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Equivalence of factorial designs with qualitative and quantitative factors
Journal of Statistical Planning and Inference 142 (2012) 79–85
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Equivalence of factorial designs with qualitative and quantitative factors Tena I. Katsaounis Department of Mathematics, The Ohio State University at Mansfield, 1680 University Drive, Mansfield, OH 44906, USA
a r t i c l e i n f o
abstract
Article history: Received 11 May 2008 Received in revised form 21 June 2011 Accepted 22 June 2011 Available online 14 July 2011
Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed. & 2011 Elsevier B.V. All rights reserved.
Keywords: Combinatorial equivalence Geometric equivalence Deseq algorithm Design isomorphism Indicator function cg-Equivalence
1. Introduction Symmetric fractional factorial designs (SFFDs) are used in designing experiments for studying the effects of multi-level factors. In the last 50 years there have been several studies on methods of detection of equivalence of SFFDs with either qualitative factors (called combinatorial equivalence) or quantitative factors (called geometric equivalence). See, for example, Chen (1992), Lin et al. (1993), Sun et al. (2002), Fang and Ge (2003), Block and Mee (2005), Stufken and Tang (2007), Lin and Sitter (2008), Srivastava and Ding (2010) and Pang and Liu (2010). Equivalent designs have the same statistical properties for estimation of factorial contrasts and for model fitting. However, non-equivalent designs may have the same statistical properties under a particular model, but different properties under different models. Combinatorially equivalent designs can be obtained from one another by reordering the treatment combinations, relabeling the factors and the levels of one or more factors (Clark and Dean, 2001; Ma et al., 2001; Xu, 2003; Katsaounis and Dean, 2008). In the case of geometric equivalence reversal of symbol order is the only permissible relabeling of the levels of one or more factors (Clark and Dean, 2001; Cheng and Ye, 2004; Evangelaras et al., 2005; Katsaounis et al., 2007). Necessary and sufficient conditions for combinatorial equivalence have been given by Clark and Dean (2001) and Katsaounis and Dean (2008), whereas for geometric equivalence by Clark and Dean (2001), Katsaounis et al. (2007) and Cheng and Ye (2004). Many experiments require the study of qualitative and quantitative factors. In this paper equivalence of symmetric fractional factorial designs with both types of factors is discussed. A symmetric fractional factorial design d having f1 qualitative factors, f2 quantitative factors and n runs (SFFD(f1, f2)) is represented by an n f design matrix T d , where the ði,kÞth element is the level at which the kth factor is observed in the ith run (i ¼ 1, . . . ,n,k ¼ 1, . . . ,f ,f ¼ f1 þ f2 ), the levels of
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each factor are coded as 0,1, . . . ,s1 and s Z2 is a prime or a power of a prime. In the following, without loss of generality, is assumed that the first f1 columns of T d are assigned to qualitative factors and the last f2 to quantitative factors, corresponding to the submatrices T cd (n f1 ) and T gd (n f2 ), respectively. The usual randomization applied to runs, factor labels and factor levels of a design before its use leads to the following definition of equivalence: Definition 1.1. Two n f symmetric fractional factorial designs d1 and d2 each having f1 qualitative factors and f2 quantitative factors are equivalent (called cg-equivalent) if one can be obtained from the other by reordering the runs, relabeling the factors of the same type, relabeling the levels of one or more qualitative factors and reversing the levels of one or more quantitative factors. For symmetric multi-level designs (s 4 2) having only qualitative factors cg-equivalence and combinatorial equivalence coincide. Similarly, for designs having only quantitative factors cg-equivalence coincides with geometric equivalence. Note that for two-level designs all three types of equivalence coincide. However, in general, two combinatorially equivalent designs (with s 4 2) might not be cg-equivalent. For example, consider the three-level 36 5 design e14 obtained as the projection with columns (1 2 3 4 6) of an orthogonal array listed as oa.36.13.3.2 on /http://www.research.att.com/ njasS. Let us assume that the first column of the design matrix of e14 is assigned to a qualitative factor while the remaining four to quantitative factors (called design e14.a). Then the design obtained after permuting symbols (0 1 2) to (1 2 0) in the first two columns and to (2 1 0) in the third column is combinatorially equivalent to e14.a but not cg-equivalent. As another example, assume that the first two columns of the design matrix of e14 are assigned to qualitative factors and the last three to quantitative factors (called design e14.b). Then the design obtained using first the column permutation (3 1 2 5 4) and then the symbol permutation (2 1 0) in the fourth column is geometrically equivalent (thus also combinatorially equivalent) to e14.b but not cg-equivalent. On the other hand, designs that are not combinatorially equivalent are not cg-equivalent. This allows one to use screening methods for combinatorial nonequivalence as screening methods for non-cg-equivalence (see Section 4). In the rest of this paper the term equivalence (non-equivalence) refers to cg-equivalence (non-cg-equivalence) unless otherwise specified. A number of alternative necessary and sufficient methods for cg-equivalence, obtained by combining existing necessary and sufficient methods for combinatorial and geometric equivalence, are given in Section 2. In Section 3 a modification of algorithm Deseq2 (Clark and Dean, 2001), called cg-Deseq2, which can be used to detect equivalence or non-equivalence of any two SFFDs(f1, f2) is discussed. In Section 4 the squared centered L2 discrepancy (Ma et al., 2001) and the moment aberration projection criterion (Xu, 2003) are proposed as fast screening methods for detecting non-cg-equivalence of any two SFFDs(f1, f2). Examples that illustrate the performance of cg-Deseq2 and the proposed screens are given in Sections 4.1 and 5. Extensions of the results of this paper to asymmetric designs with qualitative and quantitative factors having different number of levels are discussed in Section 6. Conclusions are stated in Section 7.
2. Necessary and sufficient conditions for equivalence of symmetric designs In this section alternative necessary and sufficient methods for equivalence of any two SFFDs(f1, f2) are given in Theorems 2.1–2.3. Theorem 2.1 was implemented in an algorithm (cg-Deseq2; see Section 3) which can be used to check equivalence or non-equivalence of two designs.
2.1. The cg-CD condition for equivalence Combinatorial equivalence of (n f1 ) submatrices T cd1 and T cd2 and geometric equivalence of (n f2 ) submatrices T gd1 and T gd2 are necessary conditions for equivalence of two SFFDs(f1, f2) d1 and d2. Combinatorial equivalence of T cd1 and T cd2 can be determined by the Hamming distance matrix method of Clark and Dean (2001) (Corollary 2.1, called CD1 condition; see also Katsaounis and Dean, 2008). The (n n) Hamming distance matrix H d of an (n f) design matrix T d has as ði,jÞth element equal to the Hamming distance between the ith and jth row of T d , that is, the number of factors whose levels differ in rows i and j (i,j ¼ 1, . . . ,n). On the other hand, geometric equivalence of T gd1 and T gd2 can be verified using the absolute Euclidean distance matrix method of Clark and Dean (2001) (called gCD1 condition; see also Katsaounis et al., 2007). The ði,jÞth element of the (n n) absolute Euclidean distance matrix Ed of an (n f) design matrix T d is the absolute Euclidean distance between the ith and jth row in T d . H d is invariant under column permutations and relabelings of factor levels, whereas Ed is invariant under column permutations and reversal of factor levels (but neither H d nor Ed is invariant under row permutations). As shown in Theorem 2.1, two designs d1 and d2 are equivalent if there is a row permutation, applied simultaneously to T cd2 and T gd2 , and a column permutation within each type of factors which transform T d2 to T d1 (up to relabelings of the levels of one or more qualitative factors and reversal of the levels of one or more quantitative factors). Theorem 2.1. Two symmetric factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors are equivalent if and only if there is a column permutation fa1 , . . . ,af1 g of f1, . . . ,f1 g of T cd2 , a column permutation faf1 þ 1 , . . . ,af2 g of ff1 þ 1, . . . ,f2 g of T gd2 and a common row permutation matrix R applied simultaneously to T cd2 and T gd2 such that, for all p1 ¼ 0,1,2, . . . ,f1 and
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81
p2 ¼ 0,1,2, . . . ,f2 with 1 rp ¼ p1 þ p2 r f ,f ¼ f1 þ f2 the following holds: fa1 ,...,ap1 g
1g Edfp11 þ 1,...,p2 g ¼ R½H d2 ½H df1,...,p 1
fap
E d2 1
þ 1 ,...,ap2 g
R0 ,
ð1Þ
fi1 ,...,ip1 g where H d fi ,...,i g
T cd
is the Hamming distance matrix based on the columns i1 , . . . ,ip1 of design matrix with p1 qualitative factors p2 p þ1 and Ed 1 is the absolute Euclidean distance matrix based on the columns ip1 þ 1 , . . . ,ip2 of design matrix T gd with p2 quantitative factors (and R½A BR0 ¼ ½RAR0 RBR0 for matrices R,A,B). Proof. Necessity: Follows by applying the necessity part of the proof of CD1 condition to qualitative factors, that of gCD1 condition to quantitative factors, and using the common permutation matrix R and the facts that for any n f design d: Hd ¼
f X
H fig , d
Ed ¼
i¼1
f X
Efig : d
i¼1
Sufficiency: Let ða1 , . . . ,ap1 ,ap1 þ 1 , . . . ,ap2 Þ ¼ ðc1 , . . . ,cf Þ. By condition (1), called cg-CD condition, CD1 condition holds for each column c1 , . . . ,cf1 of T cd2 with column permutation ðc1 , . . . ,cf1 Þ ¼ ða1 , . . . ,ap1 Þ and row permutation given by permutation matrix R. Also by condition (1), gCD1 condition holds for each column cf1 þ 1 , . . . ,cf2 of T gd2 with column permutation ðcf1 þ 1 , . . . ,cf2 Þ ¼ ðap1 þ 1 , . . . ,af2 Þ and the same row permutation matrix R applied simultaneously to T cd2 and T gd2 . & 2.2. The cg-CDK condition for equivalence Katsaounis et al. (2007, Theorem 2.1) showed that a necessary and sufficient condition for geometric equivalence of two symmetric fractional factorial designs d1 and d2 is that the (n k) projections of T d2 of a fixed dimension kð2 1,2, . . . f 1) are geometrically equivalent to the corresponding (n k) projections of T d1 . A similar result was given by Katsaounis and Dean (2008, Theorem 5.1) about combinatorial equivalence. By combining Theorem 2.1 of Katsaounis et al. (2007) and Theorem 5.1 of Katsaounis and Dean (2008) alternative ways of determining equivalence of any two SFFDs(f1, f2) are obtained, as in Theorem 2.2 below. Theorem 2.2. Two symmetric factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors are equivalent if and only if there is a column permutation fa1 , . . . ,af1 g of f1, . . . ,f1 g of T cd2 , a column permutation faf1 þ 1 , . . . ,af2 g of ff1 þ 1, . . . ,f2 g of T gd2 and a common row permutation matrix R applied simultaneously to T cd2 and T gd2 such that, for a given k1, 1 rk1 rf1 1, and a given k2, f1 þ 1r k2 r f2 1, the following holds: fi1 ,...,ik1 g
½H d1
fj1 ,...,jk2 g
Ed1
fai1 ,...,ai
¼ R½H d2
k1
g
faj1 ,...,aj
E d2
k2
g
R0 ,
ð2Þ fi1 ,...,ik1 g
for every selection of fi1 , . . . ,ik1 g from f1, . . . ,f1 g and every selection of fj1 , . . . ,jk2 g from ff1 þ1, . . . ,f2 g, where H d Hamming distance matrix based on the columns i1 , . . . ,ik1 of design matrix
T cd
fj1 ,...,jk2 g
with k1 qualitative factors and Ed
is the is the
absolute Euclidean distance matrix based on the columns j1 , . . . ,jk2 of design matrix T gd with k2 quantitative factors (and R½A BR0 ¼ ½RAR0 RBR0 for matrices R,A,B). Proof. Necessity: Follows by applying the necessity part of Theorem 2.2 of the proof of CD1 condition to qualitative factors, the necessity part of the proof of gCD1 condition to quantitative factors and using the common row permutation matrix R. Sufficiency: Let ða1 , . . . ,af1 ,af1 þ 1 , . . . ,af2 Þ ¼ ðc1 , . . . ,cf Þ. Theorem 5.1 of Katsaounis and Dean (2008) holds for columns 1, . . . ,f1 of T cd2 with column permutation ðc1 , . . . ,cf1 Þ ¼ ða1 , . . . ,af1 Þ and row permutation given by permutation matrix R, as in condition (2) (called cg-CDK condition). In addition, Theorem 2.1 of Katsaounis et al. (2007) holds for columns f1 þ 1, . . . ,f2 of T gd2 with column permutation ðcf1 þ 1 , . . . ,cf2 Þ ¼ ðaf1 þ 1 , . . . ,af2 Þ and the same row permutation matrix R (as in condition (2)) applied simultaneously to T cd2 and T gd2 . & 2.3. The cg-IndCD conditions for equivalence Cheng and Ye (2004) gave a different necessary and sufficient condition for geometric equivalence (Cheng and Ye, 2004, Theorem 3.1) based on the coefficients of the polynomial terms (up to a constant) in the indicator function of a design (see, for example, Fontana et al., 2000; Ye, 2003), known as signed J-characteristics of a design (see Tang, 2001). Their method, called the indicator function method, requires existence of a column permutation and an indicator vector, whose elements determine reversal of levels of one or more factors, which map the signed J-characteristics of d2 into those of d1. For the convenience of the reader, the definition of signed J-characteristics of an (n f) SFFD d with three-level quantitative factors, as given in Katsaounis et al. (2007), is provided below. It extends in a straightforward manner to SFFDs with factors having more than three levels by a suitable definition of the orthogonal contrasts (see Cheng and Ye, 2004). Let t ¼ ðt1 , . . . ,tf Þ be an indicator vector with 0 rtk rs1 for k ¼ 1, . . . ,f and let ht be defined as ð1Þ
ð2Þ
ðf Þ
ht ¼ ht1 ht2 htf ,
ð3Þ
82
T.I. Katsaounis / Journal of Statistical Planning and Inference 142 (2012) 79–85 ðkÞ
ðkÞ
ðkÞ
where denotes Kronecker product and where, for factor k, fh0 ,h1 , . . . ,hs1 g is any set of orthonormal s 1 vectors, with ðkÞ h0 a constant vector. E.g. the following is a set of orthonormal contrasts that can be used for three-level factors with levels coded as 0, 1, 2: 2 1 pffiffi ð1,1,1Þ0 if tk ¼ 0, 6 3 6 p1ffiffi ð1,0,1Þ0 if t ¼ 1, ðkÞ k htk ¼ 6 2 4 p1ffiffi ð1,2,1Þ0 if t k ¼ 2: 6 That is, ht (ta0) in (3) is one of a set of orthonormal factorial contrasts measuring the interaction between all factors corresponding to ti a0, ði ¼ 1, . . . ,f Þ. In general, let H be an sf sf matrix with columns sf =2 ht for all possible indicator vectors t listed in lexicographical order. Then ðH Þ0 H ¼ sf I, where I is the identity matrix. Following Tang (2001) and Ai and Zhang (2004), let J d ¼ ðH Þ0 N d ,
ð4Þ
where N d is a vector whose uth element is the number of times that the uth treatment combination (in lexicographical order) is to be observed in the design (u ¼ 1, . . . ,sf ). The elements of the vector J d , which we may label as Jtd in lexicographical order of t, are called the ‘‘signed J-characteristics’’ of design d. The signed J-characteristics of a design d are invariant under row and column permutations, but not under reversal of factor levels. Thus, a row permutation of T d rearranges the rows of the Hamming distance matrix of T cd but does not affect the J-characteristics of T gd . This allows us to combine the CD1 condition and the indicator function method, as described in Theorem 2.3. Theorem 2.3. Two symmetric factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors are equivalent if and only if there is a column permutation fa1 , . . . ,af1 g of f1, . . . ,f1 g of T cd2 , a column permutation faf1 þ 1 , . . . ,af2 g of ff1 þ 1, . . . ,f2 g of T gd2 , a common row permutation matrix R applied simultaneously to T cd2 and T gd2 , and an indicator vector q ¼ ðqf1 þ 1 , . . . ,qf2 Þ of 0’s and1’s such that, for all p1 ¼ 0,1, . . . ,f1 and p2 ¼ 0,1, . . . ,f2 with 1 rp ¼ p1 þ p2 r f ,f ¼ f1 þf2 , the following holds: fa1 ,...,ap1 g
1g ¼ RH d2 H df1,...,p 1
where and
fi1 ,...,ip1 g Hd
ð5Þ
is the Hamming distance matrix based on the columns i1 , . . . ,ip1 of a design matrix T cd with p1 qualitative factors 0
Jtdf1 þ 1 ,...,tf 1 2
R0 ,
¼@
f2 Y k ¼ f1 þ 1
1 ð1Þqk tak AJtda2 f
1 þ1
ð6Þ
,...,taf
2
for all possible t ¼ ðtf1 þ 1 , . . . ,tf2 Þ with tk ¼ 0,1, . . . ,s1,k ¼ f1 þ 1, . . . ,f2 , and Jtdi
p1 þ 1
based on columns ip1 þ 1 , . . . ,ip2 of a design matrix
T gd
,...,tip
is the signed J-characteristic (Tang, 2001) 2
with p2 quantitative factors.
Proof. Necessity: Follows by applying (a) the necessity part of the proof of CD1 condition to qualitative factors, the row permutation given by permutation matrix R as in (5) and the fact that for any n f design d: Hd ¼
f X
H fig d
i¼1
and (b) the necessity part of the proof of Theorem 3.1 of Cheng and Ye (2004) to quantitative factors, which applies directly since signed J-characteristics are invariant under any row permutation, and in particular under the row permutation given by R as in (5). Sufficiency: Let ða1 , . . . ,ap1 ,ap1 þ 1 , . . . ,ap2 Þ ¼ ðc1 , . . . ,cf Þ. By condition (5), CD1 condition holds for all columns c1 , . . . ,cf1 of T cd2 with column permutation ðc1 , . . . ,cf1 Þ ¼ ða1 , . . . ,ap1 Þ and row permutation given by permutation matrix R. In addition, the signed J-characteristics of the quantitative factors in T gd2 are invariant under any row permutation and thus under the row permutation given by R (as in condition (5)) applied simultaneously to T cd2 and T gd2 . Thus, Theorem 3.1 of Cheng and Ye (2004) holds for columns cf1 þ 1 , . . . ,cf2 of T gd2 with column permutation ðcf1 þ 1 , . . . ,cf2 Þ ¼ ðap1 þ 1 , . . . ,ap2 Þ and reversals as indicated by vector q (see (6)). Conditions (5) and (6) are called cg-IndCD conditions. Note that the conditions in Theorem 2.3 require a row permutation, a column permutation within each type of factors and a vector that indicates reversal of levels of one or more quantitative factors which map d2 to d1, up to relabelings of one or more qualitative factors. & 3. The cg-Deseq2 algorithm An implementation of Theorem 2.1 in a Fortran program which can be used to check equivalence or non-equivalence of any two SFFDs(f1, f2), called cg-Deseq2, is obtainable from the author. It is a modification of Deseq2 algorithm (see Clark and Dean, 2001; Katsaounis and Dean, 2008). The algorithm first searches for a row permutation that transforms the Hamming distance matrix of T cd2 to the Hamming distance matrix of T cd1 . If such a row permutation is found, then it searches for
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a column permutation so that condition (1) of Theorem 2.1 is satisfied. The algorithm does an efficient search for such permutations, similar to that of Deseq2; for a detailed description of Deseq2 see Clark and Dean (2001). Note that an exhaustive search demands the investigation of up to f1 ! f2 ! column permutations, n! row permutations and sf1 2f2 column element permutations and column element reversals. If two designs d1 and d2 are equivalent cg-Deseq2 gives a row permutation and a column permutation (within each type of factors) which transform T d2 to T d1 , up to a set of relabelings of one or more quantitative factors and reversal of levels of one or more quantitative factors which can be found by inspection. If an initial row permutation that transforms the Hamming distance of T cd2 to the Hamming distance of T cd1 is not found or a row and a column permutation which satisfy condition (1) of Theorem 2.1 does not exist then d1 and d2 are declared non-equivalent. The performance of cg-Deseq2 in detecting equivalence or non-equivalence of two SFFDs(f1, f2) is examined in Sections 4.1 and 5 via examples. In each example, 100 random row and column permutations (within each type of factors) and level relabelings (reversals) within one or more columns representing qualitative (quantitative) factors were made of one design (unless specified otherwise), while the other design remained ‘‘fixed’’. The minimum and maximum running times to the nearest second were recorded and results are discussed. The algorithm was written in Fortran 90 and run on Linux machines, running Red Hat Enterprise Linux 5.4. 4. Screening methods for non-equivalence Checking for either combinatorial non-equivalence of submatrices T cd1 and T cd2 or geometric non-equivalence of T gd1 and T gd2 is one way to check for non-equivalence of d1 and d2 (see Section 2.1). Another way is by applying a Hamming distance based screening method to qualitative factors and a Euclidean distance based screening method to quantitative factors and using a common row permutation (see, for example, the ‘‘necessity’’ part of Theorem 2.1; see also Katsaounis et al., 2007 and Katsaounis and Dean, 2008, for existing methods). Alternatively, a screening method for detecting combinatorial non-equivalence of d1 and d2 can also be used as a screening method for non-cg-equivalence, since combinatorially non-equivalent designs are not cg-equivalent. This approach is examined below. Following the recommendation of Katsaounis and Dean (2008), the squared centered L2 discrepancy (Ma et al., 2001) (called CD22 ) and the moment aberration projection method (Xu, 2003) (called momp) are evaluated in Section 4.1 as fast screens for non-equivalence of any two SFFDs(f1, f2). Note that Deseq2 (Clark and Dean, 2001) can be used as a screening method for designs with qualitative and quantitative factors, but for large designs could be very slow (as for the pair of designs (ssd.18.3.12, ssd.18.3.12.b) and (sloa25617, sloa25617b) discussed in Section 4.1; see also Katsaounis and Dean, 2008). 4.1. Detection of non-equivalence Three and four-level designs were used to evaluate the performance of cg-Deseq2, CD22 and momp algorithms. The following examples illustrate that cg-Deseq2 can detect non-equivalence quickly for designs with a small number of factors and runs. Occasionally though it can be slow depending on the size and the structure of the designs. In such cases the screening criteria for combinatorial non-equivalence CD22 and momp can be useful in detecting non-equivalence of two SFFDs(f1, f2). For the pair of 36 5 orthogonal arrays e14 (see Section 1) and e15, obtained as the projection with columns (1 2 3 4 7) of the orthogonal array listed as oa.36.13.3.2 on /http://www.research.att.com/ njasS, cg-Deseq2 with f1 ¼1 (i.e. with the first column being a qualitative factor and the remaining four quantitative factors) detects non-equivalence in less than 0.25 s; similar results hold for f1 ¼2, 3, 4 (i.e. f2 ¼ 3, 2, 1). In this example, in all four cases, screening methods CD22 and momp fail to detect combinatorial non-equivalence and thus non-equivalence. For the supersaturated 9 10 designs ssd1 and ssd2 (see Katsaounis et al., 2007, Table 4) cg-Deseq2 with f1 ¼5, 7 detects non-equivalence in less than 0.13 s. In this example, CD22 and momp also detect combinatorial non-equivalence very quickly (in less than 0.13 s). Results are similar for the supersaturated three-level 18 12 designs ssd.18.3.12 and ssd.18.3.12.b (from http://www.stat.ucla.edu/ hqxu/pub/ssd/ssd18x12h.txt, obtained via Theorems 8 and 9, respectively); cg-Deseq2 with f1 ¼1,6 detects non-equivalence very quickly and CD22 and momp also detect combinatorial non-equivalence very quickly (in less than 0.5 s). As an example that involves larger designs, the 81 40 orthogonal arrays d8140a and d8140b (see Katsaounis and Dean, 2008) were examined; cg-Deseq2 with f1 ¼15 and 30 detects non-equivalence in less than 0.11 s; CD22 and momp detect combinatorial non-equivalence in less than 0.20 s. As another example, the 256 17 four-level designs sloa25617 and sloa25617b (see Katsaounis et al., 2007, Table 6) were examined. However, cg-Deseq2 with f1 ¼8 ran at least 175,877 s for one iteration; this is considerably slower than CD22 and momp which took less than 0.20 s to detect combinatorial non-equivalence. 5. Detection of equivalence The examples in this section illustrate that cg-Deseq2 can sometimes be slow, depending on the size and structure of the designs, but is the only method capable of detecting equivalence of two SFFDs(f1, f2).
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For the three-level 36 5 orthogonal array e14 (see Section 1) and the equivalent designs obtained from e14 by 100 random row permutations, column permutations, level permutations of qualitative factors and level reversals of quantitative factors, cg-Deseq2 with f1 ¼1 and f2 ¼4 detects equivalence in less than 0.20 s, in all cases. Similar results are obtained with f1 ¼ 2, 3, 4. As another example, the 18 7 three-level orthogonal arrays dp11837 and dp41837 (see Katsaounis and Dean, 2008, Table 5) were considered; cg-Deseq2 with f1 ¼ 6 runs with minimum time 16 s and maximum time 18 s; CD22 and momp run somewhat faster, however, they cannot guarantee equivalence. Results are similar for f1 ¼2, 3, 4, 5. For the 36 8 three-level orthogonal arrays sl8 and wk8 (see Katsaounis et al., 2007, Table 1) cg-Deseq2 with f1 ¼3, 4, 5, 6, 7 detects equivalence very fast (in less than 7 s); however cg-Deseq2 with f1 ¼2 is considerably slower (with running time several days for one iteration). Thus, depending on the structure of the design, cg-Deseq2 can be slow. Screening methods CD22 and momp are fast in all cases (with f1 ¼2, 3, 4, 5, 6, 7) but unable to guarantee equivalence. 6. Asymmetric designs with qualitative and quantitative factors An n f fractional factorial design d is called asymmetric when the levels of the factors are 0,1, . . . ,sk 1,k ¼ 1, . . . f , where sk is a prime or a power of a prime (k ¼ 1, . . . ,f ). Randomization of an asymmetric design with qualitative and quantitative factors leads to the following definition of equivalence. Two n f asymmetric fractional factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors (ASFFDs(f1, f2)) are equivalent (called cg-equivalent) if one can be obtained from the other by reordering the runs, relabeling the factors of the same type and number of levels, relabeling the levels of the qualitative factors with the same number of levels and reversing the levels of the quantitative factors with the same number of levels. Theorems 2.1 and 2.2 give necessary and sufficient conditions for equivalence of any two ASFFDs(f1, f2), provided that column permutations are within factors of the same type and number of levels. Theorem 2.3 also gives necessary and sufficient conditions for equivalence of any two ASFFDs(f1, f2), provided that column permutations are within factors of the same type and number of levels (in (5) and (6)) and (in (6)) reversals of levels are within quantitative factors with the same number of levels (and with a suitable selection of orthonormal contrasts for the quantitative factors; see Cheng and Ye, 2004). Algorithm cg-Deseq2 can be modified to check equivalence or non-equivalence of any two ASFFDs(f1, f2). Algorithms CD22 and momp can be used as screens for non-equivalence of two ASFFDs(f1, f2) with no modification. 7. Conclusions Algorithm cg-Deseq2, an implementation of Theorem 2.1, can be used to detect equivalence or non-equivalence of any two symmetric fractional factorial designs d1 and d2 with qualitative and quantitative factors, up to level permutations of qualitative factors and reversal of levels of quantitative factors. If two designs are equivalent it returns a row and column permutation that transform the design matrix of d2 to the design matrix of d1. As examples showed, depending on the size and structure of the designs, occasionally cg-Deseq2 can be slow (Sections 4.1 and 5). It is conjectured that a faster algorithm could be obtained by implementing Theorem 2.2, since condition (2) involves establishing only equivalence of projections of a dimension (k) smaller than the dimension (f) of the design matrices. On the other hand, an implementation of Theorem 2.3 would lead to a slower algorithm, since in addition to a search for a row permutation and a column permutation within each type of factors, a search for a vector that indicates reversal of levels of one or more quantitative factors is required (cf. Katsaounis et al., 2007, in their application of Theorem 3.1 of Cheng and Ye, 2004). As fast screens for non-equivalence of designs with qualitative and quantitative factors, the screening methods for combinatorial non-equivalence squared centered L2 discrepancy (Ma et al., 2001) and moment aberration projection (Xu, 2003) (see Section 4.1) can be used prior to cg-Deseq2. Note that these algorithms can sometimes be very slow too (see Katsaounis and Dean, 2008). Theorems 2.1–2.3 can be extended to obtain necessary and sufficient conditions for asymmetric fractional factorial designs with qualitative and quantitative factors. Algorithm cg-Deseq2 can be modified to handle factors with different number of levels.
Acknowledgments The author would like to sincerely thank the referees for their valuable comments. References Ai, M.Y., Zhang, R.C., 2004. Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285. Block, R.M., Mee, R.W., 2005. Resolution IV designs with 128 runs. J. Quality Technol. 37, 282–293. Chen, J., 1992. Some results on 2nk fractional factorial designs and search for minimum aberration designs. Ann. Statist. 20, 2124–2141. Cheng, S.W., Ye, K., 2004. Geometric isomorphism and minimum aberration for factorial designs with quantitative factors. Ann. Statist. 32, 2168–2185. Clark, J.B., Dean, A.M., 2001. Equivalence of fractional factorial designs. Statist. Sinica 11, 537–547.
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Evangelaras, H., Koukouvinos, C., Dean, A.M., Dingus, C.A., 2005. Projection properties of certain three level orthogonal arrays. Metrika 62, 241–257. Fang, K.T., Ge, G., 2003. A sensitive algorithm for detecting the inequivalence of Hadamard matrices. Math. Comput. 73, 843–851. Fontana, R., Pistone, G., Rogantin, M.P., 2000. Classification of two-level factorial designs. J. Statist. Plann. Inference 87, 149–172. Katsaounis, T.I., Dean, A.M., 2008. A survey and evaluation of methods for determination of combinatorial equivalence of factorial designs. J. Statist. Plann. Inference 138, 245–258. Katsaounis, T.I., Dingus, C.A., Dean, A.M., 2007. On the geometric equivalence and non-equivalence of symmetric factorial designs. J. Statist. Theory Practice 1 (1), 101–105. Lin, C., Wallis, W.D., Zhu, L., 1993. Generalized 4-profiles of Hadamard matrices. J. Combin. 18, 397–400. Lin, C.D., Sitter, R.R., 2008. An isomorphism check for two-level fractional factorial designs. J. Statist. Plann. Inference 138, 1085–1101. Ma, C.X., Fang, K.T., Lin, D.K.J., 2001. On the isomorphism of fractional factorial designs. J. Complexity 17, 86–97. Pang, F., Liu, M.-Q., 2010. Indicator function based on complex contrasts and its application in general factorial designs. J. Statist. Plann. Inference 140, 189–197. Srivastava, A.K., Ding, Y., 2010. Graph based isomorph-free generation of two-level regular fractional factorial designs. J. Statist. Plann. Inference 140, 169–179. Stufken, J., Tang, B., 2007. Complete enumeration of two-level orthogonal arrays of strength d with dþ 2 constraints. Ann. Statist. 4, 423–430. Sun, D.X., Li, W., Ye, K.Q., 2002. An Algorithm for Sequentially Constructing Non-isomorphic Orthogonal Designs and its Applications. Technical Report, Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, SUNYSB-AMS-02-13. Tang, B., 2001. Theory of J-characteristic for fractional factorial designs and projection justification of minimum G2-aberration. Biometrika 88, 401–407. Xu, H., 2003. Minimum moment aberration for nonregular designs and supersaturated designs. Statist. Sinica 13, 691–708. Ye, K., 2003. Indicator function and its application in two level factorial designs. Ann. Statist. 31, 984–994.
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Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Equivalence of factorial designs with qualitative and quantitative factors Tena I. Katsaounis Department of Mathematics, The Ohio State University at Mansfield, 1680 University Drive, Mansfield, OH 44906, USA
a r t i c l e i n f o
abstract
Article history: Received 11 May 2008 Received in revised form 21 June 2011 Accepted 22 June 2011 Available online 14 July 2011
Two symmetric fractional factorial designs with qualitative and quantitative factors are equivalent if the design matrix of one can be obtained from the design matrix of the other by row and column permutations, relabeling of the levels of the qualitative factors and reversal of the levels of the quantitative factors. In this paper, necessary and sufficient methods of determining equivalence of any two symmetric designs with both types of factors are given. An algorithm used to check equivalence or non-equivalence is evaluated. If two designs are equivalent the algorithm gives a set of permutations which map one design to the other. Fast screening methods for non-equivalence are considered. Extensions of results to asymmetric fractional factorial designs with qualitative and quantitative factors are discussed. & 2011 Elsevier B.V. All rights reserved.
Keywords: Combinatorial equivalence Geometric equivalence Deseq algorithm Design isomorphism Indicator function cg-Equivalence
1. Introduction Symmetric fractional factorial designs (SFFDs) are used in designing experiments for studying the effects of multi-level factors. In the last 50 years there have been several studies on methods of detection of equivalence of SFFDs with either qualitative factors (called combinatorial equivalence) or quantitative factors (called geometric equivalence). See, for example, Chen (1992), Lin et al. (1993), Sun et al. (2002), Fang and Ge (2003), Block and Mee (2005), Stufken and Tang (2007), Lin and Sitter (2008), Srivastava and Ding (2010) and Pang and Liu (2010). Equivalent designs have the same statistical properties for estimation of factorial contrasts and for model fitting. However, non-equivalent designs may have the same statistical properties under a particular model, but different properties under different models. Combinatorially equivalent designs can be obtained from one another by reordering the treatment combinations, relabeling the factors and the levels of one or more factors (Clark and Dean, 2001; Ma et al., 2001; Xu, 2003; Katsaounis and Dean, 2008). In the case of geometric equivalence reversal of symbol order is the only permissible relabeling of the levels of one or more factors (Clark and Dean, 2001; Cheng and Ye, 2004; Evangelaras et al., 2005; Katsaounis et al., 2007). Necessary and sufficient conditions for combinatorial equivalence have been given by Clark and Dean (2001) and Katsaounis and Dean (2008), whereas for geometric equivalence by Clark and Dean (2001), Katsaounis et al. (2007) and Cheng and Ye (2004). Many experiments require the study of qualitative and quantitative factors. In this paper equivalence of symmetric fractional factorial designs with both types of factors is discussed. A symmetric fractional factorial design d having f1 qualitative factors, f2 quantitative factors and n runs (SFFD(f1, f2)) is represented by an n f design matrix T d , where the ði,kÞth element is the level at which the kth factor is observed in the ith run (i ¼ 1, . . . ,n,k ¼ 1, . . . ,f ,f ¼ f1 þ f2 ), the levels of
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each factor are coded as 0,1, . . . ,s1 and s Z2 is a prime or a power of a prime. In the following, without loss of generality, is assumed that the first f1 columns of T d are assigned to qualitative factors and the last f2 to quantitative factors, corresponding to the submatrices T cd (n f1 ) and T gd (n f2 ), respectively. The usual randomization applied to runs, factor labels and factor levels of a design before its use leads to the following definition of equivalence: Definition 1.1. Two n f symmetric fractional factorial designs d1 and d2 each having f1 qualitative factors and f2 quantitative factors are equivalent (called cg-equivalent) if one can be obtained from the other by reordering the runs, relabeling the factors of the same type, relabeling the levels of one or more qualitative factors and reversing the levels of one or more quantitative factors. For symmetric multi-level designs (s 4 2) having only qualitative factors cg-equivalence and combinatorial equivalence coincide. Similarly, for designs having only quantitative factors cg-equivalence coincides with geometric equivalence. Note that for two-level designs all three types of equivalence coincide. However, in general, two combinatorially equivalent designs (with s 4 2) might not be cg-equivalent. For example, consider the three-level 36 5 design e14 obtained as the projection with columns (1 2 3 4 6) of an orthogonal array listed as oa.36.13.3.2 on /http://www.research.att.com/ njasS. Let us assume that the first column of the design matrix of e14 is assigned to a qualitative factor while the remaining four to quantitative factors (called design e14.a). Then the design obtained after permuting symbols (0 1 2) to (1 2 0) in the first two columns and to (2 1 0) in the third column is combinatorially equivalent to e14.a but not cg-equivalent. As another example, assume that the first two columns of the design matrix of e14 are assigned to qualitative factors and the last three to quantitative factors (called design e14.b). Then the design obtained using first the column permutation (3 1 2 5 4) and then the symbol permutation (2 1 0) in the fourth column is geometrically equivalent (thus also combinatorially equivalent) to e14.b but not cg-equivalent. On the other hand, designs that are not combinatorially equivalent are not cg-equivalent. This allows one to use screening methods for combinatorial nonequivalence as screening methods for non-cg-equivalence (see Section 4). In the rest of this paper the term equivalence (non-equivalence) refers to cg-equivalence (non-cg-equivalence) unless otherwise specified. A number of alternative necessary and sufficient methods for cg-equivalence, obtained by combining existing necessary and sufficient methods for combinatorial and geometric equivalence, are given in Section 2. In Section 3 a modification of algorithm Deseq2 (Clark and Dean, 2001), called cg-Deseq2, which can be used to detect equivalence or non-equivalence of any two SFFDs(f1, f2) is discussed. In Section 4 the squared centered L2 discrepancy (Ma et al., 2001) and the moment aberration projection criterion (Xu, 2003) are proposed as fast screening methods for detecting non-cg-equivalence of any two SFFDs(f1, f2). Examples that illustrate the performance of cg-Deseq2 and the proposed screens are given in Sections 4.1 and 5. Extensions of the results of this paper to asymmetric designs with qualitative and quantitative factors having different number of levels are discussed in Section 6. Conclusions are stated in Section 7.
2. Necessary and sufficient conditions for equivalence of symmetric designs In this section alternative necessary and sufficient methods for equivalence of any two SFFDs(f1, f2) are given in Theorems 2.1–2.3. Theorem 2.1 was implemented in an algorithm (cg-Deseq2; see Section 3) which can be used to check equivalence or non-equivalence of two designs.
2.1. The cg-CD condition for equivalence Combinatorial equivalence of (n f1 ) submatrices T cd1 and T cd2 and geometric equivalence of (n f2 ) submatrices T gd1 and T gd2 are necessary conditions for equivalence of two SFFDs(f1, f2) d1 and d2. Combinatorial equivalence of T cd1 and T cd2 can be determined by the Hamming distance matrix method of Clark and Dean (2001) (Corollary 2.1, called CD1 condition; see also Katsaounis and Dean, 2008). The (n n) Hamming distance matrix H d of an (n f) design matrix T d has as ði,jÞth element equal to the Hamming distance between the ith and jth row of T d , that is, the number of factors whose levels differ in rows i and j (i,j ¼ 1, . . . ,n). On the other hand, geometric equivalence of T gd1 and T gd2 can be verified using the absolute Euclidean distance matrix method of Clark and Dean (2001) (called gCD1 condition; see also Katsaounis et al., 2007). The ði,jÞth element of the (n n) absolute Euclidean distance matrix Ed of an (n f) design matrix T d is the absolute Euclidean distance between the ith and jth row in T d . H d is invariant under column permutations and relabelings of factor levels, whereas Ed is invariant under column permutations and reversal of factor levels (but neither H d nor Ed is invariant under row permutations). As shown in Theorem 2.1, two designs d1 and d2 are equivalent if there is a row permutation, applied simultaneously to T cd2 and T gd2 , and a column permutation within each type of factors which transform T d2 to T d1 (up to relabelings of the levels of one or more qualitative factors and reversal of the levels of one or more quantitative factors). Theorem 2.1. Two symmetric factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors are equivalent if and only if there is a column permutation fa1 , . . . ,af1 g of f1, . . . ,f1 g of T cd2 , a column permutation faf1 þ 1 , . . . ,af2 g of ff1 þ 1, . . . ,f2 g of T gd2 and a common row permutation matrix R applied simultaneously to T cd2 and T gd2 such that, for all p1 ¼ 0,1,2, . . . ,f1 and
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81
p2 ¼ 0,1,2, . . . ,f2 with 1 rp ¼ p1 þ p2 r f ,f ¼ f1 þ f2 the following holds: fa1 ,...,ap1 g
1g Edfp11 þ 1,...,p2 g ¼ R½H d2 ½H df1,...,p 1
fap
E d2 1
þ 1 ,...,ap2 g
R0 ,
ð1Þ
fi1 ,...,ip1 g where H d fi ,...,i g
T cd
is the Hamming distance matrix based on the columns i1 , . . . ,ip1 of design matrix with p1 qualitative factors p2 p þ1 and Ed 1 is the absolute Euclidean distance matrix based on the columns ip1 þ 1 , . . . ,ip2 of design matrix T gd with p2 quantitative factors (and R½A BR0 ¼ ½RAR0 RBR0 for matrices R,A,B). Proof. Necessity: Follows by applying the necessity part of the proof of CD1 condition to qualitative factors, that of gCD1 condition to quantitative factors, and using the common permutation matrix R and the facts that for any n f design d: Hd ¼
f X
H fig , d
Ed ¼
i¼1
f X
Efig : d
i¼1
Sufficiency: Let ða1 , . . . ,ap1 ,ap1 þ 1 , . . . ,ap2 Þ ¼ ðc1 , . . . ,cf Þ. By condition (1), called cg-CD condition, CD1 condition holds for each column c1 , . . . ,cf1 of T cd2 with column permutation ðc1 , . . . ,cf1 Þ ¼ ða1 , . . . ,ap1 Þ and row permutation given by permutation matrix R. Also by condition (1), gCD1 condition holds for each column cf1 þ 1 , . . . ,cf2 of T gd2 with column permutation ðcf1 þ 1 , . . . ,cf2 Þ ¼ ðap1 þ 1 , . . . ,af2 Þ and the same row permutation matrix R applied simultaneously to T cd2 and T gd2 . & 2.2. The cg-CDK condition for equivalence Katsaounis et al. (2007, Theorem 2.1) showed that a necessary and sufficient condition for geometric equivalence of two symmetric fractional factorial designs d1 and d2 is that the (n k) projections of T d2 of a fixed dimension kð2 1,2, . . . f 1) are geometrically equivalent to the corresponding (n k) projections of T d1 . A similar result was given by Katsaounis and Dean (2008, Theorem 5.1) about combinatorial equivalence. By combining Theorem 2.1 of Katsaounis et al. (2007) and Theorem 5.1 of Katsaounis and Dean (2008) alternative ways of determining equivalence of any two SFFDs(f1, f2) are obtained, as in Theorem 2.2 below. Theorem 2.2. Two symmetric factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors are equivalent if and only if there is a column permutation fa1 , . . . ,af1 g of f1, . . . ,f1 g of T cd2 , a column permutation faf1 þ 1 , . . . ,af2 g of ff1 þ 1, . . . ,f2 g of T gd2 and a common row permutation matrix R applied simultaneously to T cd2 and T gd2 such that, for a given k1, 1 rk1 rf1 1, and a given k2, f1 þ 1r k2 r f2 1, the following holds: fi1 ,...,ik1 g
½H d1
fj1 ,...,jk2 g
Ed1
fai1 ,...,ai
¼ R½H d2
k1
g
faj1 ,...,aj
E d2
k2
g
R0 ,
ð2Þ fi1 ,...,ik1 g
for every selection of fi1 , . . . ,ik1 g from f1, . . . ,f1 g and every selection of fj1 , . . . ,jk2 g from ff1 þ1, . . . ,f2 g, where H d Hamming distance matrix based on the columns i1 , . . . ,ik1 of design matrix
T cd
fj1 ,...,jk2 g
with k1 qualitative factors and Ed
is the is the
absolute Euclidean distance matrix based on the columns j1 , . . . ,jk2 of design matrix T gd with k2 quantitative factors (and R½A BR0 ¼ ½RAR0 RBR0 for matrices R,A,B). Proof. Necessity: Follows by applying the necessity part of Theorem 2.2 of the proof of CD1 condition to qualitative factors, the necessity part of the proof of gCD1 condition to quantitative factors and using the common row permutation matrix R. Sufficiency: Let ða1 , . . . ,af1 ,af1 þ 1 , . . . ,af2 Þ ¼ ðc1 , . . . ,cf Þ. Theorem 5.1 of Katsaounis and Dean (2008) holds for columns 1, . . . ,f1 of T cd2 with column permutation ðc1 , . . . ,cf1 Þ ¼ ða1 , . . . ,af1 Þ and row permutation given by permutation matrix R, as in condition (2) (called cg-CDK condition). In addition, Theorem 2.1 of Katsaounis et al. (2007) holds for columns f1 þ 1, . . . ,f2 of T gd2 with column permutation ðcf1 þ 1 , . . . ,cf2 Þ ¼ ðaf1 þ 1 , . . . ,af2 Þ and the same row permutation matrix R (as in condition (2)) applied simultaneously to T cd2 and T gd2 . & 2.3. The cg-IndCD conditions for equivalence Cheng and Ye (2004) gave a different necessary and sufficient condition for geometric equivalence (Cheng and Ye, 2004, Theorem 3.1) based on the coefficients of the polynomial terms (up to a constant) in the indicator function of a design (see, for example, Fontana et al., 2000; Ye, 2003), known as signed J-characteristics of a design (see Tang, 2001). Their method, called the indicator function method, requires existence of a column permutation and an indicator vector, whose elements determine reversal of levels of one or more factors, which map the signed J-characteristics of d2 into those of d1. For the convenience of the reader, the definition of signed J-characteristics of an (n f) SFFD d with three-level quantitative factors, as given in Katsaounis et al. (2007), is provided below. It extends in a straightforward manner to SFFDs with factors having more than three levels by a suitable definition of the orthogonal contrasts (see Cheng and Ye, 2004). Let t ¼ ðt1 , . . . ,tf Þ be an indicator vector with 0 rtk rs1 for k ¼ 1, . . . ,f and let ht be defined as ð1Þ
ð2Þ
ðf Þ
ht ¼ ht1 ht2 htf ,
ð3Þ
82
T.I. Katsaounis / Journal of Statistical Planning and Inference 142 (2012) 79–85 ðkÞ
ðkÞ
ðkÞ
where denotes Kronecker product and where, for factor k, fh0 ,h1 , . . . ,hs1 g is any set of orthonormal s 1 vectors, with ðkÞ h0 a constant vector. E.g. the following is a set of orthonormal contrasts that can be used for three-level factors with levels coded as 0, 1, 2: 2 1 pffiffi ð1,1,1Þ0 if tk ¼ 0, 6 3 6 p1ffiffi ð1,0,1Þ0 if t ¼ 1, ðkÞ k htk ¼ 6 2 4 p1ffiffi ð1,2,1Þ0 if t k ¼ 2: 6 That is, ht (ta0) in (3) is one of a set of orthonormal factorial contrasts measuring the interaction between all factors corresponding to ti a0, ði ¼ 1, . . . ,f Þ. In general, let H be an sf sf matrix with columns sf =2 ht for all possible indicator vectors t listed in lexicographical order. Then ðH Þ0 H ¼ sf I, where I is the identity matrix. Following Tang (2001) and Ai and Zhang (2004), let J d ¼ ðH Þ0 N d ,
ð4Þ
where N d is a vector whose uth element is the number of times that the uth treatment combination (in lexicographical order) is to be observed in the design (u ¼ 1, . . . ,sf ). The elements of the vector J d , which we may label as Jtd in lexicographical order of t, are called the ‘‘signed J-characteristics’’ of design d. The signed J-characteristics of a design d are invariant under row and column permutations, but not under reversal of factor levels. Thus, a row permutation of T d rearranges the rows of the Hamming distance matrix of T cd but does not affect the J-characteristics of T gd . This allows us to combine the CD1 condition and the indicator function method, as described in Theorem 2.3. Theorem 2.3. Two symmetric factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors are equivalent if and only if there is a column permutation fa1 , . . . ,af1 g of f1, . . . ,f1 g of T cd2 , a column permutation faf1 þ 1 , . . . ,af2 g of ff1 þ 1, . . . ,f2 g of T gd2 , a common row permutation matrix R applied simultaneously to T cd2 and T gd2 , and an indicator vector q ¼ ðqf1 þ 1 , . . . ,qf2 Þ of 0’s and1’s such that, for all p1 ¼ 0,1, . . . ,f1 and p2 ¼ 0,1, . . . ,f2 with 1 rp ¼ p1 þ p2 r f ,f ¼ f1 þf2 , the following holds: fa1 ,...,ap1 g
1g ¼ RH d2 H df1,...,p 1
where and
fi1 ,...,ip1 g Hd
ð5Þ
is the Hamming distance matrix based on the columns i1 , . . . ,ip1 of a design matrix T cd with p1 qualitative factors 0
Jtdf1 þ 1 ,...,tf 1 2
R0 ,
¼@
f2 Y k ¼ f1 þ 1
1 ð1Þqk tak AJtda2 f
1 þ1
ð6Þ
,...,taf
2
for all possible t ¼ ðtf1 þ 1 , . . . ,tf2 Þ with tk ¼ 0,1, . . . ,s1,k ¼ f1 þ 1, . . . ,f2 , and Jtdi
p1 þ 1
based on columns ip1 þ 1 , . . . ,ip2 of a design matrix
T gd
,...,tip
is the signed J-characteristic (Tang, 2001) 2
with p2 quantitative factors.
Proof. Necessity: Follows by applying (a) the necessity part of the proof of CD1 condition to qualitative factors, the row permutation given by permutation matrix R as in (5) and the fact that for any n f design d: Hd ¼
f X
H fig d
i¼1
and (b) the necessity part of the proof of Theorem 3.1 of Cheng and Ye (2004) to quantitative factors, which applies directly since signed J-characteristics are invariant under any row permutation, and in particular under the row permutation given by R as in (5). Sufficiency: Let ða1 , . . . ,ap1 ,ap1 þ 1 , . . . ,ap2 Þ ¼ ðc1 , . . . ,cf Þ. By condition (5), CD1 condition holds for all columns c1 , . . . ,cf1 of T cd2 with column permutation ðc1 , . . . ,cf1 Þ ¼ ða1 , . . . ,ap1 Þ and row permutation given by permutation matrix R. In addition, the signed J-characteristics of the quantitative factors in T gd2 are invariant under any row permutation and thus under the row permutation given by R (as in condition (5)) applied simultaneously to T cd2 and T gd2 . Thus, Theorem 3.1 of Cheng and Ye (2004) holds for columns cf1 þ 1 , . . . ,cf2 of T gd2 with column permutation ðcf1 þ 1 , . . . ,cf2 Þ ¼ ðap1 þ 1 , . . . ,ap2 Þ and reversals as indicated by vector q (see (6)). Conditions (5) and (6) are called cg-IndCD conditions. Note that the conditions in Theorem 2.3 require a row permutation, a column permutation within each type of factors and a vector that indicates reversal of levels of one or more quantitative factors which map d2 to d1, up to relabelings of one or more qualitative factors. & 3. The cg-Deseq2 algorithm An implementation of Theorem 2.1 in a Fortran program which can be used to check equivalence or non-equivalence of any two SFFDs(f1, f2), called cg-Deseq2, is obtainable from the author. It is a modification of Deseq2 algorithm (see Clark and Dean, 2001; Katsaounis and Dean, 2008). The algorithm first searches for a row permutation that transforms the Hamming distance matrix of T cd2 to the Hamming distance matrix of T cd1 . If such a row permutation is found, then it searches for
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a column permutation so that condition (1) of Theorem 2.1 is satisfied. The algorithm does an efficient search for such permutations, similar to that of Deseq2; for a detailed description of Deseq2 see Clark and Dean (2001). Note that an exhaustive search demands the investigation of up to f1 ! f2 ! column permutations, n! row permutations and sf1 2f2 column element permutations and column element reversals. If two designs d1 and d2 are equivalent cg-Deseq2 gives a row permutation and a column permutation (within each type of factors) which transform T d2 to T d1 , up to a set of relabelings of one or more quantitative factors and reversal of levels of one or more quantitative factors which can be found by inspection. If an initial row permutation that transforms the Hamming distance of T cd2 to the Hamming distance of T cd1 is not found or a row and a column permutation which satisfy condition (1) of Theorem 2.1 does not exist then d1 and d2 are declared non-equivalent. The performance of cg-Deseq2 in detecting equivalence or non-equivalence of two SFFDs(f1, f2) is examined in Sections 4.1 and 5 via examples. In each example, 100 random row and column permutations (within each type of factors) and level relabelings (reversals) within one or more columns representing qualitative (quantitative) factors were made of one design (unless specified otherwise), while the other design remained ‘‘fixed’’. The minimum and maximum running times to the nearest second were recorded and results are discussed. The algorithm was written in Fortran 90 and run on Linux machines, running Red Hat Enterprise Linux 5.4. 4. Screening methods for non-equivalence Checking for either combinatorial non-equivalence of submatrices T cd1 and T cd2 or geometric non-equivalence of T gd1 and T gd2 is one way to check for non-equivalence of d1 and d2 (see Section 2.1). Another way is by applying a Hamming distance based screening method to qualitative factors and a Euclidean distance based screening method to quantitative factors and using a common row permutation (see, for example, the ‘‘necessity’’ part of Theorem 2.1; see also Katsaounis et al., 2007 and Katsaounis and Dean, 2008, for existing methods). Alternatively, a screening method for detecting combinatorial non-equivalence of d1 and d2 can also be used as a screening method for non-cg-equivalence, since combinatorially non-equivalent designs are not cg-equivalent. This approach is examined below. Following the recommendation of Katsaounis and Dean (2008), the squared centered L2 discrepancy (Ma et al., 2001) (called CD22 ) and the moment aberration projection method (Xu, 2003) (called momp) are evaluated in Section 4.1 as fast screens for non-equivalence of any two SFFDs(f1, f2). Note that Deseq2 (Clark and Dean, 2001) can be used as a screening method for designs with qualitative and quantitative factors, but for large designs could be very slow (as for the pair of designs (ssd.18.3.12, ssd.18.3.12.b) and (sloa25617, sloa25617b) discussed in Section 4.1; see also Katsaounis and Dean, 2008). 4.1. Detection of non-equivalence Three and four-level designs were used to evaluate the performance of cg-Deseq2, CD22 and momp algorithms. The following examples illustrate that cg-Deseq2 can detect non-equivalence quickly for designs with a small number of factors and runs. Occasionally though it can be slow depending on the size and the structure of the designs. In such cases the screening criteria for combinatorial non-equivalence CD22 and momp can be useful in detecting non-equivalence of two SFFDs(f1, f2). For the pair of 36 5 orthogonal arrays e14 (see Section 1) and e15, obtained as the projection with columns (1 2 3 4 7) of the orthogonal array listed as oa.36.13.3.2 on /http://www.research.att.com/ njasS, cg-Deseq2 with f1 ¼1 (i.e. with the first column being a qualitative factor and the remaining four quantitative factors) detects non-equivalence in less than 0.25 s; similar results hold for f1 ¼2, 3, 4 (i.e. f2 ¼ 3, 2, 1). In this example, in all four cases, screening methods CD22 and momp fail to detect combinatorial non-equivalence and thus non-equivalence. For the supersaturated 9 10 designs ssd1 and ssd2 (see Katsaounis et al., 2007, Table 4) cg-Deseq2 with f1 ¼5, 7 detects non-equivalence in less than 0.13 s. In this example, CD22 and momp also detect combinatorial non-equivalence very quickly (in less than 0.13 s). Results are similar for the supersaturated three-level 18 12 designs ssd.18.3.12 and ssd.18.3.12.b (from http://www.stat.ucla.edu/ hqxu/pub/ssd/ssd18x12h.txt, obtained via Theorems 8 and 9, respectively); cg-Deseq2 with f1 ¼1,6 detects non-equivalence very quickly and CD22 and momp also detect combinatorial non-equivalence very quickly (in less than 0.5 s). As an example that involves larger designs, the 81 40 orthogonal arrays d8140a and d8140b (see Katsaounis and Dean, 2008) were examined; cg-Deseq2 with f1 ¼15 and 30 detects non-equivalence in less than 0.11 s; CD22 and momp detect combinatorial non-equivalence in less than 0.20 s. As another example, the 256 17 four-level designs sloa25617 and sloa25617b (see Katsaounis et al., 2007, Table 6) were examined. However, cg-Deseq2 with f1 ¼8 ran at least 175,877 s for one iteration; this is considerably slower than CD22 and momp which took less than 0.20 s to detect combinatorial non-equivalence. 5. Detection of equivalence The examples in this section illustrate that cg-Deseq2 can sometimes be slow, depending on the size and structure of the designs, but is the only method capable of detecting equivalence of two SFFDs(f1, f2).
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For the three-level 36 5 orthogonal array e14 (see Section 1) and the equivalent designs obtained from e14 by 100 random row permutations, column permutations, level permutations of qualitative factors and level reversals of quantitative factors, cg-Deseq2 with f1 ¼1 and f2 ¼4 detects equivalence in less than 0.20 s, in all cases. Similar results are obtained with f1 ¼ 2, 3, 4. As another example, the 18 7 three-level orthogonal arrays dp11837 and dp41837 (see Katsaounis and Dean, 2008, Table 5) were considered; cg-Deseq2 with f1 ¼ 6 runs with minimum time 16 s and maximum time 18 s; CD22 and momp run somewhat faster, however, they cannot guarantee equivalence. Results are similar for f1 ¼2, 3, 4, 5. For the 36 8 three-level orthogonal arrays sl8 and wk8 (see Katsaounis et al., 2007, Table 1) cg-Deseq2 with f1 ¼3, 4, 5, 6, 7 detects equivalence very fast (in less than 7 s); however cg-Deseq2 with f1 ¼2 is considerably slower (with running time several days for one iteration). Thus, depending on the structure of the design, cg-Deseq2 can be slow. Screening methods CD22 and momp are fast in all cases (with f1 ¼2, 3, 4, 5, 6, 7) but unable to guarantee equivalence. 6. Asymmetric designs with qualitative and quantitative factors An n f fractional factorial design d is called asymmetric when the levels of the factors are 0,1, . . . ,sk 1,k ¼ 1, . . . f , where sk is a prime or a power of a prime (k ¼ 1, . . . ,f ). Randomization of an asymmetric design with qualitative and quantitative factors leads to the following definition of equivalence. Two n f asymmetric fractional factorial designs d1 and d2 with f1 qualitative factors and f2 quantitative factors (ASFFDs(f1, f2)) are equivalent (called cg-equivalent) if one can be obtained from the other by reordering the runs, relabeling the factors of the same type and number of levels, relabeling the levels of the qualitative factors with the same number of levels and reversing the levels of the quantitative factors with the same number of levels. Theorems 2.1 and 2.2 give necessary and sufficient conditions for equivalence of any two ASFFDs(f1, f2), provided that column permutations are within factors of the same type and number of levels. Theorem 2.3 also gives necessary and sufficient conditions for equivalence of any two ASFFDs(f1, f2), provided that column permutations are within factors of the same type and number of levels (in (5) and (6)) and (in (6)) reversals of levels are within quantitative factors with the same number of levels (and with a suitable selection of orthonormal contrasts for the quantitative factors; see Cheng and Ye, 2004). Algorithm cg-Deseq2 can be modified to check equivalence or non-equivalence of any two ASFFDs(f1, f2). Algorithms CD22 and momp can be used as screens for non-equivalence of two ASFFDs(f1, f2) with no modification. 7. Conclusions Algorithm cg-Deseq2, an implementation of Theorem 2.1, can be used to detect equivalence or non-equivalence of any two symmetric fractional factorial designs d1 and d2 with qualitative and quantitative factors, up to level permutations of qualitative factors and reversal of levels of quantitative factors. If two designs are equivalent it returns a row and column permutation that transform the design matrix of d2 to the design matrix of d1. As examples showed, depending on the size and structure of the designs, occasionally cg-Deseq2 can be slow (Sections 4.1 and 5). It is conjectured that a faster algorithm could be obtained by implementing Theorem 2.2, since condition (2) involves establishing only equivalence of projections of a dimension (k) smaller than the dimension (f) of the design matrices. On the other hand, an implementation of Theorem 2.3 would lead to a slower algorithm, since in addition to a search for a row permutation and a column permutation within each type of factors, a search for a vector that indicates reversal of levels of one or more quantitative factors is required (cf. Katsaounis et al., 2007, in their application of Theorem 3.1 of Cheng and Ye, 2004). As fast screens for non-equivalence of designs with qualitative and quantitative factors, the screening methods for combinatorial non-equivalence squared centered L2 discrepancy (Ma et al., 2001) and moment aberration projection (Xu, 2003) (see Section 4.1) can be used prior to cg-Deseq2. Note that these algorithms can sometimes be very slow too (see Katsaounis and Dean, 2008). Theorems 2.1–2.3 can be extended to obtain necessary and sufficient conditions for asymmetric fractional factorial designs with qualitative and quantitative factors. Algorithm cg-Deseq2 can be modified to handle factors with different number of levels.
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