Projection properties of certain three level orthogonal arrays

Transcription

1 Metrika (2005) 62: DOI /s ORIGINAL ARTICLE H. Evangelaras C. Koukouvinos A. M. Dean C. A. Dingus Projection properties of certain three level orthogonal arrays Springer-Verlag 2005 Abstract Two orthogonal arrays based on 3 symbols are said to be isomorphic or combinatorially equivalent if one can be obtained from the other by a sequence of row permutations, column permutations and permutations of symbols in each column. Orthogonal arrays are used as screening designs to identify active main effects, after which the properties of the subdesign for estimating these effects and possibly their interactions become important. Such a subdesign is known as a projection design. In this paper we have identified all the inequivalent projection designs of an OA(27, 13, 3, 2), anoa(18, 7, 3, 2) and an OA(36, 13, 3, 2) into k = 3, 4 and 5 factors. It is shown that the generalized wordlength pattern criterion proposed by Ma and Fang [23] can distinguish between most, but not all, inequivalent classes. We propose an extension of the Es 2 criterion (which is commonly used for measuring efficiency of 2-level designs) to distinguish further between the non-isomorphic classes and to measure the efficiency of the designs in these classes. Some concepts on generalized resolution are also discussed. Keywords Average squared correlation generalized wordlength pattern isomorphism main effect plan orthogonal arrays projection properties Mathematics Subject Classification (2000) Primary 62K15 Secondary 05B15 1 Introduction An orthogonal array OA(n, q, s, t) is an n q array with entries from a set of s distinct symbols arranged so that, for any collection of t columns of the array, each H. Evangelaras (B) C. Koukouvinos Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece A. M. Dean C. A. Dingus Department of Statistics, The Ohio State University, Columbus, OH , USA

2 242 H. Evangelaras et al. of the s t row vectors appears equally often. Thus we see that s t divides n. We call n the number of runs in the orthogonal array, q the number of factors, s the number of levels for each factor and t the strength of the array. Hedayat [17] defined an OA(n, q, s, t+) to be an OA(n, q, s, t) that is not of strength t + 1 but has one or more subarrays that form an OA(n, q,s,t + 1), with q <q. Orthogonal arrays are useful as screening designs for situations in which a large number, q, of factors is examined but only few, k, of these are expected to be important. This situation is known as factor sparsity (see Box and Meyer [4]). Orthogonal arrays with factors each having two levels have been studied extensively in the literature for identifying active main effects (ignoring interactions). In this setting they are referred to as orthogonal main effect plans, since all main effect contrasts in the levels of each of the q factors are estimable and the ordinary least squares estimators of any two main effect contrasts from two different factors are uncorrelated. For more details on the use of orthogonal arrays in factorial experiments, we refer the interested reader to Hedayat, Sloane and Stufken [18, chapter 11], or Dey and Mukerjee [13, chapter 2]. For an overview of fractional factorial designs and their properties, we refer the reader to Dean and Voss [11, chapter 15], Montgomery [25, chapters 8, 9], and Wu and Hamada [27, chapters 4, 5]. The properties of an orthogonal array for the identification of a particular set of k active factors, and possibly their interactions, are examined by investigation of the projection design consisting of the the columns that correspond to the k active factors, see Cheng [6], Cheng and Wu [7], Evangelaras, Georgiou and Koukouvinos [14], Goh and Street [16], Lin and Draper [21,22] and Wang and Wu [26]. Since the projection design of interest depends upon the outcome of the main effects analysis, it is necessary to study the properties of all possible projection designs that may arise from the original orthogonal array. The selection of k factors is equivalent to the selection of an n k subarray of the original orthogonal array with n runs. Hedayat, Sloane and Stufken [18] showed that a n k subarray of an OA(n, q, s, t) is an OA(n, k, s, t ) where t = min(k, t). Two subarrays are said to be isomorphic or combinatorially equivalent if one can be obtained from the other by a sequence of permutations of rows, columns and levels of each factor. Determination of combinatorial equivalence is computer intensive, especially when n, k and s are large. Clark and Dean [10] regarded designs as sets of points in q-dimensional space, where q is the number of factors, and they showed that two designs are combinatorially equivalent if the Hamming distances (see MacWilliams and Sloane [24]) between the points are the same in all possible dimensions. Using this property, they proposed a computer algorithm for checking combinatorial equivalence of designs. In Section 2, we determine all the combinatorially inequivalent subarrays of the orthogonal arrays OA(18, 7, 3, 2), OA(27, 13, 3, 2) and OA(36, 13, 3, 2) that are shown in Table 10 in the Appendix, checking equivalence directly by the above definition. One representative of each equivalence class is shown in Tables 1, 3, 4 and 6 in Section 2 for projections into k = 3, 4 and 5 factors, respectively. For each listed design, we have also calculated the value of the generalized wordlength pattern criterion proposed by Ma and Fang [23]. (A similar definition of generalized wordlength pattern was also given by Xu and Wu [30]). This criterion can be applied to all regular and non-regular factorial designs having factors with s levels.

3 Projection properties of certain three level orthogonal arrays 243 Wang and Wu [26] studied the projection properties of the OA(18, 7, 3, 2) when projected into 3 and 4 factors and their non-isomorphic designs are in compliance with our Table 1. Their results were later used by several authors (see Cheng and Wu [7], Xu and Wu [30], Xu [28] and Xu, Cheng and Wu [29]). Xu and Wu [30] and Xu, Cheng and Wu [29] used the generalized minimum aberration criterion proposed by Xu and Wu [30] for distinguishing designs rather than checking directly for isomorphism. Cheng and Wu [7] presented a list of non-isomorphic designs with 3 and 4 factors that arise from the OA(36, 12, 3, 2). Their list is upgraded in this work since we projected the OA(36, 13, 3, 2) listed in the Appendix having an extra 13th column. Xu, Cheng and Wu [29] also considered the projection properties of several alternative 18-run and 27-run orthogonal arrays which they obtained using Xu s [28] algorithm. In Section 2, we list the inequivalent classes of projections from the orthogonal arrays of Table 10 together with the generalized wordlength pattern for the designs in each equivalence class. We show that the generalized wordlength pattern can distinguish between many, but not all, of the inequivalent classes. Consequently, in Section 3, we propose a measure of efficiency, similar to the Es 2 criterion used for 2-level supersaturated designs (see Section 3), which is able to distinguish between most of the projection classes and can be used for comparison of the subarrays. The proposed criterion is based upon the average correlations between sets of orthogonal main effect and interaction contrasts, and is independent of the particular contrasts used. In Section 4, we discuss the relationship of this criterion to the actual models that may be fitted in a projection design. We note that some of the combinatorially equivalent designs identified in Section 2 will be inequivalent for quantitative factors (see Section 3), where the only permutation of levels that produces isomorphic designs is the permutation of low and high levels. We refer the interested reader to Cheng and Wu [7] and Cheng and Ye [8] for a study of 3-level designs with quantitative factors. Concepts on generalized resolution based on the definition of Deng and Tang [12] are discussed in Section 5. 2 Combinatorially inequivalent designs Let D be a fractional factorial design with n runs and k factors, each factor having s levels. The generalized wordlength pattern of D was defined by Ma and Fang [23] to be W g (D) ={A g 1 (D),...,Ag k (D)} (1) where, for i = 1,...,kand j = 0,...,k, where A g i (D) = 1 n(s 1) P i (j; k) = k P i (j; k)e j (D), (2) j=0 i ( 1) r (s 1) i r ( j r )( k j r=0 i r )

4 244 H. Evangelaras et al. are the Krawtchouk polynomials (see MacWilliams and Sloane [24], page 130) with ( j r) = 0ifr>j, and where Ej (D) is the distance distribution of D, defined as: E i (D) = n 1 #{(c, d), c, d D, d H (c, d) = i} where d H (c, d) is the Hamming distance between two runs c and d of D. A similar definition was given independently by Xu and Wu [30] but without the divisor (s 1) in (2). For the undefined terms in coding theory, we refer the interested reader to MacWilliams and Sloane [24], pages 8 and 151. When a fractional factorial design is regular and is governed by a defining relation, the generalized wordlength pattern reduces to the original wordlength pattern. Based on the generalized wordlength pattern, Ma and Fang [23] defined the resolution of a design D to be the smallest value of i with positive A g i (D) in W g (D). Then, if design D 1 and D 2 have resolution i, D 1 has smaller aberration than D 2 if A g t (D 1 )<A g t (D 2 ) where t is the smallest integer for which A g t (D 1 ) = A g t (D 2 ). A design has minimum generalized aberration if no other design with s levels and n runs has smaller aberration. This is a generalization of the minimum aberration criterion of Fries and Hunter [15]. In Sections , we find all the inequivalent projection designs of the orthogonal arrays in Table 10 and identify those designs with minimum generalized aberration. 2.1 Inequivalent designs that arise from the OA(27, 13, 3, 2) We now present all combinatorially inequivalent subarrays with k = 3, 4 and 5 factors arising from the orthogonal array OA(27, 13, 3, 2) of Table 10 in the Appendix. Some of the results presented in this section can also be found in Chen, Sun and Wu [5]. When k = 3, there are 286 possible subarrays consisting of k = 3 factors (columns) and these fall into 2 inequivalent classes of projection designs under combinatorial equivalence. The first class consists of columns which give the full 3 3 factorial design. One representative set of columns is the set {1, 2, 3}. The full factorial arose 234 times in the set of all 286 projections. The other 52 projections each consists of three replicates of a 3III 3 1 fractional factorial design and confound a pair of contrasts from the three-factor interaction. For example, the projection onto factors 1, 2, 4 gives the fraction with defining relation I = ABC 2. The OA(27, 13, 3, 2) can thus be regarded as an OA(27, 13, 3, 2+), since all projection designs in class 1 are permutations of an OA(n = 27,k = 3,s = 3,t = 3). When k = 4, there are 715 possible choices of k = 4 factors out of the q = 13 factors. The 715 possible projection designs fall into 3 inequivalent projections classes. The first class contains 468 projection designs, each of which consists of a 3III 4 1 fractional factorial design confounding a pair of contrasts from a three-factor interaction. One representative of this class is the projection onto factors 1, 2, 3, 4 and has defining relation I = ABD 2. All other projection designs in this class are permutations of this design and have generalized wordlength pattern (0,0,1,0). The second class consists of 234 projection designs each of which is a IV fractional factorial design confounding a pair of contrasts from the four-factor

5 Projection properties of certain three level orthogonal arrays 245 interaction and has generalized wordlength pattern (0,0,0,1). One representative of this class is the projections design consisting of columns 1, 2, 3, 6 of the OA(27, 13, 3, 2) which has defining relation I = ABCD 2. The third class of projections contains only 13 designs. Each of these consists of three copies of a 3III 4 2 fractional factorial design confounding a pair of contrasts from each of the four three-factor interactions. The generalized wordlength pattern is therefore (0,0,4,0). A representative design in this class consists of columns 1, 2, 4, 10 and has defining relation I = AB 2 D = BCD = ACD 2 = ABC 2. Clearly the second class of projection designs contain the minimum aberration designs that arise when the OA(27,13,3,2+) is projected into four factors. The 1287 possible choices of k = 5 factors out of the q = 13 factors also fall into 3 inequivalent classes under combinatorial equivalence. The first class contains 702 equivalent projection designs and consists of a 3III 5 2 fractional factorial design confounding pairs of contrasts from two three-factor interactions, one four-factor interaction and one five-factor interaction, so the generalized wordlength pattern is (0,0,2,1,1). For example, the design consisting of columns 1, 2, 3, 4, 5 has defining relation I = ABD 2 = BCE 2 = AC 2 D 2 E = AB 2 CD 2 E 2. The second class contains 468 equivalent projection designs, each of which is a fractional factorial design confounding a pair of interactions from one threefactor interaction and three five-factor interactions, so has generalized wordlength pattern (0,0,1,3,0). One representative is given by columns 1, 2, 3, 4, 7 which has defining relation I = ABD 2 = AB 2 CE 2 = AC 2 DE = BCDE 2. The third class 3III 5 2 contains 117 projection designs each of which is a 3III 5 2 fractional factorial design confounding a pair of contrasts from each of four five-factor interactions and has generalized wordlength pattern (0,0,4,0,0). For example, columns 1, 2, 3, 4, 10 give a design with defining relation I = ABD 2 = AB 2 E = ADE 2 = BDE. The second class of equivalent projection designs contains the minimum aberration design that arises when the OA(27, 13, 3, 2+) is projected into five factors. We see that the generalized wordlength pattern is able to distinguish and rank order all of the inequivalent projection designs for k = 3, 4, 5 factors. We investigate the properties of the classes further in Section Inequivalent projections of the OA(18, 7, 3, 2) In this section, we identify and examine the equivalence classes of projections into k = 3, 4 and 5 factors of the orthogonal array OA(18, 7, 3, 2) of Table 10 in the Appendix. There are 35 possible projections of the original array into k = 3 factors. Under combinatorial equivalence, this set of 35 projections falls into 3 inequivalent classes of projection designs, where the classes contain 28, 6 and 1 isomorphic projection designs, respectively. The 35 possible projections of the original array into k = 4 factors fall into 4 inequivalent classes of projections with 15, 12, 4 and 4 isomorphic projection designs, respectively, in the four classes. The 21 possible projections into k = 5 factors also fall into 4 inequivalent classes containing 6, 8, 6 and 1 design, respectively. One representative set of columns of the OA(18, 7, 3, 2) that give a design in each projection class is given in Table 1 and the corresponding value of the generalized minimum aberration criterion (1) is shown in Table 2. It can be seen that the generalized minimum aberration criterion

6 246 H. Evangelaras et al. Table 1 Inequivalent Projections of OA(18, 7, 3, 2) into k = 3, 4 and 5 factors Design Columns Design Columns Design Columns Design Columns selected selected selected selected 18p.3.1 {1 2 3} 18p.3.2 {1 2 7} 18p.3.3 {3 5 7} 18p.4.1 {1 234} 18p.4.2 {1 237} 18p.4.3 {1 247} 18p.4.4 {1 357} 18p.5.1 {1 2345} 18p.5.2 {1 2347} 18p.5.3 {1 2357} 18p.5.4 {1 2467} Table 2 Generalized wordlength pattern of inequivalent projection designs of Table 1 Design GWP Design GWP Design GWP 18p.3.1 (0, 0, 0.25) 18p.3.2 (0, 0, 0.5) 18p.3.3 (0, 0, 1) 18p.4.1 (0, 0, 1, 0.75) 18p.4.2 (0,0,1.25, 0.5) 18p.4.3 (0, 0, 1.75, 0) 18p.4.4 (0, 0, 1.75, 0) 18p.5.1 (0, 0, 2.5, 3.75, 0) 18p.5.2 (0, 0, 3.25, 2.25, 0.75) 18p.5.3 (0, 0, 3.5, 1.75, 1) 18p.5.4 (0, 0, 4, 0.75, 1.5) is able to rank order all but equivalence classes 18p.4.3 and 18p.4.4. In Section 3, we show that these classes can, however, be distinguished via an average squared correlation criterion. 2.3 Inequivalent projections of the OA(36, 13, 3, 2) Table 3 shows the generator columns and the generalized wordlength pattern of the 6 inequivalent classes of projection designs arising from the 286 projections of the OA(36, 13, 3, 2) onto k = 3 factors. The generalized wordlength pattern can rank order all of these, and we see that class 36p.3.1 with generalized wordlength pattern (0, 0, 0.063) represents the minimum generalized aberration design. Similarly, the 715 projections into k = 4 factors are classified into 27 inequivalent classes. The columns of the OA(36, 13, 3, 2) that give a representative design of each class are given in Table 4. In Table 5, these are re-ordered according to their generalized wordlength pattern. Class 36p.4.2 contains 216 isomorphic minimum aberration projection designs. Note that the generalized wordlength pattern is unable to distinguish and rank order all classes of designs. For example, classes 36p.4.1 and 36p.4.17 each have generalized wordlength pattern (0, 0, 0.625, 0.188). However, most of the non-distinguished classes can be distinguished subsequently using the average squared correlation criterion in Section 3. The 1287 possible projections of the OA(36, 13, 3, 2) into k = 5 factors are classified into 84 inequivalent classes. These are shown with one representative from each class and the generalized wordlength pattern of each design in Table 6. Again, not all classes can be distinguished by the generalized wordlength pattern. Table 3 Inequivalent projections of OA(36, 13, 3, 2) into k = 3 factors Design Columns GWP Design Columns GWP Design Columns GWP selected selected selected 36p.3.1 {1 2 3} (0, 0, 0.063) 36p.3.2 {1 2 5} (0, 0, 0.438) 36p.3.3 {1 2 13} (0, 0, 1) 36p.3.4 {1 3 8} (0, 0, 0.25) 36p.3.5 {3 4 13} (0, 0, 0.125) 36p.3.6 {3 6 13} (0, 0, 0.313)

7 Projection properties of certain three level orthogonal arrays 247 Table 4 Inequivalent Projections of OA(36, 13, 3, 2) into k = 4 factors Design Columns Design Columns Design Columns selected selected selected 36p.4.1 {1 234} 36p.4.2 {1 236} 36p.4.3 {1 238} 36p.4.4 {1 239} 36p.4.5 {1 2313} 36p.4.6 {1 2513} 36p.4.7 {1 3413} 36p.4.8 {1 3513} 36p.4.9 {1 3613} 36p.4.10 {1 3812} 36p.4.11 {1 3813} 36p.4.12 {1 3913} 36p.4.13 { } 36p.4.14 {1 4913} 36p.4.15 {1 6713} 36p.4.16 {3 4513} 36p.4.17 {3 4613} 36p.4.18 {3 4713} 36p.4.19 {3 4913} 36p.4.20 { } 36p.4.21 {3 5713} 36p.4.22 {3 5813} 36p.4.23 {3 5913} 36p.4.24 { } 36p.4.25 { } 36p.4.26 { } 36p.4.27 { } Table 5 Generalized wordlength pattern of inequivalent projection designs in Table 4 Design GWP Design GWP Design GWP 36p.4.2 (0, 0, 0.25, 0.375) 36p.4.4 (0, 0, 0.25, 0.563) 36p.4.13 (0, 0, 0.313, 0.313) 36p.4.15 (0, 0, 0.313, 0.313) 36p.4.7 (0, 0, 0.313, 0.5) 36p.4.22 (0, 0, 0.437, 0.375) 36p.4.3 (0, 0, 0.438, 0.188) 36p.4.16 (0, 0, 0.438, 0.188) 36p.4.9 (0, 0, 0.5, 0.125) 36p.4.11 (0, 0, 0.5, 0.125) 36p.4.12 (0, 0, 0.5, 0.313) 36p.4.19 (0, 0, 0.625, 0) 36p.4.25 (0, 0, 0.625, 0) 36p.4.1 (0, 0, 0.625, 0.188) 36p.4.17 (0, 0, 0.625, 0.188) 36p.4.21 (0, 0, 0.625, 0.375) 36p.4.8 (0, 0, 0.688, 0.125) 36p.4.14 (0, 0, 0.688, 0.125) 36p.4.20 (0, 0, 0.812, 0) 36p.4.18 (0, 0, 0.813, 0.188) 36p.4.26 (0, 0, 0.813, 0.375) 36p.4.23 (0, 0, 1, 0) 36p.4.10 (0, 0, 1, 0.75) 36p.4.5 (0, 0, 1.188, 0) 36p.4.27 (0, 0, 1.188, 0) 36p.4.24 (0, 0, 1.375, 0.375) 36p.4.6 (0, 0, 2.313, 0) Class 36p.5.12, which contains 72 isomorphic projections, represents the minimum aberration design. 3 Average correlation criterion The non-isomorphic projection designs identified in Section 2 can be evaluated via a criterion similar to the Es 2,orAve(s 2 ), efficiency criterion that was introduced by Booth and Cox [3] for determining efficient 2-level supersaturated designs. If c i and c j are column vectors of the model matrix X, representing the main effect contrasts in the levels of two factors (or representing interactions), then the correlation between the contrasts is calculated as the angle between the two vectors (see Johnson and Wichern [19], page 118). Thus, the correlation between contrasts c i and c j is ρ i;j = c i c j (c i c i)(c j c j ) = s ij sii s jj, (3) where s ij is the (ij)th element of the matrix X X. For 2-level designs, s ii = s jj = n for all i, j = 1,...,nand the design with minimum average ρ 2 i;j is the design with minimum average s 2 ij and is called Es2 -optimal (Booth and Cox [3], Lin [20]). For three-level factors, we may use the same type of definition, except that the model matrix X contains two orthogonal contrasts for each main effect and four orthogonal contrasts for each two-factor interaction (see, for example, Dean and

8 248 H. Evangelaras et al. Table 6 Inequivalent projections of OA(36, 13, 3, 2) into k = 5 factors Design Columns selected GWP Design Columns selected GWP 36p.5.1 {1,2,3,4,5} (0, 0, 1.375, 1.313, 0.188) 36p.5.2 {1,2,3,4,6} (0, 0, 1, 1.5, 0.375) 36p.5.3 {1,2,3,4,7} (0, 0, 1, 1.5, 0.375) 36p.5.4 {1,2,3,4,8} (0, 0, 1.187, 1.125, 0.563) 36p.5.5 {1,2,3,4,13} (0, 0, 2, 0.813, 0.625) 36p.5.6 {1,2,3,5,13} (0, 0, 2.75, 0.438, 0.25) 36p.5.7 {1,2,3,6,7} (0, 0, 0.813, 1.688, 0.375) 36p.5.8 {1,2,3,6,13} (0, 0, 1.813, 0.625, 0.437) 36p.5.9 {1,2,3,8,10} (0, 0, 0.813, 1.5, 0.563) 36p.5.10 {1,2,3,8,12} (0, 0, 1.375, 1.5, 0) 36p.5.11 {1,2,3,8,13} (0, 0, 1.813, 0.625, 0.437) 36p.5.12 {1,2,3,9,10} (0, 0, 0.625, 2.063, 0.187) 36p.5.13 {1,2,3,9,13} (0, 0, 1.813, 1.188, 0.437) 36p.5.14 {1,2,3,10,13} (0, 0, 1.625, 1, 0.25) 36p.5.15 {1,3,4,5,13} (0, 0, 1.188, 1.5, 0.187) 36p.5.16 {1,3,4,6,13} (0, 0, 1, 1.5, 0.375) 36p.5.17 {1,3,4,7,13} (0, 0, 1.188, 1.313, 0.375) 36p.5.18 {1,3,4,8,13} (0, 0, 1, 1.313, 0.563) 36p.5.19 {1,3,4,9,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.20 {1,3,4,11,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.21 {1,3,4,12,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.22 {1,3,5,6,13} (0, 0, 1.375, 0.75, 0.75) 36p.5.23 {1,3,5,7,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.24 {1,3,5,8,13} (0, 0, 1.375, 0.938, 0.563) 36p.5.25 {1,3,5,9,13} (0, 0, 1.75, 0.75, 0.375) 36p.5.26 {1,3,5,10,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.27 {1,3,6,7,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.28 {1,3,6,8,13} (0, 0, 1.375, 0.563, 0.938) 36p.5.29 {1,3,6,9,13} (0, 0, 1.188, 1.313, 0.375) 36p.5.30 {1,3,6,10,13} (0, 0, 1.375, 0.938, 0.563) 36p.5.31 {1,3,6,11,13} (0, 0, 1.75, 0.938, 0.187) 36p.5.32 {1,3,6,12,13} (0, 0, 1.375, 0.938, 0.563) 36p.5.33 {1,3,7,10,13} (0, 0, 1.375, 0.938, 0.563) 36p.5.34 {1,3,7,11,13} (0, 0, 1.563, 0.75, 0.563) 36p.5.35 {1,3,8,10,13} (0, 0, 1.188, 0.938, 0.75) 36p.5.36 {1,3,8,12,13} (0, 0, 1.563, 1.125, 0.188) 36p.5.37 {1,3,9,10,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.38 {1,3,9,11,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.39 {1,3,10,11,13} (0, 0, 1.188, 1.313, 0.375) 36p.5.40 {1,4,5,6,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.41 {1,4,5,9,13} (0, 0, 1.563, 0.75, 0.563) 36p.5.42 {1,4,5,10,13} (0, 0, 1.562, 0.938, 0.375) 36p.5.43 {1,4,6,7,13} (0, 0, 1, 1.313, 0.563) 36p.5.44 {1,4,6,9,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.45 {1,4,6,11,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.46 {1,4,7,9,13} (0, 0, 1.375, 0.938, 0.563) 36p.5.47 {1,4,7,11,13} (0, 0, 1.563, 0.563, 0.75) 36p.5.48 {1,4,10,11,13} (0, 0, 1.563, 0.563, 0.75) 36p.5.49 {1,5,6,7,13} (0, 0, 1.188, 1.313, 0.375) 36p.5.50 {1,6,7,9,13} (0, 0, 1, 1.125, 0.75) 36p.5.51 {3,4,5,6,13} (0, 0, 1.188, 0.938, 0.75) 36p.5.52 {3,4,5,7,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.53 {3,4,5,8,13} (0, 0, 1, 1.313, 0.563) 36p.5.54 {3,4,5,9,13} (0, 0, 1.563, 0.563, 0.75) 36p.5.55 {3,4,5,10,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.56 {3,4,5,11,13} (0, 0, 1.563, 0.75, 0.563) 36p.5.57 {3,4,5,12,13} (0, 0, 1.187, 1.125, 0.563) 36p.5.58 {3,4,6,7,13} (0, 0, 1.562, 0.938, 0.375) 36p.5.59 {3,4,6,8,13} (0, 0, 1.75, 0.563, 0.563) 36p.5.60 {3,4,6,9,13} (0, 0, 1.375, 0.938, 0.563) 36p.5.61 {3,4,6,10,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.62 {3,4,6,11,13} (0, 0, 2.125, 0.938, 0.375) 36p.5.63 {3,4,7,8,13} (0, 0, 1.562, 0.938, 0.375) 36p.5.64 {3,4,7,9,13} (0, 0, 1.562, 0.938, 0.375) 36p.5.65 {3,4,7,10,13} (0, 0, 1.938, 0.75, 0.75) 36p.5.66 {3,4,7,11,13} (0, 0, 1.75, 0.75, 0.375) 36p.5.67 {3,4,8,10,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.68 {3,4,8,11,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.69 {3,4,8,12,13} (0, 0, 1.375, 0.75, 0.75) 36p.5.70 {3,4,9,10,13} (0, 0, 1.563, 0.563, 0.75) 36p.5.71 {3,4,10,11,13} (0, 0, 1.938, 0.563, 0.375) 36p.5.72 {3,5,6,7,13} (0, 0, 1.375, 1.125, 0.375) 36p.5.73 {3,5,6,9,13} (0, 0, 1.75, 0.75, 0.375) 36p.5.74 {3,5,6,11,13} (0, 0, 1.938, 0.563, 0.375) 36p.5.75 {3,5,7,8,13} (0, 0, 1.75, 1.125, 0.563) 36p.5.76 {3,5,7,10,13} (0, 0, 1.562, 0.938, 0.375) 36p.5.77 {3,5,8,10,13} (0, 0, 1.188, 1.313, 0.375) 36p.5.78 {3,5,8,12,13} (0, 0, 1.188, 1.313, 0.375) 36p.5.79 {3,5,9,10,13} (0, 0, 1.75, 0.375, 0.75) 36p.5.80 {3,5,10,11,13} (0, 0, 1.562, 0.938, 0.375) 36p.5.81 {3,6,7,9,13} (0, 0, 1.75, 0.563, 0.563) 36p.5.82 {3,6,9,12,13} (0, 0, 1.75, 1.313, 0.375) 36p.5.83 {3,7,8,10,13} (0, 0, 1.938, 0.563, 0.375) 36p.5.84 {4,5,10,11,13} (0, 0, 2.313, 1.875, 0.375) Voss [11], Montgomery [25], and Wu and Hamada [27, section 5.6]). For quantitative factors, the contrasts of interest are often the linear and quadratic main effect contrasts and the linear linear etc. interaction trend contrasts. For qualitative factors, we may use the same contrasts although they have no physical meaning. We may also include all of the contrasts in the X matrix even if some are aliased together. To evaluate each three-level projection design identified in Section 2, we calculate the average of the squared correlations between each of the two orthogonal trend contrasts for the sth main effect and each of the four orthogonal trend contrasts for the two-factor interaction between factors t and u (for all s = t<u). We use the notation Ave(ρs;tu 2 ) (s = t<u) and call these the average squared correlations of order 3. Similarly, we calculate the average squared correlation of order 4,Ave(ρqs;tu 2 ), between each of the four orthogonal trend contrasts for the (qs)th two-factor interaction (between factors q and s) and those for the (tu)th two-factor interaction, for all q<s = t<u.

9 Projection properties of certain three level orthogonal arrays 249 Although we have used the trend contrasts in our calculations, Theorem 1 shows that the average squared correlations of order 3 and 4 for any design are, in fact, invariant to the particular choice of orthonormal contrasts spanning each factorial space. We first note that the correlation (3) can be defined equivalently by the inner product of the normalized contrast vectors c i /(c i c i) 1/2 and c j /(c j c j ) 1/2. Theorem 1 The average squared correlation between complete sets of orthonormal contrasts for any two factorial effects is invariant to the particular sets of orthonormal contrasts selected. Proof Let C i be a v m i matrix whose columns form a set of orthonormal contrast vectors corresponding to the factorial effect F i with m i degrees of freedom, where F i denotes either a main effect or an interaction, (i = 1, 2,...,v 1). For any 1 i v 1, let G i be an m i m i orthogonal matrix, then (C i G i ) (C i G i ) = G i C i C ig i = G i G i = I mi. Since the columns of C i G i span the same space as the columns of C i, it follows that the columns of C i G i forms an alternative set of orthonormal contrasts for measuring the factorial effect F i. Let Ave(ρi;j 2 ) denote the average squared correlation between the set of orthonormal contrasts C i measuring F i and the set of orthonormal contrasts C j measuring F j, then Ave(ρi;j 2 ) = (m im j ) 1 trace[c i C j ] [C i C j ] = (m i m j ) 1 trace[c j C ic i C j ]. Similarly, let Ave(ρi;j ) be the average squared correlation between the sets of orthonormal contrasts C i G i and C j G j, then Ave(ρi;j ) = (m im j ) 1 trace[g j C j C ig i G i C i C j G j ] = (m i m j ) 1 trace[c j C ic i C j ] where the last line follows from the fact that trace (AB) = trace (BA) and that G i and G j are square nonsingular orthogonal matrices. The average squared correlations of order 3 and 4 are summarized in Table 7 for the designs in each of the inequivalent projection classes arising from the OA(18, 7, 3, 2) of Section 2.2. As an example, consider the projection design consisting of the array columns 1, 2, 7 which is the representative of class 18p.3.2 listed in Table 1. This has generalized wordlength pattern (0, 0, 0.5). Suppose that we use the linear and quadratic trend contrasts for the main effects and interactions in the model matrix, X, and associate the orthogonal array codes 0, 1, 2 with the low, medium and high levels for each factor. If the factors are qualitative, the designation is arbitrary. For this example, we list the contrasts in the order A L,A Q,B L,B Q,C L,C Q,A L B L,..., (where A L denotes the linear contrast associated with the first factor in the projection design etc.) and number the contrasts, in order, 1, 2, 3, etc. The mean column of the model matrix is numbered zero.

10 250 H. Evangelaras et al. The average squared correlation of order 3 between the main effect of factor C and the AB interaction is calculated as follows. First, the correlation (3) between the C L contrast and the A L B L interaction is ρ CL ;A L B L = s 5,7 s5,5 s 7,7 = = , where s ij is the (i, j)th element of X X. Similarly, the correlations between the other pairs of trend contrasts from C and AB are ρ CL ;A L B Q = 0/ = ρ CL ;A Q B L = 0/ = ρ CL ;A Q B Q = 18/ = ρ CQ ;A L B L = 6/ 36 8 = ρ CQ ;A L B Q = 0/ = ρ CQ ;A Q B L = 0/ = ρ CQ ;A Q B Q = 18/ = Thus, the average squared correlation Ave(ρC;AB 2 ) between C and AB is the average of the squares of the above 8 individual correlations, giving It can be verified that the average squared correlations Ave(ρA;BC 2 ) for A with BC and Ave(ρB;AC 2 ) for B with AC are each 0.125, also. The average squared correlations between a pair of two-factor interactions are based on the correlations of the 16 pairs of interaction contrasts (A L B L ; A L C L, etc.) and it can be verified that Ave(ρAB;AC 2 ) =Ave(ρ2 AB;BC ) =Ave(ρ2 AC;BC ) = Therefore, the isomorphic designs in class 18p.3.2 each have three average squared correlations of order 3 equal to and three average squared correlations of order 4 equal to This information is summarized in Table 7, together with the average squared correlations for all the designs of Section 2.2. It can be seen that the inequivalent classes Table 7 Average squared correlation values for the inequivalent projection designs from OA(18, 7, 3, 2) Class Number of order 3 Number of order 4 Ave(ρ 2 ) equal to Ave(ρ 2 ) equal to p p p p p p p p p p p

11 Projection properties of certain three level orthogonal arrays 251 Table 8 Average squared correlation values for the inequivalent projection designs from OA(27, 13, 3, 2) Class Number of order 3 Number of order 4 Ave(ρ 2 ) equal to Ave(ρ 2 ) equal to p p p p p p p p of projection designs have very different properties and that projection designs in class have smaller correlations between the trend contrasts than designs in classes 18p.3.2 and 18p.3.3. The ranking of desirability of the 4- and 5-factor projection classes is less obvious. We stress once more that the average squared correlations are not dependent upon the particular (complete) sets of orthonormal contrasts selected. Interestingly, if we take an average over all order 3 correlations and over all order 4 correlations for each equivalence class of projections, we lose the distinction between class 18p.4.3 and 18p.4.4, as we do with the generalized wordlength pattern. Tables 8 and 9 give similar information for the inequivalent projections from the orthogonal arrays OA(27, 13, 3, 2) and OA(36, 13, 3, 2) of Sections 2.1 and 2.3. The generalized wordlength pattern and the average squared correlations of order 3 and 4 are able to distinguish between all of the inequivalent classes of projection designs from the OA(27, 13, 3, 2) and between all of the inequivalent classes of 3-factor projection designs from the OA(36, 13, 3, 2). In addition, the average squared correlations of order 3 and 4 are able to distinguish between 25 of the 27 inequivalent classes of 4-factor projection designs from the OA(36, 13, 3, 2) and between 77 of the 84 inequivalent classes of 5-factor projection designs, whereas the generalized wordlength pattern can only distinguish between 20 of the 27 and between 35 of the 84 classes, respectively. 4 Use of projection designs If a model involving main effects and interactions among the k factors of a projection design is to be fitted (c.f. Cheng and Wu [7]), it is necessary to know which interaction contrasts can be included in the model. In the following discussion, we restrict attention to qualitative factors and postpone an investigation of quantitative factors to future work (also see Cheng and Wu [7]; Chen and Ye [8]). The projection designs from the OA(27,13,3,2) of Table 10 are all regular fractional factorial designs and examples of their defining relations were given in Section 2.1. Some of the pairs of main effect contrasts are aliased with pairs of interaction contrasts and some pairs of interaction contrasts are aliased together. Only one pair of contrasts from each alias set can be fitted in the model (see, for example, Dean and Voss [11, section 15.3] or Montgomery [25, section 9.3]).

12 252 H. Evangelaras et al. Table 9 Average squared correlation values for the inequivalent projection designs from OA(36,13,3,2) Class Number of order 3 Number of order Ave (ρ 2 ) 10 3 equal to Ave(ρ 2 ) 10 3 equal to Nevertheless, the average squared correlations calculated in Section 3 still provide a method by which projection designs (and, consequently, orthogonal arrays) can be distinguished and ranked. The OA(18,7,3,2) and OA(36,13,3,2) give rise to nonregular projection designs and at most 17 and 35 factorial contrasts, respectively, can be estimated in a model. In Section 2, we listed only one representative from each equivalence class and, in the following, we investigate the models that can be fitted for the representative designs. We discuss models involving only main effects and 2-factor interactions, and we associate the factor names A, B, C, with the projection design columns listed in the order given in Tables 2, 3, 5 and 6. For the OA(18,7,3,2), at most two of the three 2-factor interactions can be fitted in the 3-factor projection designs, but only projection design 18p.3.1 is able to fit any selected pair of interactions. Design 18p.3.2 can fit one of AC or BC, but not AB. In design 18p.3.3, every main effect is aliased with two degrees of freedom from a 2-factor interaction, so no 2-factor interaction can be completely estimated along with the main effects. This ordering of desirability of designs is the same as that suggested by both the generalized wordlength pattern and the average squared correlations. In the 4-factor projection design 18p.4.1, any single interaction can be fitted. Also any pair of interactions can be fitted except for the pairs (AB, CD), (AC, BD) or (AD, BC). In 18p.4.2, any single interaction apart from AB can be fitted, as can any pair not of the form (AB, xy), (xc,yc)or(xd,yd). Projection designs 18p.4.3 and 18p.4.4, which have the same generalized wordlength pattern, can fit only a single 2-factor interaction involving D or A, respectively. The average

13 Projection properties of certain three level orthogonal arrays 253 Table 9 (Contd.) Class Number of order 3: Number of order 4: Ave(ρ 2 ) 10 3 equal to Ave(ρ 2 ) 10 3 equal to squared correlations shown in Table 7 indicate that 18p.4.3 should be a better design than 18p.4.4 and, indeed, this is the case in the following sense. If a single 2-factor interaction, or a pair of interactions, listed above as not able to be fitted were to be included in the model, then design 18p.4.3 would have one degree of freedom from a main effect aliased with one degree of freedom from a 2-factor interaction, whereas design 18p.4.4 would alias two degrees of freedom from each. In the 5-factor projection designs from the OA(18,7,3,2), any single interaction can be fitted in 18p.5.1; any single interaction except for AB, AD or BD in 18p.5.2; any single interaction except for AB, CD, CE, DE in 18p.5.3; and any single interaction that involves E in 18p.5.3. Thus, the number of estimable models increases in the same order as the rank of the generalized wordlength patterns. The average squared correlations suggest that 18p.5.2 and 18p.5.4 should be better designs than 18p.5.3. As with the the 4-factor projections, this is true in the number of aliased contrasts that would result when alternative models are fitted. Designs 18p.5.2 and 18p.5.4 never alias more than one degree of freedom of a main effect and two-factor interaction in a model with only one interaction, whereas design 18p.5.3 may alias two. For the three-factor projection designs from the OA(36,7,3,2), designs 36p.3.1, 36p.3.5 and 36p.3.6 allow all three 2-factor interactions to be fitted; designs 36p.3.2 and 36p.3.4 allow a specific two out of three interactions, but design 36p.3.3 allows only models with main effects. Design 36p.3.3 is very clearly not as good as the oth-

14 254 H. Evangelaras et al. Table 9 (Contd.) Class Number of order 3: Number of order 4: Ave(ρ 2 ) 10 3 equal to Ave(ρ 2 ) 10 3 equal to ers under the generalized wordlength pattern and the average squared correlation criteria. In the four-factor projection designs, all six 2-factor interactions can be fitted in designs , , , and Any selected five of the six interactions can be fitted in designs and ; any four of the six in designs and ; any three of the six in , and ; any one interaction in designs and ; no interactions in and ; and any two interactions in the remaining 11 designs. From the average squared correlations of order 3 and 4, designs and stand out as being the worst designs whereas, under the generalized wordlength pattern, design is indistinguishable from and better than The average squared correlations of order 4 also correctly suggest that designs and are not good designs, whereas the generalized wordlength pattern ranks as the second best design. For the five-factor projection designs, the average squared correlations of order 3 and 4, suggest that projection designs 36p.5.5, 6, 8, 11, 13 and 14 are not good designs. This is indeed the case, since none of these can estimate all contrasts measuring any selected 2-factor interaction. All projection designs except for numbers 1, 5 8, 10 15, 27, 36, 39, 61, 63, 68, 77, 82 and 84 can estimate models with two 2-factor interactions. All of these, except for 36p.5.61 and 36p.5.63 (which can estimate 44 of the 45 sets of two factor interactions) can be identified from the average squared correlations of order 4.

15 Projection properties of certain three level orthogonal arrays Concepts on generalized resolution A regular fractional factorial design is said to be of resolution R, ifnop-factor interaction is aliased with any other interaction between fewer than R p factors. For designs with two-level factors and governed by a defining relation, two factorial effects are either fully aliased or completely uncorrelated. Thus, in a resolution R design, the contrast corresponding to a p-factor interaction is indistinguishable from at least one (R p)-factor interaction contrast. For example, in a 2III 3 1 fractional factorial design with defining relation I=ABC, the contrast for the main effect of A cannot be distinguished from that of interaction BC, and so the A and BC contrasts are fully aliased. The information matrix of such a design contains off-diagonal elements equal to zero for uncorrelated contrasts and equal to the total number of runs for fully aliased contrasts. The positions of the nonzero elements provide information about the defining relation of the design. To continue the above example, if the contrasts in the model matrix correspond to the factorial effects A, B, C, AB, AC, BC, ABC in order, then, in the 7 7 information matrix, the elements [1,6], [2,5], [3,4] in the upper triangular block will be equal to n = 4. This method works reasonably well for two-level factorial designs and has been generalized by Deng and Tang [12] to compare non-regular factorial designs where effects are not fully aliased with others. They defined the J-values of the factorial design, as the off-diagonal elements of the information matrix corresponding to a model consisting of all factorial effects, to calculate the correlation (times n) of specific set of effects. Clearly, 0 J nfor any pair of factorial effects. For orthogonal designs, all pairs of main effect contrasts are orthogonal and, as in Section 3, the correlations of order r (for r 3) between factorial contrasts are the correlations of importance. If some of the correlations of order r = 3 (correlations of main effects with two factor interactions) are non-zero then the corresponding design is of resolution at least 3 but not 4. If all the correlations of order r = 3 are zero and some of the correlations of order r = 4 ( correlations between pairs of two-factor interactions) are non-zero then the design is of resolution at least 4 but not 5, and so on. Let r be the smallest order of the non-zero correlations and let ρ max (r) be the maximum absolute value of the rth order correlations. Then, similar to Deng and Tang s [12] definition in 2-level arrays, we can obtain a measure of generalized resolution using GRes = r + (1 ρ max (r) ). (4) Equation (4) can be used to obtain a generalized resolution value for any orthogonal design having factors at three levels using a specified set of contrasts. For quantitative factors, the contrasts would naturally be the linear and quadratic main effect contrasts and the linear linear etc. interaction trend contrasts. For instance, using the correlations obtained in Section 3 for the projection design based on columns {1, 2, 7} in the class 18p.3.2, we have r = 3 and ρ max (3) = Then, using equation (4) we obtain, for this design, GRes = for the design 18p.3.2. For qualitative factors, it is preferable to define generalized resolution in such a way that it does not depend on the set of contrasts used. In this case, we use for ρ max (r) the square root of the maximum order r correlation. In this case the

16 256 H. Evangelaras et al. generalized resolution for all projection designs in the class 18p.3.2 for qualitative factors is GRes = 3 + (1 ρ max ) (3) = , while that of class 18p.3.1 is 3.75 and that of class 18p.3.3 is 3.5. Acknowledgements We would like to thank Yanxing Lynne Zhao for help with the computing. Appendix Table 10 OA(18, 7, 3, 2), OA(27, 13, 3, 2) and OA(36, 13, 3, 2) OA(18, 7, 3, 2) OA(27, 13, 3, 2) OA(36, 13, 3, 2) References 1. Abraham, B., Chipman, H., Vijayan, K.: Some risks in the construction and analysis of supersaturated designs. Technometrics 41, (1999) 2. Beattie, S.D., Fong, D.K.H., Lin, D.K.J.: A two-stage Bayesian model selection strategy for supersaturated designs. Technometrics 44, (2002) 3. Booth, K.H.V., Cox, D.R.: Some systematic supersaturated designs. Technometrics 4, (1962) 4. Box, G.E.P., Meyer, D.: An analysis for unreplicated fractional factorials. Technometrics 28, (1986) 5. Chen, J., Sun, D.X., Wu, C.F.J.: A catalogue of two-level and three-level fractional factorial designs with small runs. Int. Stat. Rev. 61, (1993) 6. Cheng, C.S.: Some projection properties of orthogonal arrays. Ann. Stat. 23, (1995)