Feng-Shun Chai

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

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.

Statistical designs for two-color microarray experiments involving technical replication

In a two-color microarray experiment, we consider the issues of determination of which mRNA samples are to be labeled with which fluorescent dye and which mRNA samples are to be hybridized together on the same slide. Specific attention is given to the test-control experiments whose primary interest lies in comparing several test treatments with a control treatment. A statistical linear model is proposed to characterize two major sources of systematic variation: the variation among distinct slides and that between fluorescent dyes. Furthermore, the possible correlation due to technical replication is also incorporated into the model. A series of A-optimal or highly efficient designs are generated from a heuristic algorithm based on the proposed model. It is shown that the obtained designs are robust not only to the variation of the correlation because of technique replication, but also to the loss of one or two slides. In addition, the comparative experiments involving technical replication are also discussed.

Further results on orthogonal array plus one run plans

This paper considers the issue of optimality of orthogonal array plus one run plans under generalized criteria of type 1 which include the D-, A-, and E-criteria. Our results, in conjunction with those in Mukerjee (Ann. Statist.) cover all orthogonal arrays of strength two and involving up to 100 rows, except perhaps those having 72 rows, and thus almost completely settle the problem for resolution III plans over a practical range.

A-optimal block designs for parallel line assays

Abstract Consider a parallel line assay, in which m 1 given doses of standard preparation and m 2 given doses of test preparation are used. For such an assay, an alternative definition of contrasts of major importance is given. A-optimal connected block designs for estimating preparation and combined regression contrasts are obtained for symmetric and asymmetric assays. Tables of A-optimal designs are provided.

Optimal designs for nearest-neighbor analysis

For field trials of v varieties in blocks of size k, in the presence of a random trend, data may be analyzed by the nearest-neighbor method based on first differences. The variance components model for this analysis is the linear variance (LV) model. Optimal designs for the case k⩽v were given by Martin et al. (J. Statist Plann. Inference 34 (1993) 433–450) and the case v<k⩽2v by Martin (1998). In this paper, for the general case of any k and v we show that a universally optimal design can be constructed from a semibalanced array if the varieties within blocks are arranged in a particular order. The order does not depend on the magnitudes of the variance components.

Construction and optimality of nearly linear trend-free designs

Abstract The necessary condition that a proper block design can be converted to a (linear) trend-free design is r(k + 1) ≡ 0 (mod 2) (Yeh and Bradley, Comm. Statist. Theory Methods 12 (1983) 1–24). When this condition does not hold, the best we can do is try to convert the design to a nearly (linear) trend-free design (Yeh et al., J. Amer. Statist. Assoc. 80 (1985) 985–992). Our results include: 1. (i) construction of the nearly trend-free version of BIB designs; 2. (ii) A-, D- and E-optimality of BIB designs for models that include trend effects.

Generalized E(s 2) Criterion for Multilevel Supersaturated Designs

Several extensions of the popular E(s 2) criterion of Booth and Cox (1962) to multilevel supersaturated designs have been advanced in literature. These extensions are not unique due to different ways they measure overall nonorthogonality between all pairs of the columns of the model matrix. We exploit the connection of the E(s 2) criterion with A- and D-optimality that naturally lends itself to a generalized criterion for the multilevel situation in a unified way. The extensions provided in literature follow as special cases of the generalized criterion. A lower bound to the generalized criterion is derived for a wide class of designs, and a method of construction for the symmetrical case is discussed.

Three Parallel Flats Designs for Two‐level Factorial Experiments

This paper investigates the properties of the class of three parallel flats designs for two-level factorial experiments. It shows that the designs constructed from this class of designs can have a very simple correlation structure. The correlation of any pair of best linear unbiased estimators of factorial effects is 0, ⅓ or ¼. Furthermore, the designs obtained also have high D-efficiency. Finally, a class of designs is generated with run-size N= 12 to illustrate the use of the theorem.

A note on universally optimal row-column designs with empty nodes

Identification of universally optimal row-column designs is investigated. This paper shows that Kunerts (1993) examples of universally optimal generalized non-binary designs are not special cases. One can construct an universally optimal generalized non-binary design by use of a binary one.

Optimal two-level choice designs for estimating main and specified two-factor interaction effects

Under the multinomial logit model, designs for choice experiments are usually based on an a priori assumption that either only the main effects of the factors or the main effects and all two-factor interaction effects are to be estimated. However, in practice, there are situations where interest lies in the estimation of main plus some two-factor interaction effects. For example, interest on such specified two-factor interaction effects arise in situations when one or two factor(s) like price and/or brand of a product interact individually with the other factors of the product. For two-level choice experiments with n factors, we consider a model involving the main plus all two-factor interaction effects, with our interest lying in the estimation of the main effects and a specified set of two-factor interaction effects. The two-factor interaction effects of interest are either (i) one factor interacting with each of the remaining n − 1 factors or (ii) each of the two factors interacting with each of the remaining n − 2 factors. For the two models, we first characterize the information matrix and then construct universally optimal choice designs for choice set sizes 3 and 4.