Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143 Construction of optimal supersaturated designs by the packing method ¾ ¿ FANG Kaitai½ , GE Gennian & LIU Minqian 1. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China (email: [email protected]); 2. Department of Mathematics, Zhejiang University, Hangzhou 310027, China (email: [email protected]); 3. Department of Statistics, Nankai University, Tianjin 300071, China (email: [email protected]) Received July 31, 2002 Abstract A supersaturated design is essentially a factorial design with the equal oc- currence of levels property and no fully aliased factors in which the number of main ef- fects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently un- related kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersatu- rated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties. Keywords: Kirkman triple systems, orthogonality, packing design, resolvability, supersaturated design. DOI: 10.1360/02ys0271 Many preliminary studies in industrial and scientific experiments contain a large num- ber of potentially relevant factors, but often only a few are believed to have significant effects. When experiments are expensive, the problem is how to identify these few sig- nificant effects with a relatively small number of runs. One approach is to use a so-called supersaturated design, namely, a factorial design in which the number of main effects is greater than the number of runs. Developing such screening designs has received a great deal of attention: Satterthwaite [½] introduced the supersaturated design as a random bal- ance design. Booth and Cox [¾] first examined this problem systematically. Since then, most studies, such as refs. [3—14], have focused on two-level supersaturated designs. These two-level designs can be used for screening the factors in simple linear models. When the relationship between a set of factors and a response is nonlinear, or approxi- mated by a polynomial response surface model, designs with multi-level and mixed-level are often required, e.g., to explore nonlinear effects of the factors. Recently, Yamada and [½6] Lin[½5] and Yamada et al. presented two construction methods for three-level supersat- urated designs; Fang, Lin and Ma [½7] obtained a new class of multi-level supersaturated Copyright by Science in China Press 2004 Optimal supersaturated designs 129 designs through a global optimization algorithm, while Fang, Lin and Liu [½8] provided a class of universally optimal mixed-level supersaturated designs. Ñ ´Ò; Õ µ Ò Ñ Õ Let Í denote a design with runs and -level factors, which corresponds = ´ ; ¡¡¡; µ Ò ¢ Ñ X Ü Ü ½ to an matrix Ñ and satisfies the following conditions: ; ¾; ¡¡¡;Õ 1. all the levels, denoted by ½ , of each factor appear equally often; X 2. no two columns of X are fully aliased, i.e., no column of can be obtained from another column of X by permuting levels. Ñ ´Ò; Õ µ Ò The set of all such designs is denoted by Í . Obviously, must be a multiple Ô = Ò=Õ of Õ . Throughout the paper let . Two columns are called orthogonal if all of Ñ ´Ò; Õ µ their level-combinations appear equally often. And a design in Í is called an Ñ Ä ´Õ µ orthogonal array of strength 2, denoted by Ò , if any two columns are orthogonal. ´Õ ½µ 6 Ò ½ In this case, Ñ . A comprehensive study of orthogonal arrays can be ´Õ ½µ = Ò ½ Ñ´Õ ½µ > found in ref. [19]. When Ñ , the design is saturated. When ½ Ò , the design does not have enough degrees of freedom to estimate all the factorial Ñ ´Ò; Õ µ main effects simultaneously, it is called a supersaturated design, denoted by Ë . Ñ Ò; Õ ; ѵ Ë ´Ò; Õ µ For given ´ , different ’s may have different performances in statistical ´f µ inference. In this paper we employ the E criterion proposed by Fang, Lin and ÆÇD Ñ [½8] ´Ò; Õ µ E ´f µ Liu for evaluating such designs. An Ë is called -optimal, denoted by ÆÇD Ñ Ñ Ë ´Õ µ E ´f µ Í ´Ò; Õ µ Ò , if it minimizes over all possible designs in . And we will ÆÇD ÑaÜ f Æ propose two additional criteria, i.e. and ÑaÜ , for making further comparisons ÆÇD ´f µ among E -optimal designs. ÆÇD Ñ Ë ´Õ µ Then, how to generate Ò ’s? In fact, one can employ some powerful optimization Ñ Ñ [½¼] Ë ´Õ µ Í ´Ò; Õ µ methods to search for Ò ’s over , like Yamada et al. and Fang, Lin and [½7] Ñ Ma . However, when Ò; Õ and are moderate, it is difficult to search a complete list of designs. The main purpose of the paper is to propose a new way for constructing Ñ Ñ Ë ´Õ µ Í ´Ò; Õ µ Ò ’s. We will establish a connection between designs in a subset of and resolvable packing designs which are well-studied in combinatorial design theory. Then the construction approach just follows from this connection. ¾ < Õ When Ò , the orthogonality between any two columns could not be satisfied. ; Ü Ü i Naturally, we hope that the factorial level-combinations in any two columns j of Ñ ´Ò; Õ µ an Ë will be distributed as evenly as possible. Therefore, we define a subset of Ñ Ñ Ñ ¾ ´Ò; Õ µ Í ´Ò; Õ ; Ö µ Í ´Ò; Õ ; Ö µ Õ Í , denoted by , such that for each design in , any of the Ö level-combinations in any two columns appears at most Ö times. The number is called > ½ ´Ò; Õ ; ѵ the largest frequency of the design. Obviously Ö . For given , we expect to Ñ ¾ ´Ò; Õ ; Ö µ Ö Ò = Õ choose a design in Í such that is as small as possible. When , for given Ñ ´Ò; Õ ; ѵ Ä ´Õ µ , we will choose an orthogonal array Ò if it exists as its largest frequency ¾ ½ Ö = ½ Ò 6 Õ Ö takes the minimum value , i.e. When , the following natural questions arise: Ñ Ò; Õ ; Ö µ Í ´Ò; Õ ; Ö µ Question 1. For given ´ , we can find many designs belonging to www.scichina.com 130 Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143 Ñ with different number of columns, Ñ. What is the largest value of ? Or equivalently, Ñ Í ´Ò; Õ ; Ö µ ´Ò; Õ ; ѵ what is the smallest value of Ö in for given ? ´f µ Question 2. How do we construct E -optimal supersaturated designs for given ÆÇD Ò; Õ ; Ñ; Ö µ Ö = ½; ¾ Ñ ´ , especially for and achieving its largest value in Questions 1? Ñ ´Ò; Õ ; Ö µ With the connection between designs in Í and resolvable packing designs, the exact answer for Question 1 is obtained (see Theorem 3 in Section 2.1), and a class Ñ Ë ´Õ µ of Ò ’s that satisfy the requirement in Question 2 is constructed without any com- 4× Ë ´¿ µ × = ½; ¡¡¡; 7 puter search. For example, a class of 9 for are obtained by the new ¾8 Ë ´¿ µ approach, which are listed in table 1. There are 7 parts each having 4 columns in 9 . 4 Ä ´¿ µ Ö = ½ This design has the following properties: (a) each part is an 9 , i.e., ; (b) for 4× × = ¾; ¡¡¡; 7; 4× Ë ´¿ µ any the design formed by the first columns is an 9 with the largest = ¾ ´Ò; Õ ; Ö µ = ´9; ¿; ¾µ Ñ frequency Ö ; (c) for given , the largest value of is 28, which is ¾8 Ë ´¿ µ ´Ò; Õ ; ѵ = ´9; ¿; 4×µ;× = ¾; ¡¡¡; 7 attained by this 9 ; and (d) for given , the small- 4× Ö Ë ´¿ µ E ´f µ ÑaÜ f est value of is 2 which is attained by the 9 . Under criteria , ÆÇD ÆÇD Æ and ÑaÜ , these designs have a slightly better performance than that obtained by Yamada [½7] et al. [½6] and Fang, Lin and Ma through computer searches. More detailed comparisons will be given in Section 3. In fact, many new designs are obtained by our new approach. 4× Ë ´¿ µ´½ 6 × 6 7µ Table 1 9 Row1234567 1 1321 3111 2231 2123 1213 2312 1132 2 1132 1321 3111 2231 2123 1213 2312 3 2312 1132 1321 3111 2231 2123 1213 4 1213 2312 1132 1321 3111 2231 2123 5 2123 1213 2312 1132 1321 3111 2231 6 2231 2123 1213 2312 1132 1321 3111 7 3111 2231 2123 1213 2312 1132 1321 8 3222 3222 3222 3222 3222 3222 3222 9 3333 3333 3333 3333 3333 3333 3333 ´f µ ÑaÜ f The paper is organized as follows. In Section 1, the definitions of E , ÆÇD ÆÇD Æ and ÑaÜ are introduced, some justification for using them as design criteria is provided, and some properties of these three criteria are obtained. In Section 2, the connection Ñ ´Ò; Õ ; Ö µ between designs in Í and resolvable packing designs is established, and the ´f µ combinatorial approach, i.e. the packing method, for constructing E -optimal su- ÆÇD Ñ ´Ò; Õ ; Ö µ persaturated designs in Í is proposed.