Generalized resolution for orthogonal arrays

Transcription

1 Submitted to he Annals of Statistis Generalized resolution for orthogonal arrays y Ulrike Grömping and Hongquan u 2 euth University of Applied Sienes erlin and University of alifornia Los Angeles he generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration riterion (u and Wu 200). We provide a statistial interpretation for the number of shortest words of an orthogonal array in terms of sums of 2 values (based on orthogonal oding) or sums of squared anonial orrelations (based on arbitrary oding). Diretly related to these results, we derive two versions of generalized resolution for qualitative fators, both of whih are generalizations of the generalized resolution by Deng and ang (999) and ang and Deng (999). We provide a suffiient ondition for one of these to attain its upper bound, and we provide expliit upper bounds for two lasses of symmetri designs. Fator wise generalized resolution values provide useful additional detail.. Introdution Orthogonal arrays (OAs) are widely used for designing experiments. One of the most important riteria for assessing the usefulness of an array is the generalized word length pattern (GWLP) as proposed by u and Wu (200): A 3, A 4, are the numbers of (generalized) words of lengths 3, 4,, and the design has resolution, if A i = 0 for all i < and A > 0. Analogously to the well-known minimum aberration riterion for regular frational fatorial designs (Fries and Hunter 980), the quality riterion based on the GWLP is generalized minimum aberration (GMA; u and Wu 200): a design D has better generalized aberration than a design D 2, if its resolution is higher or if both designs have resolution if its number A of shortest words is smaller; in ase of ties in A, frequenies of suessively longer words are ompared, until a differene is enountered. he definition of the A i in u and Wu is very tehnial (see Setion 2). One of the key results of this paper is to provide a statistial meaning for the number of shortest words, A : we will show that A is the sum of 2 values from linear models with main effets model matrix olumns in orthogonal oding as dependent variables and full models in other fators on the explanatory side. For Supported in part by Deutshe Forshungsgemeinshaft Grant G 3843/-. 2 Supported by National Siene Foundation Grant DMS AMS 2000 subjet lassifiations: Primary 62K5, seondary 055, 62K05, 62J99, 62H20 Key words and phrases: orthogonal arrays, generalized word length pattern, generalized minimum aberration, generalized resolution, qualitative fators, omplete onfounding, anonial orrelation, weak strength t

2 arbitrary fator oding, the sum of 2 interpretation annot be upheld, but it an be shown that A is the sum of squared anonial orrelations (Hotelling 936) between a fator s main effets model matrix olumns in arbitrary oding and the full model matrix from other fators. hese results will be derived in Setion 2. For regular frational fatorial 2-level designs, the GWLP oinides with the well-known word length pattern (WLP). An important differene between regular and non-regular designs is that fatorial effets in regular frational fatorial designs are either ompletely aliased or not aliased at all, while non-regular designs an have partial aliasing, whih an lead to non-integer entries in the GWLP. In fat, the absene of omplete aliasing has been onsidered an advantage of non-regular designs (e.g. those by Plakett and urman 946) for sreening appliations. Deng and ang (999) and ang and Deng (999) defined generalized resolution (G) for non-regular designs with 2-level fators, in order to apture their advantage over omplete onfounding in a number. For example, the 2 run Plakett-urman design has G=3.67, whih indiates that it is resolution III, but does not have any triples of fators with omplete aliasing. Evangelaras et al. (2005) have made a useful proposal for generalizing G (alled Ges by them) for designs in quantitative fators at 3 levels; in onjuntion with heng and Ye (2004), their proposal an easily be generalized to over designs with quantitative fators in general. However, there is so far no onvining proposal for designs with qualitative fators. he seond goal of this paper is to lose this gap, i.e. to generalize Deng and ang s / ang and Deng s G to OAs for qualitative fators. Any reasonable generalization of G has to fulfill the following requirements: (i) it must be oding-invariant, i.e. must not depend on the oding hosen for the experimental fators (this is a key differene vs. designs for quantitative fators), (ii) it must be appliable for symmetri and asymmetri designs (i.e. designs with a fixed number of levels and designs with mixed numbers of levels), (iii) like in the 2-level ase, + > G must hold, and G = must be equivalent to the presene of omplete aliasing somewhere in the design, implying that + > G > indiates a resolution design with no omplete aliasing among projetions of fators. We offer two proposals that fulfill all these requirements and provide a rationale behind eah of them, based on the relation of the GWLP to regression relations and anonial orrelations among the olumns of the model matrix. he paper is organized as follows: Setion 2 formally introdues the GWLP and provides a statistial meaning to its number of shortest words, as disussed above. Setion 3 briefly introdues generalized resolution by Deng and ang (999) and ang and Deng (999) and generalizes it in two meaningful ways. Setion 4 shows weak strength (in a version modified from u 2003 to imply strength ) to be suffiient for maximizing one of the generalized resolutions in a resolution design. Furthermore, it derives an expliit upper bound for the proposed generalized resolutions for two lasses of symmetri designs. Setion 5 derives fator wise versions of both types of generalized resolution and demonstrates that these provide useful additional detail to the overall values. he paper loses with a disussion and an outlook on future work. hroughout the paper, we will use the following notation: An orthogonal array of resolution = strength in N runs with n fators will be denoted as OA(N, s, ), with s,, s n the numbers of levels of the n fators (possibly but not neessarily distint), or as OA(N, s n s k n k, ) with n fators at s levels,, n k fators at s k levels (s,, s k possibly but not neessarily distint), whihever is more suitable for the purpose at hand. A subset of k indies that identifies a k-fator projetion is denoted by {u,,u k } ( {,,n}). he unsquared letter always refers to the resolution of a design, while 2 denotes the oeffiient of determination. 2

3 2. Projetion frequenies and linear models onsider an OA(N, s, ). he resolution implies that main effets an be onfounded with interations among fators, where the extent of onfounding of degree an be investigated on a global sale or in more detail: Following u and Wu (200), the fators are oded in orthogonal ontrasts with squared olumn length normalized to N. We will use the expression normalized orthogonal oding to refer to this oding; on the ontrary, the expressions orthogonal oding or orthogonal ontrast oding refer to main effets model matrix olumns that have mean zero and are pairwise orthogonal, but need not be normalized. For later referene, note that for orthogonal oding (whether normalized or not) the main effets model matrix olumns for an OA (of strength at least 2) are always unorrelated. We write the model matrix for the full model in normalized orthogonal oding as M = (M 0, M,, M n ), () where M 0 is a olumn of + s, M ontains all main effets model matries, and M k is the matrix of ( ) all n k k-fator interation model matries, k = 2,,n. he portion u u k of M k = ( k,, n k+ n ) denotes the model matrix for the partiular k-fator interation indexed by {u,,u k } and is obtained by all produts from one main effets ontrast olumn eah from the k fators in the interation. Note that the normalized orthogonal oding of the main effets implies that all olumns of M k have squared length N for k. Now, on the global sale, the overall number of words of length k an be obtained as the sum of squared olumn averages of M k, i.e., A k = N M k M k N / N 2. Obviously, this sum an be split into ontributions from individual k-fator projetions for more detailed onsiderations, i.e., ( ) A = N = : a u,..., u, (2) 2 k N u... uk u... uk N k k { u,..., uk} { u,..., uk} {,..., n} {,..., n} where a k (u,,u k ) is simply the A k value of the k-fator projetion {u,,u k }. he summands a k (u,,u k ) are alled projetion frequenies. Example. For 3-level fators, normalized polynomial oding has the linear ontrast oeffiients 3 2, 0, 3 2 and the quadrati ontrast oeffiients 2, 2, 2. For the regular design OA(9, 3 3, 2) with the defining relation =A+ (mod 3), the model matrix M has dimensions 9x27, inluding one olumn for M 0, six for M, twelve for M 2 and eight for M 3. Like always, the olumn sum of M 0 is N (here: 9), and like for any orthogonal array, the olumn sums of M and M 2 are 0, whih implies A 0 =, A =A 2 =0. We now take a loser look at M 3, arranging fator A as ( ), fator as ( ) and fator as their sum (mod 3), denoting linear ontrast olumns by the subsript l and quadrati ontrast olumns by the subsript q. hen, 3

4 ontrast A l l l Aq l l A l q l Aq q l A l l q Aq l q A l q q Aq q q M3 = olumn sum Half of the squared olumn sums of M 3 are 243/8 and 8/8, respetively. his implies that the sum of the squared olumn sums is A 3 = a 3 (,2,3) = (4*243/8+4*8/8)/8 = 2. able. A partially onfounded OA(8, 2 3 2, 2) (transposed) A Example 2. able displays the only OA(8, 2 3 2, 2) that annot be obtained as a projetion from the L8 design that was popularized by aguhi (of ourse, this triple is not interesting as a stand-alone design, but as a projetion from a design in more fators only; the aguhi L8 is for onveniene displayed in able 4 below). he 3-level fators are oded like in Example, for the 2-level fator, normalized orthogonal oding is the ustomary -/+ oding. Now, the model matrix M has dimensions 8x8, inluding one olumn for M 0, five for M, eight for M 2 and four for M 3. Again, A 0 =, A =A 2 =0. he squared olumn sums of M 3 are 9 (x), 27 (2x) and 8 (x), respetively. hus, A 3 = a 3 (,2,3) = (9+2*27+8)/324 = 4/9. he projetion frequenies a k (u,,u k ) from Equation (2) are the building bloks for the overall A k. he a (u ) will be instrumental in defining one version of generalized resolution. heorem provides them with an intuitive interpretation. he proof is given in Appendix A. heorem. In an OA(N, s, ), denote by the model matrix for the main effets of a partiular fator {u } {,,n} in normalized orthogonal oding, and let = {u } \ {}. hen, a (u ) is the sum of the 2 -values from the s regression models that explain the olumns of by a full model in the fators from. emark. (i) heorem holds regardless whih fator is singled out for the left-hand side of the model. (ii) he proof simplifies by restrition to normalized orthogonal oding, but the result holds whenever the fator is oded by any set of orthogonal ontrasts, whether normalized or not. (iii) Individual 2 values are oding dependent, but the sum is not. (iv) In ase of normalized orthogonal oding for all fators, the full model in the fators from an be redued to the fator interation 4

5 only, sine the matrix is orthogonal to the model matries for all lower degree effets in the other fators. Example ontinued. he overall a 3 (,2,3) = 2 is the sum of two 2 values whih are, regardless whih fator is singled out as the main effets fator for the left-hand sides of regression. his reflets that the level of eah fator is uniquely determined by the level ombination of the other two fators. Example 2 ontinued. he 2 from regressing the single model matrix olumn of the 2-level fator on the four model matrix olumns for the interation among the two 3-level fators is 4/9. Alternatively, the 2 -values for the regression of the two main effets olumns for fator on the A interation olumns are /9 and 3/9 respetively, whih also yields the sum 4/9 obtained above for a 3 (,2,3). For fator in dummy oding with referene level 0 instead of normalized polynomial oding, the two main effets model matrix olumns for fator have orrelation 0.5; the sum of the 2 values from full models in A and for explaining these two olumns is /3 + /3 = 2/3 a 3 (,2,3) = 4/9. his demonstrates that heorem is not appliable if orthogonal oding (see emark (ii)) is violated. orollary. In an OA(N, s, ), let {u } {,,n}, with s min = min i=,, (s ui ). (i) A fator {u } in s levels is ompletely onfounded by the fators in = {u } \ { }, if and only if a (u ) = s. (ii) a (u ) s min. (iii) If several fators in {u } have s min levels, either all of them are or none of them is ompletely onfounded by the respetive other fators in {u }. (iv) A fator with more than s min levels annot be ompletely onfounded by the other fators in {u }. Part (i) of orollary follows easily from heorem, as a (u ) = s if and only if all 2 values for olumns of the fator main effets model matrix are 00%, i.e. the fator main effets model matrix is ompletely explained by the fators in. Part (ii) follows, beause the sum of 2 values is of ourse bounded by the minimum number of regressions onduted for any single fator, whih is s min. Parts (iii) and (iv) follow diretly from parts (i) and (ii). For symmetri s-level designs, part (ii) of the orollary has already been proven by u, heng and Wu (2004). able 2. An OA(8, 4 2 2, 2) (transposed) A Example 3. For the design of able 2, s min = 2, and a 3 (,2,3) =, i.e. both 2-level fators are ompletely onfounded, while the 4-level fator is only partially onfounded. he individual 2 values for the separate degrees of freedom of the 4-level fator main effet model matrix depend on the oding (e.g. 0.2, 0 and 0.8 for the linear, quadrati and ubi ontrasts in normalized orthogonal polynomial oding), while their sum is, regardless of the hosen orthogonal oding. heorem 2. In an OA(N, s, ), let {u } {,,n} with s min = min i=,, (s u i ). Let {u } with s = s min, = {u } \ {}. Under normalized orthogonal oding denote by the main effets model matrix for fator and by the fator interation model matrix for the fators in. 5

6 If a (u ) = s min, an be orthogonally transformed (rotation and or swithing) suh that s min of its olumns are ollinear to the olumns of. Proof. a (u ) = s min implies all s min regressions of the olumns of on the olumns of have 2 =. hen, eah of the s min olumns an be perfetly mathed by a linear ombination b of the olumns; sine all olumns have the same length, this linear transformation involves rotation and/or swithing only. If neessary, these s min orthogonal linear ombinations an be supplemented by further length-preserving orthogonal linear ombinations so that the dimension of remains intat. /// heorems and 2 are related to anonial orrelation analysis, and the redundany index disussed in that ontext (Stewart and Love 968). In order to make the following omments digestible, a brief definition of anonial orrelation analysis is inluded without going into any tehnial detail about the method; details an e.g. be found in Härdle and Simar (2003, h.4). It will be helpful to think of the olumns of the main effets model matrix of fator as the Y variables and the olumns of the full model matrix in the other fators from the set (exluding the onstant olumn of ones for the interept) as the variables of the following definition and explanation. As it would be unnatural to onsider the model matries from experimental designs as random variables, we diretly define anonial orrelation analysis in terms of data matries and Y (N rows eah) and empirial ovariane matries S xx = * * /(N ), S yy = Y * Y * /(N ), S xy = * Y * /(N ) and S yx = Y * * /(N ), where the supersript * denotes olumnwise entering of a matrix. We do not attempt a minimal definition, but prioritize suitability for our purpose. Note that our S xx and S yy are nonsingular matries, sine the designs we onsider have strength ; the ovariane matrix ( * Y * ) ( * Y * )/(N ) of the ombined set of variables may, however, be singular, whih does not pose a problem to anonial orrelation analysis, even though some aounts request this matrix to be nonsingular. Definition. onsider a set of p -variables and q Y-variables. Let the Nxp matrix and the Nxq matrix Y denote the data matries of N observations, and S xx, S yy, S xy and S yx the empirial ovariane matries obtained from them, with positive definite S xx and S yy. (i) anonial orrelation analysis reates k = min(p, q) pairs of linear ombination vetors u i = a i and v i = Yb i with px oeffiient vetors a i and qx oeffiient vetors b i, i =,,k, suh that a) the u,,u k are unorrelated to eah other b) the v,,v k are unorrelated to eah other ) the pair (u, v ) has the maximum possible orrelation for any pair of linear ombinations of the and Y olumns, respetively d) the pairs (u i, v i ), i=2,,k suessively maximize the remaining orrelation, given the onstraints of a) and b). (ii) he orrelations r i = or(u i, v i ) are alled anonial orrelations, and the u i and v i are alled anonial variates. emark 2. (i) If the matries and Y are entered, i.e. = * and Y = Y *, the u and v vetors also have zero means, and the unorrelatedness in a) and b) is equivalent to orthogonality of the vetors. (ii) It is well-known that the anonial orrelations are the eigenvalues of the matries Q = S xx S xy S yy S yx and Q 2 = S yy S yx S xx S xy (the first min(p, q) eigenvalues of both matries are 6

7 the same; the larger matrix has the appropriate number of additional zeroes), and the a i are the orresponding eigenvetors of Q, the b i the orresponding eigenvetors of Q 2. Aording to the definition, the anonial orrelations are non-negative. It an also be shown that u i and v j, i j, are unorrelated, and orthogonal in ase of entered data matries; thus, the pairs (u i, v i ) deompose the relation between and Y into unorrelated omponents, muh like the prinipal omponents deompose the total variane into unorrelated omponents. In data analysis, anonial orrelation analysis is often used for dimension redution. Here, we retain the full dimensionality. For unorrelated Y variables like the model matrix olumns of in heorem, it is straightforward to see that the sum of the 2 values from regressing eah of the Y variables on all the variables oinides with the sum of the squared anonial orrelations. It is well-known that the anonial orrelations are invariant to arbitrary nonsingular affine transformations applied to the - and Y- variables, whih translate into nonsingular linear transformations applied to the entered - and Y- matries (f. e.g. Härdle and Simar 2003, heorem 4.3). For our appliation, this implies invariane of the anonial orrelations to fator oding. Unfortunately, this invariane property does not hold for the 2 values or their sum: aording to Lazraq and léroux (200, Setion 2) the afore-mentioned redundany index whih is the average 2 value alulated as a (u )/(s ) in the situation of heorem is invariant to linear transformations of the entered matrix, but only to orthonormal transformations of the entered Y matrix or salar multiples thereof. For orrelated Y-variables, the redundany index ontains some overlap between variables, as was already seen for example 2, where the sum of the 2 values from dummy oding exeeded a 3 (,2,3); in that ase, only the average or sum of the squared anonial orrelations yields an adequate measure of the overall explanatory power of the -variables on the Y-variables. Hene, for the ase of arbitrary oding, heorem has to be restated in terms of squared anonial orrelations: heorem 3. In an OA(N, s, ), denote by the model matrix for the main effets of a partiular fator {u } in arbitrary oding, and let = {u } \ {}. hen, a (u ) is the sum of the squared anonial orrelations from a anonial orrelation analysis of the olumns of and the olumns of the full model matrix F in the fators from. Example, ontinued. s min =3, a 3 (,2,3) = 2, i.e. the assumptions of heorems 2 and 3 are fulfilled. oth anonial orrelations must be, beause the sum must be 2. he transformation of from heorem 2 an be obtained from the anonial orrelation analysis: For all fators in the role of Y, v i y i (with y i denoting the i-th olumn of the main effets model matrix of the Y-variables fator) an be used. For the first or seond fator in the role of Y, the orresponding anonial vetors on the side fulfill u q l l q 3 l l 3 q q, u 2 3 l q 3 q l l l q q (or replaed by A for the seond fator in the role of Y), with the indies l and q denoting the normalized linear and quadrati oding introdued above. For the third fator in the role of Y, u 3 A l l + A q l + A l q + 3 A q q, u 2 A l l 3 A l q 3 A q l + A q q. Example, now with dummy oding. When using the design of Example for an experiment with qualitative fators, dummy oding is muh more usual than orthogonal ontrast oding. his example shows how heorem 3 an be applied for arbitrary non-orthogonal oding: A is for A= and 0 7

8 otherwise, A 2 is for A=2 and 0 otherwise, and are oded analogously; interation matrix olumns are obtained as produts of the respetive main effets olumns. he main effet and twofator interation model matrix olumns in this oding do not have olumn means zero and have to be entered first by subtrating /3 or /9, respetively. As anonial orrelations are invariant to affine transformations, dummy oding leads to the same anonial orrelations as the previous normalized orthogonal polynomial oding. We onsider the first fator in the role of Y; the entered model matrix olumns y = A /3 and y 2 = A 2 /3 are orrelated, so that we must not hoose both anonial variates for the Y side proportional to the original variates. One instane of the anonial variates for the Y side is v = y 2, v2 = ( y+ 2y 2) 6 ; these anonial vetors are unique up to rotation only, beause the two anonial orrelations have the same size. he orresponding anonial vetors on the side are obtained from the entered full model matrix (( ),( ),( ),( ),( ),( ),( ),( )) F = as u = ( f f + f + 2f f + f ) 2 and ( + + ) u2 = 2f +f2 f3 2f4 3f5 3f7 3f 8 6, with f j denoting the j-th olumn of F. Note that the anonial vetors u and u 2 now ontain ontributions not only from the interation part of the model matrix but also from the main effets part, i.e. we do indeed need the full model matrix as stated in heorem 3. Example 2, ontinued. s min = 2, a 3 (,2,3) = 4/9, i.e. the assumption of heorem 2 is not fulfilled, the assumption of heorem 3 is. he anonial orrelation using the one olumn main effets model matrix of the 2-level fator A in the role of Y is 2/3, the anonial orrelations using the main effets model matrix for the 3-level fator in the role of Y are 2/3 and 0; in both ases, the sum of the squared anonial orrelations is a 3 (,2,3) = 4/9. For any other oding, for example the dummy oding for fator onsidered earlier, the anonial orrelations remain unhanged (2/3 and 0, respetively), sine they are oding invariant; thus, the sum of the squared anonial orrelations remains 4/9, even though the sum of the 2 values was found to be different. Of ourse, the linear ombination oeffiients for obtaining the anonial variates depend on the oding (see e.g. Härdle and Simar 2003 heorem 4.3). anonial orrelation analysis an also be used to verify that a result analogous to heorem 2 annot be generalized to sets of fators for whih a (u ) < s min. For this, note that the number of non-zero anonial orrelations indiates the dimension of the relationship between the - and the Y- variables. able 3 displays the 2 values from two different orthogonal odings and the squared anonial orrelations from the main effets matrix of the first fator (Y-variables) vs. the full model matrix of the other two fators (-variables) for the ten non-isomorphi GMA OA(32,4 3,2) obtained from Eendebak and Shoen (203). hese designs have one generalized word of length 3, i.e. they are nonregular. here are ases with one, two and three non-zero anonial orrelations, i.e. neither is it generally possible to ollapse the linear dependene into a one-dimensional struture nor does the linear dependene generally involve more than one dimension. 8

9 able 3. Main effets matrix of fator A regressed on full model in fators and for the 0 nonisomorphi GMA OA(32, 4 3, 2) 2 values from polynomial oding 2 values from Helmert oding Squared anonial orrelations Designs L Q A /3 / /8 3/24 / /4 5/2 / ,6,8, /4 5/2 / ,5, /6 7/48 / Generalized resolution efore presenting the new proposals for generalized resolution, we briefly review generalized resolution for symmetri 2-level designs by Deng and ang (999) and ang and Deng (999). For 2- level fators, eah effet has a single degree of freedom (df) only, i.e. all the s in any M k (f. Equation ()) are one-olumn matries. Deng and ang (999) looked at the absolute sums of the olumns of M, whih were termed J-harateristis by ang and Deng (999). Speifially, for a resolution design, these authors introdued G as max J G = +, (3) N where J = N M is the row vetor of the J-harateristis N u u obtained from the n - fator interation model olumns u u. For 2-level designs, it is straightforward to verify the following identities: G = + max a ( u,..., u ) = + max ρ(, ), (4) u u2... u ( u,..., u ) (,..., ) u u where ρ denotes the orrelation; note that the orrelation in (4) does not depend on whih of the u i takes the role of u. Deng and ang (999, prop. 2) proved a very onvining projetion interpretation of their G. Unfortunately, Prop. 4.4 of Diestelkamp and eder (2002), in whih a partiular OA(8, 3 3, 2) is proven to be indeomposable into two OA(9, 3 3, 2), implies that Deng and ang s result annot be generalized to more than two levels. he quantitative approah by Evangelaras et al. (2005, their eq. (4)) generalized the orrelation version of (4) by applying it to single df ontrasts for the quantitative fators. For the qualitative fators onsidered here, any approah based on diret usage of single df ontrasts is not aeptable beause it is oding dependent. he approah for qualitative fators taken by Evangelaras et al. is unreasonable, as will be demonstrated in Example 5. Pang and Liu (200) also proposed a generalized resolution based on omplex ontrasts. For designs with more than 3 levels, permuting levels for one or more fators will lead to different generalized resolutions aording to their definition, whih is unaeptable for qualitative fators. For 2-level designs, their approah boils down to omitting the square root from max a ( u,..., u ) in (4), whih implies that their proposal does not simplify to ( u,..., u ) the well-grounded generalized resolution of Deng and ang (999) / ang and Deng (999) for 2- level designs. his in itself makes their approah unonvining. Example 5 will ompare their approah to ours for 3-level designs. he results from the previous setion an be used to reate two 9

10 adequate generalizations of G for qualitative fators. hese are introdued in the following two definitions. For the first definition, an fator projetion is onsidered as ompletely aliased, whenever all the levels of at least one of the fators are ompletely determined by the level ombination of the other fators. hus, generalized resolution should be equal to, if and only if there is at least one fator projetion with a (u ) = s min. he G defined in Definition 2 guarantees this behavior and fulfills all requirements stated in the introdution: Definition 2. For an OA(N, s, ), ( u,..., u ) a G = + max. { u,..., u } min s {,..., n} u i=,..., i In words, G inreases the resolution by one minus the square root of the worst ase average 2 obtained from any fator projetion, when regressing the main effets olumns in orthogonal oding from a fator with the minimum number of levels on the other fators in the projetion. It is straightforward to see that (4) is a speial ase of the definition, sine the denominator is for 2-level designs. egarding the requirements stated in the introdution, (i) G from Def. 2 is oding invariant beause the a () are oding invariant aording to u and Wu (200). (ii) he tehnique is obviously appliable for symmetri and asymmetri designs alike, and (iii) G < + follows from the resolution, G follows from part (ii) of orollary, G = is equivalent to omplete onfounding in at least one -fator projetion aording to part (i) of orollary. Example 4. he G values for the designs from Examples and 3 are 3 (G=), the G value for the design from Example 2 is = 3. 33, and the G values for all designs from able 3 are = Now, omplete aliasing is onsidered regarding individual degrees of freedom (df). A oding invariant individual df approah onsiders a fator s main effet as ompletely aliased in an fator projetion, whenever there is at least one pair of anonial variates with orrelation one. A projetion is onsidered ompletely aliased, if at least one fator s main effet is ompletely aliased in this individual df sense. Note that it is now possible that fators with the same number of levels an show different extents of individual df aliasing within the same projetion, as will be seen in Example 5 below. Definition 3. For an OA(N, s, ) and tuples (, ) with = {u } \ {}, ( ) G = + max max r ; F ind { u,..., u } {,..., n} { u,..., u } with r ( ; F ) the largest anonial orrelation between the main effets model matrix for fator and the full model matrix of the fators in. In words, G ind is the worst ase onfounding for an individual main effets df in the design that an be obtained by the worst ase oding (whih orresponds to the v vetor assoiated with the worst anonial orrelation). Obviously, G ind is thus a striter riterion than G. Formally, heorem 3 implies that G from Def. 2 an be written as 0

11 G = + max ( ) u rj( u ; F { u }) 2,..., u s 2 j= u,..., u : min isui { u,..., u } {,..., n}. (5) Note that maximization in (5) is over tuples, so that it is ensured that the fator with the minimum number of levels does also get into the first position. omparing (5) with Def. 3, G ind G is obvious, beause r 2 2 annot be smaller than the average over all r i (but an be equal, if all anonial orrelations have the same size). his is stated in a heorem: heorem 4. For G from Def. 2 and G ind from Def. 3, G ind G. emark 3. (i) Under normalized orthogonal oding, the full model matrix F in Definition 3 an again be replaed by the fator interation matrix. (ii) Definition 3 involves alulation of n anonial orrelations ( orrelations for eah fator projetion). In any projetion with at least one 2-level fator, it is suffiient to alulate one single anonial orrelation obtained with an arbitrary 2- level fator in the role of Y, beause this is neessarily the worst ase. Nevertheless, alulation of G ind arries some omputational burden for designs with many fators. Obviously, (4) is a speial ase of G ind, sine the average 2 oinides with the only squared anonial orrelation for projetions of 2-level fators. G ind also fulfills all requirements stated in the introdution: (i) G ind is oding invariant beause the anonial orrelations are invariant to affine transformations of the and Y variables, as was disussed in Setion 2. (ii) he tehnique is obviously appliable for symmetri and asymmetri designs alike, and (iii) G ind < + again follows from the resolution, G ind follows from the properties of orrelations, and G ind = is obviously equivalent to omplete onfounding of at least one main effets ontrast in at least one fator projetion, in the individual df sense disussed above. able 4. he aguhi L8 (transposed) ow ol Example 5. We onsider the three non-isomorphi OA(8, 3 3, 2) that an be obtained as projetions from the well-known aguhi L8 (see able 4) by using olumns 3, 4 and 5 (D ), olumns 2, 3 and 6 (D 2 ) or olumns 2, 4 and 5 (D 3 ). We have A 3 (D ) = 0.5, A 3 (D 2 ) = and A 3 (D 3 ) = 2, and onsequently G(D ) = 3.5, G(D 2 ) = 3.29 and G(D 3 ) = 3. For alulating G ind, the largest anonial orrelations of all fators in the role of Y are needed. hese are all 0.5 for D and all for D 3, suh that G ind = G for these two designs. For D 2, the largest anonial orrelation is with the first fator (from olumn 2 of the L8) in the role of Y, while it is 0. 5 with either of the other two fators in the role of Y;

12 thus, G ind = 3 < G = he ompletely aliased df ontrast of the first fator is the ontrast of the third level vs. the other two levels, whih is apparent from able 5: the ontrast A = 2 vs A in (0,) is fully aliased with the ontrast of one level of vs. the other two, given a partiular level of. egardless of fator oding, this diret aliasing is refleted by a anonial orrelation one for the first anonial variate of the main effets ontrast matrix of fator A. able 5. Frequeny table of olumns 2(=A), 3(=) and 6(=) of the aguhi L8,, = 0 A ,, = A ,, = 2 A Using this example, we now ompare the G introdued here to proposals by Evangelaras et al. (2005) and Pang and Liu (200): he Ges values reported by Evangelaras et al. (2005) for designs D, D 2 and D 3 in the qualitative ase are 3.75, , 3.5, respetively; espeially the 3.5 for the ompletely aliased design D 3 does not make sense. Pang and Liu reported values 3.75, 3.75 and 3, respetively; here, at least the ompletely aliased design D 3 is assigned the value 3. Introduing the square root, as was disussed in onnetion with equation (4), their generalized resolutions beome 3.5, 3.5 and 3, respetively, i.e. they oinide with our G results for designs D and D 3. For design D 2, their value 3.5 is still different from our 3.29 for the following reason: our approah onsiders A 3 = a 3 (,2,3) as a sum of two 2 -values and subtrats the square root of their average or maximum (G or G ind, respetively), while Pang and Liu s approah onsiders it as a sum of 2 3 =8 summands, refleting the potentially different linear ombinations of the three fators in the Galois field sense, the (square root of the) maximum of whih they subtrat from Properties of G Let G be the set of all runs of an s s n full fatorial design, with G = i = s i the ardinality of G. For any design D in N runs for n fators at s,, s n levels, let N x be the number of times that a point x G appears in D. N = N G denotes the average frequeny for eah point of G in the design D. We an measure the goodness of a frational fatorial design D by the uniformity of the design points of D in the set of all points in G, that is, the uniformity of the frequeny distribution N x. One measure, suggested by ang (200) and Ai and Zhang (2004), is the variane V( D ) = G N N = G G N N x x G ( ) x x. Let N = q G + r with nonnegative integer q and r and 0 r < G (often q=0), i.e. r = N mod G is the remainder of N when divided by G. Note that x G N x = N, so V(D) is minimized if and only if eah N x takes values on q or q + for any x G. When r points in G appear q + times and the remaining G r points appear q times, V(D) reahes the minimal value r( G r) / G 2. Ai and Zhang (2004) showed that V(D) is a funtion of GWLP. In partiular, if D has strength n, their 2 result implies that V ( D) = N A ( d). ombining these results, and using the following definition, n n 2

13 we obtain an upper bound for G for some lasses of designs and provide a neessary and suffiient ondition under whih this bound is ahieved. Definition 4 (modified from u 2003). (i) A design D has maximum t-balane, if and only if the possible level ombinations for all projetions onto t olumns our as equally often as possible, i.e. either q or q+ times, where q is an integer suh that N = q G proj + r with G proj the set of all runs for the full fatorial design of eah respetive t-fator-projetion and 0 r < G proj. (ii) An OA(N, s, t ) with n t has weak strength t if and only if it has maximum t- balane. We denote weak strength t as OA(N, s, t ). emark 4. u (2003) did not require strength t in the definition of weak strength t, i.e. the u (2003) definition of weak strength t orresponds to our definition of maximum t-balane. For the frequent ase, for whih all t-fator projetions have q=0 or q= and r=0 in Def. 4 (i), maximum t- balane is equivalent to the absene of repeated runs in any projetion onto t fators. In that ase, maximum t-balane implies maximum k-balane for k > t, and weak strength t is equivalent to strength t with absene of repeated runs in any projetion onto t or more fators. heorem 5. Let D be an OA(N, s s, ). hen A ( D) ( i= i ) r s r, where r is the remainder 2 N when N is divided by s i= i. he equality holds if and only if D has weak strength. As all fator projetions of any OA(N, s, ) fulfill the neessary and suffiient ondition of heorem 5, we have the following orollary: orollary 2. Suppose that an OA(N, s... s n, ) does not exist. hen any OA(N, s, ) has maximum G among all OA(N, s, ). orollary 3. Suppose that an OA(N, s n, ) does not exist. Let D be an OA(N, s n, ). hen ( r) r s G( D) +, where r is the remainder when N is divided by s. he equality holds if 2 N ( s ) and only if D has weak strength. Example 6. () Any projetion onto three 3-level olumns from an OA(8, 6 3 6, 2) has 8 distint runs (q=0, r=n=8) and is an OA of weak strength 3, so it has A 3 =/2 and G = ( 8 2 2) =3.5. (2) Any projetion onto three or more s-level olumns from an OA(s 2, s s+, 2) has G = 3, sine N = r = s 2, so that the upper limit from the orollary beomes G = = 3. Using the following lemma aording to Mukerjee and Wu 995, orollary 3 an be applied to a further lass of designs. Lemma (Mukerjee and Wu 995). For a saturated OA(N, s n s 2 n 2, 2) with n (s ) + n 2 (s 2 ) = N, let δ i (a, b) be the number of oinidenes of two distint rows a and b in the n i olumns of s i levels, for i =, 2. hen s δ (a, b) + s 2 δ 2 (a, b) = n + n 2. onsider a saturated OA(2s 2, (2s) s 2s, 2), where r = N = 2s 2, s = 2s, s 2 = s, n =, n 2 = 2s. From Lemma, we have 2δ (a, b) + δ 2 (a, b) = 2. So any projetion onto three or more s-level olumns has 3

14 no repeated runs, and thus it ahieves the upper limit = 4 ( s 2) ( 2s 2) orollary 3. G aording to orollary 4. For a saturated OA(2s 2, (2s) s 2s, 2), any projetion onto three or more s-level olumns G = 4 s 2 2s 2, whih is optimum among all possible OAs in 2s 2 runs. has ( ) ( ) Example 7. Design of able 3 is isomorphi to a projetion from a saturated OA(32, 8 4 8, 2). A 3 attains the lower bound from heorem 5 (32 (64 32)/32 2 = ), and thus G attains the upper bound 4 (/3) ½ = 3.42 from the orollary. eause of heorem 4, any upper bound for G is of ourse also an upper bound for G ind, i.e. orollaries 3 and 4 also provide upper bounds for G ind. However, for G ind the bounds are not tight in general; for example, G ind = 3 for the design of Example 7 (see also Example 9 in the following setion). utler (2005) previously showed that all projetions onto s-level olumns of OA(s 2, s s+, 2) or OA(2s 2, (2s) s 2s, 2) have GMA among all possible designs. 5. Fator wise G values In Setion 3, two versions of overall generalized resolution were defined: G and G ind. hese take a worst ase perspetive: even if a single projetion in a large design is ompletely onfounded in the ase of mixed level designs or G ind affeting perhaps only one fator within that projetion the overall metri takes the worst ase value. It an therefore be useful to aompany G and G ind by fator speifi summaries. For the fator speifi individual df perspetive, one simply has to omit the maximization over the fators in eah projetion and has to use the fator of interest in the role of Y only. For a fator speifi omplete onfounding perspetive, one has to divide eah projetion s a () value by the fator s df rather than the minimum df, in order to obtain the average 2 value for this partiular fator. his leads to Definition 5. For an OA(N, s, ), define (i) G tot( i) = + a 2 max { u,..., u } s {,..., n} \ {} i i 2 ( i, u,..., u ) (ii) + max r ( ; ) G ind( i) i u... u { i, u,..., u } {,..., n} 2 2 =, with i the model matrix of fator i and u2 u the fator interation model matrix of the fators in {u 2 } in normalized orthogonal oding, and r (Y;) the first anonial orrelation between matries and Y. It is straightforward to verify that G and G ind an be alulated as the respetive minima of the fator speifi G values from Definition 5: heorem 6. For the quantities from Definitions 2, 3 and 5, we have (i) G = min i G tot(i) (ii) G ind = min i G ind(i) Example 8. he aguhi L8 has G = G ind = 3, and the following G ind(i) and G tot(i) values (G ind(i) = G tot(i) for all i): 3.8, 3, 3.29, 3, 3, 3.29, 3.29, When omitting the seond olumn, the remaining seven olumns have G = G ind = 3.8, again with G ind(i) = G tot(i) and the value for all 3- level fators at When omitting the fourth olumn instead, the then remaining seven olumns 4

15 have G = 3.8, G ind = 3, G tot(i) values 3.8, 3.29, 3.29, 3.42, 3.29, 3.29, 3.29 and G ind(i) values the same, exept for the seond olumn, whih has G ind(2) = 3. G from Def. 2 and G ind from Def. 3 are not the only possible generalizations of (4). It is also possible to define a G tot, by delaring only those fator projetions as ompletely onfounded for whih all fators are ompletely onfounded. For this, the fator wise average 2 values for eah projetion also used in G tot(i) need to be onsidered. A projetion is ompletely onfounded, if these are all one, whih an be formalized by requesting their minimum or their average to be one. he average appears more informative, leading to G tot = + max { u,..., u } {,..., n} i= a ( u,..., u ) s u i. (6) It is straightforward to see that G tot G, and that G tot = G for symmetri designs. he asymmetri design of able 2 (Example 3) has G = 3 and G tot = 3 + ( + + 3) 3 = 3. 2 > 3, in spite of the fat that two of its fators are ompletely onfounded. Of ourse, mixed level projetions an never be ompletely onfounded aording to (6), whih is the main reason why we have not pursued this approah. he final example uses the designs of able 3 to show that G ind and the G ind(i) an introdue meaningful differentiation between GMA designs. Example 9. All designs of able 3 had A 3 = and G=3.42. he information provided in able 3 is insuffiient for determining G ind. able 6 provides the neessary information: the largest anonial orrelations are the same regardless whih variable is hosen as the Y variable for seven designs, while they vary with the hoie of the Y variable for three designs. here are five different G ind values for these 0 designs that were not further differentiated by A 3 or G, and in ombination with the G ind(i), seven different strutures an be distinguished. able 6. Largest anonial orrelations, G ind(i) and G ind values for the GMA OA(32, 4 3, 2) r (;23) r (2;3) r (3;2) G ind() G ind(2) G ind(3) G ind he differentiation ahieved by G ind is meaningful, as an be seen by omparing frequeny tables of the first, third and ninth design (see able 7). he first and third design have G ind =3, whih is due to a very regular onfounding pattern: in the first design, dihotomizing eah fator into a 0/ vs. 2/3 design yields a regular resolution III 2-level design (four different runs only), i.e. eah main effet 5

16 ontrast 0/ vs. 2/3 is ompletely onfounded by the two-fator interation of the other two 0/ vs. 2/3 ontrasts; the third design shows this severe onfounding for fator only, whose 0/ vs. 2/3 ontrast able 7. Frequeny tables of designs, 3 and 9 from able 6 Design,, = 0,, =,, = 2,, = 3 A A A A Design 3,, = 0,, =,, = 2,, = 3 A A A A Design 9,, = 0,, =,, = 2,, = 3 A A A A is likewise ompletely onfounded by the interation between fators A and. Design 9 is the best of all GMA designs in terms of G ind. It does not display suh a strong regularity in behavior. G ind treats Designs and 3 alike, although Design is learly more severely affeted than Design 3, whih an be seen from the individual G ind(i). However, as generalized resolution has always taken a worst ase perspetive, this way of handling things is appropriate in this ontext. 6

17 6. Disussion We have provided a statistially meaningful interpretation for the building bloks of GWLP and have generalized generalized resolution by Deng and ang (999) and ang and Deng (999) in two meaningful ways for qualitatitve fators. he omplete onfounding perspetive of G of Definition 2 appears to be more sensible than the individual df perspetive of G ind as a primary riterion. However, G ind provides an interesting new aspet that may provide additional understanding of the struture of OAs and may help in ranking tied designs. he fator wise values of Setion 5 add useful detail. It will be interesting to pursue onepts derived from the building bloks of G tot(i) and G ind(i) for the ranking of mixed level designs. As was demonstrated in Setion 5, G from Def. 2 and G ind from Def. 3 are not the only possible generalizations of (4) for qualitative fators. he alternative given in Equation (6) appears too lenient and has therefore not been pursued. he onept of weak strength deserves further attention: For symmetri designs with weak strength t aording to Def. 4, u (2003, heorem 3) showed that these have minimum moment aberration (MMA) and onsequently GMA (as MMA is equivalent to GMA for symmetri designs) if they also have maximum k-balane for k = t+,,n. In partiular, this implies that an OA(N, s n, t ) with N s t has GMA, beause of emark 4. Here, we showed that designs of the highest possible resolution maximize G if they have weak strength. It is likely that there are further benefiial onsequenes from the onept of weak strength. Appendix A: Proof of heorem Proof. Let M = ( N M ; M ; ), with M k; the model matrix for all k-fator interations, k=,,. he assumption that the resolution of the array is and the hosen orthogonal ontrasts imply M k; = 0 for k <, with as defined in the theorem. Denoting the -fator interation matrix M ; as, the preditions for the olumns of an be written as ( ) ˆ = = N, sine = N I df(). As the olumn averages of ˆ are 0 beause of the oding, the nominators for the 2 values are the diagonal elements of the matrix ˆ ˆ = 2 = NIdf ( ) =. N N Analogously, the orresponding denominators are the diagonal elements of = NI df ( ), whih are all idential to N. hus, the sum of the 2 values is the trae of an be written as tr = N N ( ) ( ) 2 ve 2 ve N 2, whih, (7) where the ve operator staks the olumns of a matrix on top of eah other, i.e. generates a olumn vetor from all elements of a matrix (see e.g. ernstein 2009 for the rule onneting trae to ve). Now, realize that 7

18 ve N ( ) ve ( i, f ) ( i, g ) = N u..., u =, i= ( f, g) where an index pair (i, j) stand for the i-th row and j-th olumn, respetively, and the olumns in ve (w.l.o.g.). hen, are assumed to appear in the order that orresponds to that in ( ) u...,u (7) beomes N u u u u N a ( u..., u ) 2...,..., =, N whih proves the assertion. /// eferenes Ai, M.-Y. and Zhang,.. (2004). Projetion justifiation of generalized minimum aberration for asymmetrial frational fatorial designs. Metrika 60, ernstein, D.S. (2009). Matrix Mathematis: heory, Fats, and Formulas with Appliation to Linear Systems heory. 2 nd ed. Prineton University Press, Prineton. utler,n.a. (2005). Generalised minimum aberration onstrution results for symmetrial orthogonal arrays. iometrika 92, heng, S.W. and Ye, K.Q. (2004). Geometri Isomorphism and minimum aberration for fatorial designs with quantitative fators. Ann. Statist. 32, Deng, L.Y. and ang,. (999). Generalized resolution and minimum aberration riteria for Plakett-urman and other nonregular fatorial designs. Statist. Sinia 9, Diestelkamp, W.S. and eder, J.H. (2002). On the deomposition of orthogonal arrays. Util. Math. 6, Eendebak, P. and Shoen, E. (203). omplete series of non-isomorphi orthogonal arrays. UL: Aessed May 24, 203. Evangelaras, H., Koukouvinos,., Dean, A.M. and Dingus,.A. (2005). Projetion properties of ertain three level orthogonal arrays. Metrika 62, Fries, A.; Hunter, W.G. (980). Minimum aberration 2**(k-p) designs. ehnometris 22, Härdle, W. and Simar, L. (2003). Applied Multivariate Statistial Analysis. Springer, New York. Hotelling, H. (936). elations etween wo Sets of Variates. iometrika 28, Lazraq, A. and léroux,. (200). Statistial Inferene onerning Several edundany Indies. J. Multivariate Anal. 79, Mukerjee,. and Wu,.F.J. (995). On the existene of saturated and nearly saturated asymmetrial orthogonal arrays. Ann. Statist. 23, Pang, F. and Liu, M. Q. (200). Indiator funtion based on omplex ontrasts and its appliation in general fatorial designs. J. Statist. Plann. Inferene 40, Plakett,.L.; urman, J.P. (946). he design of optimum multifatorial experiments. iometrika 33, Stewart, D. and Love, W. (968). A general anonial orrelation index. Psyhologial ulletin 70, ang,. (200), heory of J-harateristis for frational fatorial designs and projetion justifiation of minimum G 2 aberration. iometrika 88, ang,. and Deng, L.Y. (999). Minimum G 2 -aberration for nonregular frational fatorial designs. Ann. Statist. 27, u, H. (2003). Minimum moment aberration for non-regular designs and supersaturated designs. Statist. Sinia 3,