-factor interaction is forced to include one or two of a limited number of specific factors. More recently, Liao and Chai (2009) investigated PFDs with repeated runs for estimating a user-specified requirement set of factorial effects. Also, Tang and Zhou (2009) considered two-level orthogonal arrays that allow estimation of the grand mean, all main effects and certain important two-factor interactions.…”

D-optimal partially replicated two-level factorial designs

“…Therefore, a practical compromise is to carry out a partial replication. Partially replicated designs usually work well regardless of the effect sparsity, see Liao and Chai (2009). Dykstra (1959) proposed some high-resolution designs including repeated runs.…”

“…Liao and Chai (2004) first investigated the parallel-flats designs with a replicated flat. Most recently, Liao and Chai (2009) proposed a set of sufficient conditions and an algorithm for constructing D-optimal designs over the class of parallel-flats designs. However, there are still some limitations in their study.…”