General weighted optimality of designed experiments

Abstract

I would also like to extend my gratitude to Drs. Xinwei Deng, Klaus Hinkelmann, and Brad Jones, who were gracious enough to serve on my committee. Their advice and suggestions are greatly appreciated and I hope to work with them again in the future. Finally, I want to thank my family, especially my father, Wade, brother, Robert, and sister, Amy, and beloved partner, Sarah, who provided constant support during many personal challenges, including my mother's unexpected passing. I would like to dedicate this dissertation to my mother, Delores Tiller Stallings, who would have been immensely proud to see it completed. iii List of Tables 3.1 Weights assigned by W Hw in Example 3.2.1 to pairwise comparisons for treatments between and within treatment groups V 1 and V 2 with w 0 = 20.. .. . 3.2 Replication proportions for treatments of a given heft for A-optimal approximate CRD with respect to estimation of all non-normalized BP effects.. .. 3.3 A-optimal CRDs for estimation of all non-normalized BP effects for m = 3 factors. The last row indicates whether the design is a BA of strength 3.. . 3.4 A-optimal CRDs for estimation of all normalized BP effects for m = 3 factors. The last row indicates whether the design is a BA of strength 3.. .. .. . .

Equivalence of weighted and partial optimality of experimental designs

“…Two weight matrices W 1 and W 2 are said to be estimation equivalent if q T W −1 1 q = cq T W −1 2 q for all q ∈ E and for some constant c. Lemma 3 by Stallings and Morgan [2015] shows that W 1 and W 2 are estimation equivalent if and only if P τ W −1…”

“…When the objective of the experiment is to estimate a set of s normalized functions Q T τ , where s is equal to the dimension of E and r = s, Stallings and Morgan [2015] propose a corresponding weight matrix W Q = I − P τ + QQ T , where P τ is the orthogonal projector on E. The inverse of such a matrix is W −1 Q = I − P τ + (QQ T ) + , and W Q places weight 1 on each of the functions of interest q T i τ , i.e., (q T i W −1 Q q i ) −1 = 1 for i = 1, . .…”

“…However, Stallings and Morgan [2015] found the approach of Morgan and Wang too restrictive, and therefore extended the theory of the weighted optimality. They propose a weighted information matrix for any, generally non-diagonal, positive definite weight matrix.…”

“…In this paper, we build upon the theory presented by Stallings and Morgan [2015]. In Section 2, we compare the weighted optimality and the 'standard' optimality for estimating a system of interest.…”

“…We propose to relax the definition of the weight matrix so that a weight matrix can be constructed for any system of estimable functions. Moreover, we define a weight matrix corresponding to a system Q T τ , which is of a slightly simpler form than that by Stallings and Morgan [2015], although these two forms are equivalent with respect to the implied weighting of estimable functions. Finally, we note that it is advisable to differentiate between primary weights specified by the experimenters and secondary weights given by the weight matrix that corresponds to the system of interest Q T τ .…”