Inference for the Optimum Using Linear Regression Models with Discrete Inputs

We present a multiple-comparison-with-the-best procedure to provide inference for the optimum from regression models with discrete inputs. Two applications are given to illustrate the methodology: two-level factorial designs to identify the best drug combination and order-of-addition experiments where the primary objective is to identify the sequence with the largest mean response. The methods easily accommodate restrictions limiting the inference set of conditions. We use simulation to determine the critical values. While the methods apply to any linear regression model, we identify cases that require just a single critical value, and we also show where approximations and upper bounds mitigate the need for intensive computation. We tabulate the required critical values for a variety of common applications: the main-effect model and two-factor interaction model estimated by certain two-level factorial designs, and the pairwise order model and several component-position models for estimation based on optimal order-of-addition designs. Our work greatly simplifies the problem of rigorous inference for the optimum from regression models with discrete inputs.</p

Structure of Nonregular Two-Level Designs

Two-level fractional factorial designs are often used in screening scenarios to identify active factors. This article investigates the block diagonal structure of the information matrix of nonregular two-level designs. This structure is appealing since estimates of parameters belonging to different diagonal submatrices are uncorrelated. As such, the covariance matrix of the least squares estimates is simplified and the number of linear dependencies is reduced. We connect the block diagonal information matrix to the parallel flats design (PFD) literature and gain insights into the structure of what is estimable and/or aliased using the concept of minimal dependent sets. We show how to determine the number of parallel flats for any given design, and how to construct a design with a specified number of parallel flats. The usefulness of our construction method is illustrated by producing designs for estimation of the two-factor interaction model with three or more parallel flats. We also provide a fuller understanding of recently proposed group orthogonal supersaturated designs. Benefits of PFDs for analysis, including bias containment, are also discussed.</p

Using Individual Factor Information in Fractional Factorial Designs

<p>While literature on constructing efficient experimental designs has been plentiful, how best to incorporate prior information when assigning factors to the columns of a nonregular design has received little attention. Following Li, Zhou, and Zhang (<a href="#cit0009" target="_blank">2015</a>), we propose the individual generalized word length pattern (iGWLP) for ranking columns of a nonregular design. Taking examples from the literature of recommended orthogonal arrays, we illustrate how iGWLP helps to identify important differences in the aliasing that is likely otherwise missed. Given the complexity of characterizing partial aliasing for nonregular designs, iGWLP will help practitioners make more informed assignment of factors to columns when using nonregular fractions. We provide theoretical justification of the proposed iGWLP. A theorem is given to relate the proposed iGWLP criterion to the expected bias caused by model misspecifications. We also show that the proposed criterion may lead to designs having better projection properties in the factors considered most likely to be important. Furthermore, we discuss how iGWLP can be used for design selection. We propose a criterion for choosing best designs when the focus is on a small set of important factors, for which the aliasing of effects involving these factors is minimized.</p

Augmenting definitive screening designs: Going outside the box

Definitive screening designs (DSDs) have grown rapidly in popularity since their introduction by Jones and Nachtsheim (2011). Their appeal is that the second-order response surface (RS) model can be estimated in any subset of three factors, without having to perform a follow-up experiment. However, their usefulness as a one-step RS modeling strategy depends heavily on the sparsity of second-order effects and the dominance of first-order terms over pure quadratic terms. To address these limitations, we show how viewing a projection of the design region as spherical and augmenting the DSD with axial points in factors found to involve second-order effects remedies the deficiencies of a stand-alone DSD. We show that augmentation with a second design consisting of axial points is often the Ds-optimal augmentation, as well as minimizing the average prediction variance. Supplemented by this strategy, DSDs are highly effective initial screening designs that support estimation of the second-order RS model in three or four factors.</p