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Lower Bounds for the Uniformity Patternof Asymmetric Fractional FactorialsHong Qin a & Kashinath Chatterjee ba Department of Statistics , Central China Normal University ,Wuhan, Chinab Department of Statistics , Visva-Bharati University , Santiniketan,IndiaPublished online: 27 Apr 2009.

To cite this article: Hong Qin & Kashinath Chatterjee (2009) Lower Bounds for the Uniformity Patternof Asymmetric Fractional Factorials, Communications in Statistics - Theory and Methods, 38:9,1383-1392, DOI: 10.1080/03610920802439161

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Communications in Statistics—Theory and Methods, 38: 1383–1392, 2009Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920802439161

Lower Bounds for the Uniformity Patternof Asymmetric Fractional Factorials

HONG QIN1 AND KASHINATH CHATTERJEE2

1Department of Statistics, Central China Normal University,Wuhan, China2Department of Statistics, Visva-Bharati University,Santiniketan, India

The objective of this article is to study the issue of the projection discrepancy (Fangand Qin, 2005; Ma et al., 2003), which has wide application to the field of fractionalfactorials. Fang and Qin (2005) and Zhang and Qin (2006) derived some propertiesof the projection properties of two-level factorials. Here we extend their results toasymmetric factorials. Some lower bounds of projection discrepancy of asymmetricalfactorials are also obtained.

Keywords Asymmetrical fractional factorial; Lower bound; Projectiondiscrepancy.

Mathematics Subject Classification Primary 62K15; Secondary 62K05.

1. Introduction

The projection properties of fractional factorial designs have recently received muchattention in the literature. Ma et al. (2003) pioneered the projection discrepancies oftwo-level fractional factorials in terms of the centered L2-discrepancy (Hickernell,1998a). Liu (2002) used the projection of discrete discrepancy to evaluate theuniformity of symmetrical fractional factorials and found that orthogonality anduniformity are strongly related to each other. Zhang et al. (2005) proposed thecategorical discrepancy which generalizes the concept of discrete discrepancy givenby Hickernell and Liu (2002) and defined the categorical discrepancy patternto measure the uniformity for all possible low-dimensional projection designs.Following the projection discrepancies discussed by Ma et al. (2003), Fang andQin (2005) proposed the uniformity pattern and a minimum projection uniformitycriterion, which is different from that of Hickernell and Liu (2002), to assess and

Received December 10, 2007; Accepted August 27, 2008Address correspondence to Hong Qin, Department of Statistics, Central China Normal

University, Wuhan 430079, China; E-mail: qinhong@mail.ccnu.edu.cn

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1384 Qin and Chatterjee

compare two-level factorials. Zhang and Qin (2006) further developed the theoryof minimum projection uniformity criterion and established relationships with othercriteria. Song and Qin (2009) explored some applications of minimum projectionuniformity criterion to two-level complementary designs.

Most studies of projection discrepancies are related to symmetric factorials.Recently, Chatterjee et al. (2005) discussed the overall uniformity in terms of wrap-around L2-discrepancy (Hickernell, 1998b) for a class of factorials with two- andthree-mixed levels. Chatterjee et al. (2006) and Wang et al. (2007) gave lower boundsfor the centered L2-discrepancy and the symmetric L2-discrepancy (Hickernell,1998b) for asymmetric factorials. We note that these studies only consideredthe overall uniformity of an asymmetric design among all possible projections,but ignored different contribution to overall uniformity under different lowerdimensional projections. Based on the so-called discrete discrepancy, Hickernelland Liu (2002) defined a projection discrepancy pattern to measure the uniformityof low-dimensional projections of asymmetric designs and proposed a minimumprojection uniformity criterion in terms of this pattern, which is equivalent togeneralized minimum aberration proposed by Xu and Wu (2001). Liu et al.(2006) investigated connections among minimum projection uniformity and otheroptimality criteria. Lower bounds of projection discrepancy pattern/categoricaldiscrepancy pattern, however, have not yet been obtained in the literature for theasymmetric case. The present article aims at studying further properties of minimumprojection uniformity proposed by Fang and Qin (2005) and obtaining some lowerbounds for the uniformity pattern for a class of asymmetric factorials.

The article is organized as follows. In Sec. 2, we briefly introduce the projectiondiscrepancy, uniformity pattern, and minimum projection uniformity criterion.Section 3 gives lower bounds of uniformity pattern of asymmetrical designs.Section 4 shows four example designs for which our lower bound is achievable. Weclose in the Remarks section with some notes and comments.

The remainder of this section describes some notations which are usedthroughout this article. Consider the set, denoted by ��n� p× 2s�, of anasymmetric p× 2s factorial having n runs, where p is any positive integer (≥2)and n treatment combinations are not necessarily distinct. A typical treatmentcombination of d ∈ ��n� p× 2s� is defined by z = �u�1�� u�2��, where 0 ≤ u�1� ≤p− 1 and u�2� = �u

�2�1 � � � � � u�2�

s �, u�2�j = 0� 1, 1 ≤ j ≤ s. Define the sets V �1� =

�0� 1� � � � � p− 1�, for any fixed q, 1 ≤ q ≤ s, V �2q� = ��u�2�1 � � � � � u�2�

q � � u�2�j = 0� 1� 1 ≤

j ≤ q� and V �3q� = ��u�1�� u�2�1 � � � � � u�2�

q � � 0 ≤ u�1� ≤ p− 1� u�2�j = 0� 1� 1 ≤ j ≤ q� of

treatment combinations lexicographically ordered. For any treatment combinationz belonging to any of the above sets, let yd�z� be the number of times the treatmentcombination z occurs in the corresponding projected design of d ∈ ��n� p×2s� and y

�1�d , y�2q�d , y�3q�d be, respectively, the corresponding vector of appropriate

orders in V �1�, V �2q�, and V �3q�. Let y�q�d �u�1�� be a 2q × 1 vector with elements

yd�u�1�� u�2�� for all elements u�2� in V �2q� arranged in the lexicographic order. Define

V �1j� = �0� 1� � � � � j − 1� and V ∗�1j� = V �1� − V �1�p−j��. Let, for 1 ≤ l ≤ s, yd�V�1jl�� be

a �j × 2l�× 1 vector with elements yd�u�1�� u�2��, where u�1� ∈ V �1j� and u�2� ∈ V �2l�.

Similarly, we define �j × 2l�× 1 vector yd�V∗�1jl��. For any positive integer t, let et

denote the t × 1 vector with all elements unity and It denote an identity matrix oforder t.

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Minimum Projection Uniformity 1385

2. Projection Discrepancy for ����n� p× 2s�

Consider a design d ∈ ��n� p× 2s�, where p ≥ 3 and the levels the first factor occurequally often in the entire design. Define

hij =2dij + 1

2pj

�

where dij is the ith run, jth factor entry of d, and pj is the number of levels forfactor j. Note that p1 = p and p2 = · · · = ps+1 = 2.

Following Ma et al. (2003), the projection discrepancy value of a design d ∈��n� p× 2s� on wl, a subset of �1� 2� � � � � s + 1� of cardinality l (1 ≤ l ≤ s + 1), canbe defined as

CL�d�wl� ={(

112

)l

− 2n

n∑k=1

∏j∈wl

(12

∣∣∣∣hkj −12

∣∣∣∣− 12

∣∣∣∣hkj −12

∣∣∣∣2)

+ 1n2

n∑k=1

n∑t=1

∏j∈wl

(12

∣∣∣∣hkj −12

∣∣∣∣+ 12

∣∣∣∣htj −12

∣∣∣∣− 12�hkj − htj�

)} 12

�

For 1 ≤ l ≤ s + 1, define

MIl�d� =∑

wl∈Wl

CL�d�wl�2 + c�p� s� l�� (1)

for all l ∈ �1� 2� � � � � s + 1�, where Wl is the set of all possible wl and c�p� s� l� is thefollowing constant which does not depend on the design d. That is, if l = 1,

c�p� s� l� = 4+ sp2

48p2�

if 2 ≤ l ≤ s + 1 and p is an even integer,

c�p� s� l� = −[(

sl− 1

)[(112

)l

− 3l−1�2p2 + 1�3p225l−3

+ 2�p2 + 2�3p28l

]

−(sl

)[(112

)l

− 3l

25l−1+ 1

8l

]]�

and finally, if 2 ≤ l ≤ s + 1 and p is an odd integer,

c�p� s� l� = −[(

sl− 1

)[(112

)l

− 3l−1�p2 − 1�3p225l−4

+ 2�p2 − 1�3p28l

]

−(sl

)[(112

)l

− 3l

25l−1+ 1

8l

]]�

The function(sl

) = 0 if s < l and s�s − 1� · · · �s − l+ 1�/l! otherwise.MIl�d� provides a measure of overall uniformity of the l-factor

projections of design d. For any design d ∈ ��n� p× 2s�, the vector

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1386 Qin and Chatterjee

�MI1�d��MI2�d�� � � � �MIs+1�d�� is referred to as the uniformity pattern of the designd as in Fang and Qin (2005), which provides a measure of the projection uniformityof d onto different dimensions. The minimum projection uniformity criterion is tosequentially minimize MIl�d�’s.

The next section deals with a derivation of a lower bound of uniformity patterndefined above.

3. Lower Bounds of Uniformity Pattern

Zhang and Qin (2006) provided some lower bounds of uniformity pattern forsymmetric factorials with two levels. Here we provide a lower bound of uniformitypattern for asymmetric factorials.

Define

m�1� =

m

�1�0���

m�1�p−1

� M�1� =

m�1�00 � � � m

�1�0�p−1�

������

���

m�1��p−1�0 � � � m

�1��p−1��p−1�

�

m�2� =(3/323/32

)� M�2� =

(1/4 00 1/4

)�

where, for 0 ≤ i� j ≤ p− 1,

m�1�j = 1

2

∣∣∣∣2j + 12p

− 12

∣∣∣∣− 12

∣∣∣∣2j + 12p

− 12

∣∣∣∣2

� and

m�1�ij = 1

2

∣∣∣∣2i+ 12p

− 12

∣∣∣∣+ 12

∣∣∣∣2j + 12p

− 12

∣∣∣∣− 12

∣∣∣∣2i+ 12p

− 2j + 12p

∣∣∣∣ �Let m�2�

t and M�2�t be, respectively, the t-fold (1 ≤ t ≤ s) Kronecker products of m�2�

and M�2�.Define, for 1 ≤ t ≤ s, m�1�t� = m�1� ⊗m

�2�t and M�1�t� = M�1� ⊗M

�2�t .

As earlier, W ∗l denotes the collection of all possible subsets of �2� � � � � s + 1�

having l elements. Following the notations stated in the earlier sections and Fangand Qin (2003), we can express MIl�d� in (1) as

MI1�d� =∑

w1∈W ∗1

[112

− 2nm�2�′y

�21�d + 1

n2y�21�′d M�2�y

�21�d

]− s

48� (2)

for any fixed l, 2 ≤ l ≤ s + 1,

MIl�d� =∑

wl−1∈W ∗l−1

[(112

)l

− 2nm�1� l−1�′y

�3�l−1��d + 1

n2y�3�l−1��′d M�1� l−1�y

�3�l−1��d

]

+ ∑wl∈W ∗

l

[(112

)l

− 2nm

�2�′l y

�2l�d + 1

n2y�2l�′d M

�2�l y

�2l�d

]+ c�p� s� l�� (3)

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Minimum Projection Uniformity 1387

For 0 ≤ i ≤ s, let �i be the largest integer contained in �n/2i� and for any integerj, 1 ≤ j ≤ p/2, �ij be the largest integer contained in �nj/�p2i��. Define

�i = n− 2i�i� �∗i = n�i + �i�1+ �i�

�ij =nj

p− 2i�ij� �∗ij =

nj

p�ij + �ij�1+ �ij��

The following theorem provides lower bound for MIt�d�’s.

Theorem 3.1. Let d ∈ ��n� p× 2s�, where the levels of the first factor occur equallyoften in d. Then

(i)MI1�d� ≥ s

[�∗14n2

− 18

]� (4)

(ii) For any fixed l� 2 ≤ l ≤ s + 1� p = 2r,

MIl�d� ≥(

sl− 1

)[2

n24l−1p

r−1∑j=1

�∗�l−1�j +1

n24l−1p�∗�l−1�r −

2�p2 + 2�3p28l

]

+(sl

)[�∗ln24l

− 18l

]� (5)

and, for p = 2r + 1,

MIl�d� ≥(

sl− 1

)[2

n24l−1p

r∑j=1

�∗�l−1�j −2�p2 − 1�3p28l

]+(sl

)[�∗ln24l

− 18l

]� (6)

A proof of the theorem is in the Appendix.

4. Numerical Studies

This sections provides four examples to illustrate that the lower bound given inTheorem 3.1 are tight.

Example 4.1. Consider a design d1, given below, with n = 16, p = 4, s = 3.

d1 =

0000 1111 2222 33330011 1100 0011 11000101 1010 0101 10100110 1001 0110 1001

�

where the treatment combinations are written as columns. The lower bound for theuniformity pattern of a design in ��16� 4× 23�, obtained through Theorem 3.1, is

�0� 0� 0� 6�104× 10−5��

For the design d1, the uniformity pattern is �MI1�d1��MI2�d1��MI3�d1��MI4�d1�� =�0� 0� 0� 6�104× 10−5�.

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1388 Qin and Chatterjee

Example 4.2. Consider n = 20, p = 5, s = 3 and the design d2 ∈ ��20� 5× 23�

d2 =

0000 1111 2222 3333 44440011 1100 0011 1100 00110101 1010 0101 1010 01010110 1001 0110 1001 0110

�

where the treatment combinations are written as columns. The design d2

has minimum projection uniformity, as its uniformity pattern �0� 0� 7�813×10−5� 3�125× 10−5� attains the lower bounds given in Theorem 3.1.

Example 4.3. The design d3 ∈ ��48� 3× 25�, given below, has minimum projectionuniformity. The design is

d3 =

0000000000000000 1111111111111111 22222222222222221111111100000000 0000000011111111 11111111000000001111000011110000 0000111100001111 11110000111100001100110011001100 0011001100110011 11001100110011001010101010101010 0101010101010101 10101010101010101001011001101001 0110100110010110 1001011001101001

�

where the treatment combinations are written as columns. This is becauseits uniformity pattern �MI1�d3��MI2�d3��MI3�d3��MI4�d3��MI5�d3��MI6�d3�� =�0� 0� 0� 0� 3�391× 10−6� 2�261× 10−6� and attains the lower bound given inTheorem 3.1.

Example 4.4. Consider three designs in ��6� 3× 22�, given in Table 1. It is tobe noted that these three designs have distinct uniformity patterns and only d4�3

attains the lower bound given in Theorem 3.1. Thus, the design d4�3 is optimal. It isinteresting to note that d4�3 is obtained by adding two runs to an OA�4� 3� 2� 2� suchthat the added runs have hamming distance 2 �=3− �3/2�, where �a is the largest

Table 13× 22 designs in 6 runs

d4�1 d4�2 d4�3

0 0 0 0 0 1 0 1 00 0 1 0 1 0 0 0 11 1 0 1 0 1 1 0 01 1 1 1 1 0 1 1 12 0 1 2 0 1 2 0 12 1 0 2 1 0 2 1 0

d4�1 � �0� 6�36574074074� 1�15740740741��×10−3�d4�2 � �0� 15�6250000000� 1�15740740741��×10−3�d4�3 � �0� 1�73611111111� 1�15740740741��×10−3�

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Minimum Projection Uniformity 1389

integer less than or equal a and then replacing the levels of the first factor with thelevel 2 to the added runs. Jacroux et al. (1983) showed that if a design is obtained byadding two runs to an OA�n�m� 2� 2� such that the hamming distance between themis m− �m/2�, then the design will be Type 1 optimal according to Cheng (1978).This example justifies a connection between minimum projection uniformity andother optimality measures.

5. Remarks

In this article, we discussed the issue of minimum projection uniformity criterionon asymmetric factorials. Our tools involve application of the Kronecker calculusfor factorial arrangements and an identification with a hypothetical full factorial.Using these tools, we obtained lower bounds of uniformity pattern for a class ofasymmetric factorials, which are highly useful for measuring projection discrepancyvalues of designs obtained by different methods against theoretical lower bounds.

Appendix

The following three lemmas are later used to prove Theorem 3.1. Proofs of thelemmas are omitted, since they are straightforward.

Lemma A.1.

(A1) m�1�j = m

�1�p−j−1, 0 ≤ j ≤ k, where k = r − 1 when p = 2r and k = r when p =

2r + 1.(A2) The vector m�1� can be expressed as

m�1� = ep +k∑

j=1

j�j�

where = �2p− 1�/�8p2� or = 0 according as p is even or odd. Also, for 1 ≤j ≤ k, j = j/p2 and �j = �e′j� 0

′� e′j�′ is a vector of order p× 1.

Lemma A.2.

(B1) m�1�i�r+j� = 0, 0 ≤ i� j ≤ k, where k= r − 1 when p= 2r and k= r when p= 2r + 1.

(B2) m�1�ij = m

�1��2r−i−1��2r−j−1�, 0 ≤ i� j ≤ r − 1, where p = 2r.

(B3) m�1�ij = m

�1��2r−i��2r−j�, 0 ≤ i� j ≤ r − 1, where p = 2r + 1.

(B4) If p = 2r, the matrix M�1� can be expressed as

M�1� =r∑

j=1

�j�j�

where

�j =eje

′j O O

O O O

O O eje′j

�

�j = 1/p, for 1 ≤ j ≤ r − 1, and �r = 1/�2p�.

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1390 Qin and Chatterjee

(B5) If p = 2r + 1, the matrix M�1� can be expressed as

M�1� =r∑

j=1

�j�j�

where �j’s are as defined above and �j = 1/p, for 1 ≤ j ≤ r.

Lemma A.3.

(C1) For 1 ≤ l ≤ s, m�2�l = �le

�l�2 , where �l = �3/32�l and e

�l�2 is the l-fold Kronecker

product of e2.(C2) For 1 ≤ l ≤ s,

m�1�l� = �l ep ⊗ e�l�2 + �l

k∑j=1

j�j ⊗ e�l�2 �

where �l is as defined in (C1) and k, , j’s are as defined in Lemma A.1.(C3) For 1 ≤ l ≤ s, M�2�

l = �1/4�lI�l�2 , where I�l�2 is the l-fold Kronecker product of I2.

(C4) For 1 ≤ l ≤ s and p = 2r,

M�1�l� =r∑

j=1

�j�1/4�l�

�∗�j �

where

��∗�j =

�eje

′j�⊗ I

�l�2 O O

O O O

O O �eje′j�⊗ I

�l�2

�

where �j’s are as stated in (B4) of Lemma A.2, and(C5) for 1 ≤ l ≤ s and p = 2r + 1,

M�1�l� =r∑

j=1

�j�1/4�l�

�∗�j �

where ��∗�j is as defined in (C4) above and �j’s are the same as in (B5) of

Lemma A.2.

Proof of Theorem 3.1. (ii) For 2 ≤ l ≤ s + 1 and for p = 2r, it is to be noted fromLemma A.3 that m�1�l−1� can be expressed as

m�1�l−1� = �l−1 ep ⊗ e�l−1�2 + �l−1

k∑j=1

j�j ⊗ e�l−1�2 �

where = �2p− 1�/�8p2�, k = r − 1, j = j/p2� 1 ≤ j ≤ k, and �l−1 = �3/32�l−1.Since the levels of the first factor are assumed to occur equally often in the design,

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Minimum Projection Uniformity 1391

we get

m�1�l−1�′y�3�l−1��d = �l−1 �ep ⊗ e

�l−1�2 �′y�3�l−1��

d + �l−1

k∑j=1

j��j ⊗ e�l−1�2 �′y�3�l−1��

d

= 2p2 + 124p2

n�l−1� (7)

Now from (C4) of Lemma A.3, we get

y�3�l−1��′d M�1�l−1�y

�3�l−1��d = y

�3�l−1��′d

( r∑j=1

�j�1/4�l−1�

�∗�j

)y�3�l−1��d

=r∑

j=1

�j�1/4�l−1y

�3�l−1��′d �

�∗�j y

�3�l−1��d

=r∑

j=1

�j�1/4�l−1[yd�V

�1j�l−1���′�eje′j ⊗ I

�l−1�2 �yd�V

�1j�l−1���

+ yd�V∗�1j�l−1���′�eje

′j ⊗ I

�l−1�2 �yd�V

∗�1j�l−1���]

=r∑

j=1

�j�1/4�l−1[�yd�V

�1j�l−1���′�ej ⊗ I�l−1�2 ��

× ��e′j ⊗ I�l−1�2 �yd�V

�1j�l−1����+ �yd�V∗�1j�l−1���′�ej ⊗ I

�l−1�2 ��

× ��e′j ⊗ I�l−1�2 �yd�V

∗�1j�l−1����]�

It is to be noted that for each j, 1 ≤ j ≤ r, the elements of the vectors�e′j ⊗ I

�l−1�2 �yd�V

�1j�l−1��� and �e′j ⊗ I�l−1�2 �yd�V

∗�1j�l−1��� are non negative integers withsum �nj/p�. Thus,

yd�V�1j�l−1���′�ej ⊗ I

�l−1�2 ���e′j ⊗ I

�l−1�2 �yd�V

�1j�l−1��� ≥ �∗�l−1�j

and

yd�V∗�1j�l−1���′�ej ⊗ I

�l−1�2 ���e′j ⊗ I

�l−1�2 �yd�V

∗�1j�l−1��� ≥ �∗�l−1�j �

Therefore, we have

y�3�l−1��′d M�1� l−1�y

�3�l−1��d ≥ 2

r∑j=1

�j�1/4�l−1�∗�l−1�j � (8)

Again, it is easy to check that

m�2�′l y

�2l�d = �le

�l�′2 y

�2l�d = n�l (9)

and

y�2l�′d M

�2�l y

�2l�d = �1/4�ly�2l�

′d y

�2l�d ≥ �1/4�l�∗l � (10)

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1392 Qin and Chatterjee

Thus, from Eqs. (3), (7)–(10), one obtains, for 2 ≤ l ≤ s and for p = 2r,

MIl�d� ≥(

sl− 1

)[2

n24l−1

r∑j=1

�j�∗�l−1�j −

2�p2 + 2�3p28l

]+(sl

)[�∗ln24l

− 18l

]�

which completes the proof of (5).The proof of other cases will follow in a similar manner.

Acknowledgments

The authors would like to thank the referee and Chief Editor for their valuablecomments and suggestions that lead to the improvement in the presentationof the article. This research was partially supported by the NNSF of China(No. 10671080), NCET (No. 06-672), and the Key Project of Chinese Ministry ofEducation (No. 105119).

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