D-optimal 2 × 2 × s 3 × s 4 saturated factorial designs

In this paper, Resolution III saturated s 1 × s 2 × s 3 × s 4 , s 4 ≥ s 3 ≥ s 2 ≥ s 1 ≥ 2 factorial designs and specially the cases 2 × (s − k) × s, s − k ≥ 2, k = 0, 1 are studied, in order to obtain D-optimal plans. Subjects: Science; Mathematics & Statistics; Statistics & Probability; Statistics; Mathematical Statistics; Statistical Computing; Statistical Theory & Methods


Introduction
Saturated factorial plans is a very interesting issue in theory of exeprimental designs, since the reduced number of observations is very usefull in practise especially in screening experiments, where are used to determine which of many factors affects the measure of pertinent quality characteristics. In saturated designs the number of observation is equal to the number of parameters, so all degrees of freedom are consumed by the estimation of parameters, leaving no degrees of freedom for error variance estimation. The purpose of this paper is to give saturated resolution III designs, minimizing the generalized variance of the main effects and the general mean, that is, D-optimal designs. In recent years, there has been a considerable interest in optimal saturated main effect designs with two or three factors. Mukerjee et al. (1986) and Kraft (1990) showed all two-factor designs are equivalent with respect to D-optimality criterion. Later Mukerjee and Sinha (1990) ABOUT THE AUTHORS The problem of finding optimal designs under different types of criteria preoccupies many researchers the last decades. Most of the work on constructing optimal designs for the estimation of parameters in fractional factorials is concentrated on factors at two levels. Chatzopoulos, Kolyva-Machera, and Chatterjee (2009), studied the optimality of designs which are obtained by adding p runs to an orthogonal array for experiments involving m factors each at s levels. Chatterjee, Kolyva-Machera, and Chatzopoulos (2011), considered the issue of optimality of fractional factorial experiments involving m factors each at two levels. Pericleous, Chatzopoulos, Kolyva-Machera and Kounias, study the problem of estimating the standardized linear and quadratic contrasts in fractional factorials with k factors, each at 3 levels, when the number of runs or assemblies is N = 3 and introduced a different notion of Balanced Arrays. Chatzopoulos and Kolyva-Machera (2005), studied the saturated m 1 × m 2 × m 3 designs and Chatzopoulos & Kolyva-Machera (2008), considered the problem of finding D-optimal saturated 4 × m 2 × m 3 designs.

PUBLIC INTEREST STATEMENT
An issue of interesting in experimental designs is the saturated designs. An experimental design is called saturated if all the degrees of freedom are consumed by the estimation of the parameters without leaving degrees of freedom for error variance estimation. The saturated factorial designs, where the interest is to estimate the general mean and the main effects while all higher order interactions are negligible (resolution III plans), are commonly used in screening experiments. In recent years, there has been a considerable interest in optimal saturated main effect designs. Most researchers have dealt with the case where two or three factors are involved in the experiment on two levels. The problem is different and becomes more difficult when three or four factors are involved in the experiment on three or more levels.
considered, for the two-factor case, the optimality results on almost saturated main effect designs. Pesotan and Raktoe (1988) worked also in the special case for s 2 factorials and a subclass of s 3 factorials. Chatterjee and Mukerjee (1993) were the first who attempted to extend the two factor results to three factors. They consider 2 × s 2 × s 3 , s 2 ≥ 2, s 3 ≥ s 2 factorial to derive D-optimal saturated main effect designs. Later Chatterjee and Narasimhan (2002), using techniques from Graph Theory and Combinatorics, claimed about the upper bound of the determinant of the saturated 3 × s 2 × s 3 , s 2 ≥ 3, s 3 ≥ s 2 factorials when s 2 is odd. Chatzopoulos and Kolyva-Machera (2006) extend the results concerning D-optimal saturated main effect designs for 2 × s 2 × s 3 to 3 × s 2 × s 3 factorials, when 3 ≤ s 2 ≤ 6 and s 3 ≥ s 2 . Karagiannis and Moyssiadis (2005) and Karagiannis and Moyssiadis (2008) extend the Graph theoretic approach of Chatterjee and Narasimhan (2002) and the results of Chatzopoulos and Kolyva-Machera (2006), and give the D-optimal saturated 3 × s 2 × s 3 , s 2 ≥ 3, s 3 ≥ s 2 designs. In this paper, we study the D-optimality for saturated s 1 × s 2 × s 3 × s 4 factorials. Moreover, we give the upper bound of the determinant for the 2 2 × (s − k) × s, s − k ≥ 2, k = 0, 1, saturated designs and the corresponding design, which attains this bound. The paper is organized as follows. Some notations and preliminaries are first presented in Section 2. Section 3 deals with the main results of this paper.

Notations and preliminaries
In this paper, we follow the same notations as in Chatzopoulos and Kolyva-Machera (2006) adapted for four factors. Let us consider the setup of an s 1 × s 2 × s 3 × s 4 ,s 4 ≥ s 3 ≥ s 2 ≥ s 1 ≥ 2 saturated factorial experiment, involving four factors F 1 , F 2 , F 3 and F 4 appearing at s 1 , s 2 , s 3 and s 4 levels, respectively, with N = s 1 + s 2 + s 3 + s 4 − 3 runs. For 1 ≤ i ≤ 4 let the levels of F i be denoted by i and coded as 0, 1, … s i − 1 . Our interest is to find D-optimal resolution III designs. There are altogether s 1 s 2 s 3 s 4 treatment combinations denoted by 1 2 3 4 , that will hereafter be assumed to be lexicographically ordered.
Let, for 1 ≤ i ≤ 4, 1 i be the s i × 1 vector with each element unity, I i the identity matrix of order s i , ⊗ denotes the Kronecker product of matrices and P i be an (s i − 1) × s i matrix such that (s The usual fixed effect model under the absence of interactions is Y = W + , where Y is the response vector of the experiment, is the vector of uncorrelated random errors with zero mean and the same variance 2 and is the vector of unknown parameters, is consider. In our case = ( , � 1 , � 2 , � 3 , � 4 ) � , where is the unknown general mean and the elements of the (s i − 1) × 1 vectors i are unknown parameters representing a full set of mutually orthogonal contrasts belonging to the main effects It is easy to see that the D-optimal design does not depend on the choice of P i , 1 ≤ i ≤ 4.
Following Mukerjee and Sinha (1990) , which has full column rank. The u rows of matrix U like those of W, correspond to the lexicographically ordered treatment combinations. Moreover the columns of U span those of X 0 and hence those of W, which also has full column rank.
Hence, one may obtain W = UH, where matrix H is a nonsingular matrix of order s 1 + s 2 + s 3 + s 4 − 3. For any design d in the class  of the saturated resolution III designs with N = s 1 + s 2 + s 3 + s 4 − 3 runs, the design matrix is W d = U d H, where U d is a square matrix of order s 1 + s 2 + s 3 + s 4 − 3 such that for 1 ≤ j ≤ s 1 + s 2 + s 3 + s 4 − 3 if the i-th run in d is given by the treatment combination 1 2 3 4 then the j-th row of U d is the row of U corresponding to the treatment combination 1 2 3 4 . A design d is said to be D-optimal in the class , if it maximizes the quantity |det(W � d W d )|. Since matrix H is nonsingular a design is D-optimal if it maximizes the quantity |det(U d )|, where: The matrices Z (1) i , 1 ≤ i ≤ 3 and Z 4 are obtain.ed from the matrices X (1) i and X 4 in a similar way, as U d is obtained from U.
if the i-th factor enters the experiment at level 0 then the corresponding row of the matrix Z (1) i is a row vector with s i − 1 elements zero. On the other hand if the i-th factor enters the experiment at level p, 1 ≤ p ≤ (s i − 1), then the corresponding row of the matrix Z (1) i equals the p-th row of the identity matrix of order (s i − 1). Similarly, if the fourth factor enters the experiment at level p, 0 ≤ p ≤ s 4 − 1, then the corresponding row of the matrix Z 4 equals to the (p + 1)-th row of the identity matrix I s 4 . Let n , denote the number of these rows. It holds that , denote the number of runs where the i-th factor appears at level p and the j-th factor appears at level q. It holds that ijk , denote the number of runs where the i-th factor appears at p level, the j-th factor appears at level q and the k-th factor appears at level r. It holds that , since the design matrix of a saturated design has full column rank.
Remark 2.2 By the choice of the labels for the levels one can always assume, without loss of generality (w.l.g), that n 0 The following lemmas are crucial for the main results of our paper and can be founded in Chatterjee and Mukerjee (1993) and Chatzopoulos and Kolyva-Machera (2006). (1) Proof See Chatzopoulos and Kolyva-Machera (2006) Remark 2.3 For s 1 = s 2 = 2 we have to study only the cases where 0 ≤ s 4 − s 3 ≤ 1, that is the cases 2 2 × s 2 , s ≥ 2 and 2 2 × (s − 1) × s, s ≥ 3.
Let 1×k be a 1 × k vector with all elements equal to zero, I k be the identify matrix of order k. For m < k, it can be easily seen that: Matrix U d as given in (1), can be written as: Using relation (8), and after permutation of columns matrix U d can be written as: Now subtract the (2u)-th column from the (2u + 1)-th column and add the last s − u columns to the (2u + 1)-th column. Matrix U d , as given in (9), can be written as: Matrix U d 1 is the design matrix of the saturated u × u × 2 design d 1 with N 1 = 2u runs. Matrix U 2 is not design matrix as its first column is (2, … , 2, 0, … , 0) � , but |det(U 2 )| = 2|det(U d 2 )|, where matrix U d 2 is the design matrix of the saturated 2 × (s − u) × (s − u) design d 2 with N 2 = 2(s − u) runs. Hence, Recalling that u = s∕2 if s ≡ 0 mod 2, or u = (s + 1)∕2 if s ≡ 1 mod 2, we get that (7) holds. ✷ Theorem 3.2 Let u = (s 3 + 1)∕2 if s ≡ 1 mod 2, or u = s 3 ∕2 + 1 if s ≡ 0 mod 2. The saturated 2 2 × s 3 × s 4 , is a D-optimal design in the class of all 2 2 × s 3 × s 4 , s 4 > s 3 ≥ 2 saturated designs.

D-optimality of 2 2 × s 2 saturated designs
Lemma 3.7 Consider the saturated 2 2 × s 2 design d with N = 2s + 1 runs and corresponding matrix U d as given in (1). For the D-optimal design, it holds that: Proof The proof is similar as lemma 3.3. ✷ n 0 i = s + 1 and n 1 i = s, or n 0 i = s and n 1 i = s + 1, i = 1, 2. n 0 i = 3, n p i = 2, i = 3, 4, 1 ≤ p ≤ s − 1.