# Minimum Aberration Split-Plot Designs When the Whole Plot and Subplot Factors Do Not Have the Same Importance

Authors: Shengli Zhao; Qianqian Zhao
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

In practical factorial experiments, we sometimes find that complete randomization of the order of the runs is infeasible because it is more difficult to change the levels of some factors than the others, especially in some engineering experiments. Then, fractional factorial split-plot (FFSP) designs represent a practical option in such situations. The difficult-to-change factors are called whole plots (WP) factors, and the other factors are called subplot (SP) factors. The WP and SP factors do not have the same importance in many experiments. Then, the popular minimum aberration criterion is not suitable any more for choosing FFSP designs. This paper proposes two criteria for selecting FFSP designs. Algorithms for constructing the optimal FFSP designs under the two criteria are proposed. Some optimal designs under the two criteria are tabulated.


---

## Body

## 1. Introduction

Fractional factorial (FF) designs are widely used in various experiments, say agriculture, medicine, and engineering (see [1, 2]). Randomization is one of the fundamental principles in the design of an experiment. It provides protection against variables that are unknown to the experimenter but may impact the response. It reduces the unwanted influence of subjective judgment in treatment allocation [2]. However, complete randomization is sometimes inadvisable since the levels (settings) of some factors are very difficult or expensive to change. In such situations, fractional factorial split-plot (FFSP) designs, which involve a two-phase randomization, can be conveniently used to reduce costs and hence represent a practical design option.In an FFSP design, the factors whose levels are difficult or expensive to change are called whole plot (WP) factors and the rest are called subplot (SP) factors. In the following, a “WP factor setting” means a combination of levels of the WP factors, and an “SP factor setting” means a combination of levels of the SP factors. The two-phase randomization process in an FFSP design is as follows (see [3]): (i) randomly choose any of the WP factor settings; (ii) run the experiment with the associated SP factor settings in a random order while holding the WP factor setting fixed; (iii) repeat steps (i) and (ii) till all the WP factor settings are covered. The two-step randomization results in two error sources in the analysis of variance (ANOVA), the WP and SP error terms. In general, the latter is smaller than the former. This means that the power to detect significant effects in data analysis is different for these two kinds of factors. So, the WP and SP factors cannot be treated equally.Some practical examples also illustrate the difference of the WP and SP factors. Reference [4] cited an experiment from [1] (p. 629) as example in which the WP factors are more important than the SP factors. Reference [5] mentioned an example in which the SP factors (product design factors or control factors) are more important than the WP factors (environmental factors or noise factors). Reference [6] considered the following five scenarios for more flexible split-plot design choices: (i) basic screening; (ii) screening, with emphasis on SP effects; (iii) screening, with emphasis on WP effects; (iv) robust parameter design, with control factors at the SP level; (v) robust parameter design, with control factors at the WP level. Clearly, in scenarios (ii) and (iv), the SP factors are more important than the WP factors, and in (iii) and (v), the WP factors are more important than the SP factors.One of the most important things for an experimenter is to choose the optimal experimental design according to some criterion. The minimum aberration (MA) criterion, which was first proposed by Fries and Hunter for regular two-level designs (see [7]), is popular and efficient for assessing designs. Up to now, many statisticians have been involved in extending MA criterion into wider applications such as FFSP designs. Reference [8] gave an algorithm to search MA FFSP designs. Reference [9] investigated some theoretical results and emphasized some of the differences between FFSP designs and FF designs. Reference [10] showed how the split-plot structure affects estimation, precision, and the use of resources. Reference [11] explored a minimum secondary aberration criterion called MSA-FFSP criterion. Reference [12] constructed MSA-FFSP designs in terms of consulting designs.In the MA criterion, the WP and SP factors are treated with the same importance. From the above discussion, the WP and SP factors do not have the same importance in many experiments, thus new criterion needs to be developed. By adjusting the word lengths, reference [6] proposed different criteria for the five scenarios. These criteria are suitable for both regular and nonregular orthogonal designs. However, there are some defects in these criteria. For example, in scenario (ii), the words WW and SSS have the same length of 3, which contradicts the effect hierarchy principle somewhat, where WW and SSS denote the defining words involving 2 WP factors and 3 SP factors, respectively. Reference [4] proposed a WP-MA criterion for FFSP designs in which the WP factors are important and the experimenter is not interested in the SP effects. The WP-MA criterion, which first sequentially minimizes the WP wordlength pattern and then sequentially minimizes the SP wordlength pattern, gives full priority to the WP factors. Reference [13] constructed WP-MA FFSP designs via complementary designs. Conversely, in terms of the case that the SP factors are more important than the WP factors and the experimenter is not interested in the WP effects, Dang et al. [14] proposed the SP-MA criterion and studied the construction of SP-MA FFSP designs via complementary designs. Both the WP-MA and SP-MA criteria consider the different status of WP and SP factors, but these two criteria violate the effect hierarchy principle too when the SP/WP effects are also interested. Focusing on the WP factors, [15] proposed a WS-MA criterion for selecting FFSP designs and discussed the construction method of the optimal FFSP designs under the WS-MA criterion.Considering scenarios (ii) and (iii) in [6], this paper proposes two wordlength patterns and two corresponding criteria, called W-MA criterion and S-MA criterion, respectively, for selecting FFSP designs. By intertwining the WP and SP wordlength patterns together in different forms, both the W-MA and S-MA criteria follow the effect hierarchy principle. As described above, there is a fundamental difference between the new proposed W-MA (S-MA) criterion and the WP-MA (SP-MA) criterion. In addition, the optimal designs in [13, 14] are confined to the case that the complementary sets are relatively small than the design sets themselves. To some extent, the new W-MA and S-MA criteria can help selecting optimal designs with parameters that are not within the scope of [13, 14]. The W-MA criterion proposed in this paper is consistent with the WS-MA criterion in [15] and the S-MA criterion is new. This paper proposes 3 algorithms for constructing W-MA and S-MA designs and these algorithms provide a supplement to the theoretical construction methods in [13–15].The rest of the paper is organized as follows. Section2 introduces some related definitions and notation and proposes the W-MA and S-MA criteria for selecting FFSP designs. Section 3 gives some theoretical results and algorithms that enable us to find the optimal FFSP designs under the W-MA and S-MA criteria. Section 4 tabulates some FFSP designs for practical use. Section 5 gives a conclusion.


## 2. Optimality Criteria for Regular Split-Plot Designs

Consider an experiment which involvesn two-level factors with the two levels coded as 0 and 1, and suppose the experimenter can only afford 2q runs. From 2n runs of the full design, we select 2q ones which satisfy Bx=0 to constitute an FF design, where B is an n−q×n matrix over the Galois field GF2 with full row rank and x is a row of the full design. Such an FF design is denoted as 2n−n−q or 2n−kk=n−q. The matrix B is called the generating matrix of the design. The vectors in the row space ℛB constitute the defining contrast subgroup, denoted as G. A nonzero vector b∈ℛB is called a defining vector. A fraction of the full design is called a regular design if it is determined by a defining contrast subgroup.A vectorb′=b1,…,bn with i nonzero entries represents an ith order effect of the factors corresponding to the i nonzero entries. It is said to be of the WP type if it involves only the WP factors, and of the SP type otherwise. The cosets of the defining contrast subgroup are called alias sets. An alias set is said to be a WP alias set if it contains at least a WP type effect, and an SP alias set if it contains only SP type effects. A nonzero vector b′=b1,…,bn can also be denoted by 1b12b2⋯nbn with the convention that ibi is dropped for any i with bi=0. This system of notation is referred to as the compact notation, popularized by [16]. A defining vector b in such a compact notation is called a defining word and the number of nonzero entries of b is called its wordlength.A regular two-level FFSP design withn1 WP factors and n2 SP factors is denoted as 2n1+n2−k1+k2, which is determined by k1 independent WP type defining words and k2 independent SP type defining words. A defining word is of WP type if it involves only WP factors and SP type if it involves at least one SP factor.Take the construction of25−2 design d as an example. Let(1)B=1110010011.Then,x’s satisfied Bx=0 over GF2 constitute a 25−2 design d. The nonzero vectors in ℛB, b1′=1,1,1,0,0, b2′=1,0,0,1,1, and b3′=0,1,1,1,1 are the defining vectors of design d, and corresponding w1=123, w2=145, and w3=2345 in the compact notation, are the defining words of d. Then, the defining contrast subgroup of d is(2)G=I,123,145,2345,where I is the unity element, which corresponds to the zero vector in the row space ℛB. If 1,2, and 3 are WP factors, and 4 and 5 are SP factors, then d corresponds to an FFSP 23+2−1+1 design with WP type defining word w1 and SP type defining words s1=w2 and s2=w3, where w1 and s1 (or s2) are two independent defining words.Two2n1+n2−k1+k2 FFSP designs are isomorphic if the defining contrast subgroup of one design can be obtained from that of the other by permuting the WP factor labels and/or the SP factor labels. Isomorphic FFSP designs will be treated as the same in the following.For a2n−k design d, let Aid be the number of defining words with wordlength i. The resolution of an FF design d is defined as the smallest i such that Aid≠0. A 2n−k design with resolution R is often denoted as 2Rn−k. A 2n−k design d is said to have minimum aberration (MA) if it sequentially minimizes(3)Wd=A3d,A4d,…,And.Regarding an FFSP2n1+n2−k1+k2 design d0 as an FF 2n−k design d, where n=n1+n2 and k=k1+k2. If the 2n−k design d has MA, the 2n1+n2−k1+k2 design d0 is said to have MA.For a regular FFSP2n1+n2−k1+k2 design d, let Ai,0d be the number of WP type defining words with wordlength i, and Ai,1d be the number of SP type defining words with wordlength i. The resolution of an FFSP design d is defined as the smallest i such that Ai,0d≠0 or Ai,1d≠0. A design with resolution one or two fails to ensure the estimability of all the main effects, even under the absence of all interactions. Since the main effects are almost invariably the objects of interest in a factorial experiment, only designs with A1,0=A1,1=A2,0=A2,1=0 are considered. Define the following two n−2-dimensional sequences(4)W1d=A3,0d,A4,0d,…,An1,0d,0,…,0,W2d=A3,1d,A4,1d,…,An1,1d,…,An1+n2,1d.Among them, W1d and W2d are called the WP and SP wordlength patterns, respectively.Considering the experiments, in which the WP factors are important and the experimenter is not interested in the SP effects, Wang et al. [4] proposed the following WP-MA criterion.Definition 1.
Letd1 and d2 be the two 2n1+n2−k1+k2 designs. Under the condition that WP factors are more likely to be significant than SP factors, d1 is said to have less aberration of type WP than d2 if either (i) Ai,0d1<Ai,0d2 for the smallest integer i such that Ai,0d1≠Ai,0d2 or (ii) Ai,0d1=Ai,0d2 for any i but Aj,1d1<Aj,1d2 for the smallest integer j such that Aj,1d1≠Aj,1d2. An FFSP design d is called a minimum aberration design of type WP (WP-MA) if no other design has less aberration of type WP than d.The WP-MA criterion is suitable for selecting FFSP designs, in which the WP factors are more important than the SP factors and the experimenter is not interested in the SP effects. However, when the experimenter is interested in both the WP and SP effects, the WP-MA criterion violates the effect hierarchy principle, i.e., (i) lower order factorial effects are more likely to be important than higher order ones, and (ii) factorial effects of the same order are equally likely to be important.Considering two of the five scenarios in [6], i.e., (ii) screening with emphasis on SP effects and (iii) screening with emphasis on WP effects, we propose the following two ranking wordlength patterns for FFSP designs:(5)Wwd=A3,0,A3,1,…,An1,0,An1,1,An1+1,1,…,An1+n2,1,(6)Wsd=A3,1,A3,0,…,An1,1,An1,0,An1+1,1,…,An1+n2,1.Based on (5) and (6), we propose the following two criteria for selecting FFSP designs.Definition 2.
Letd1 and d2 be the two 2n1+n2−k1+k2 designs. Suppose i is the smallest integer such that Ai,jd1≠Ai,jd2.(1)
Designd1 is said to have less aberration of type W than design d2 if either (i) Ai,0d1<Ai,0d2 or (ii) Ai,0d1=Ai,0d2 but Ai,1d1<Ai,1d2. An FFSP design d is said to have minimum aberration of type W (W-MA) if no other design has less aberration of type W than d.(2)
Designd1 is said to have less aberration of type S than design d2 if either (i) Ai,1d1<Ai,1d2 or (ii) Ai,1d1=Ai,1d2 but Ai,0d1<Ai,0d2. An FFSP design d is said to have minimum aberration of type S (S-MA) if no other design has less aberration of type S than d.Clearly, a W-MA designd sequentially minimizes the components in Wwd and an S-MA design d sequentially minimizes the components in Wsd. In fact, the MA FFSP designs are not always unique, the W-MA and S-MA designs are further discrimination of the MA FFSP designs.


## 3. Construction of the Optimal Split-Plot Designs

This section discusses the construction of the optimal FFSP designs under the W-MA and S-MA criteria. We first give a direct result for the case ofk2=0.Theorem 1.
An FFSP2n1+n2−k1+0 design d has W-MA and S-MA if and only if its WP settings constitute an MA 2n1−k1 design.Proof.
Note that whenk2=0, the SP factors are independent of the WP factors, which results in Ai,1=0 and i=3,…,n1+n2. Then, both (5) and (6) reduce to A3,0,…,An1,0, which is just the wordlength pattern of the 2n1−k1 design formed by the WP factors of the 2n1+n2−k1+0 design d. Theorem 1 follows directly from Definition 2 and the definition of MA 2n1−k1 design. □
The construction of MA FF designs has been studied by many researchers in the past three decades. Reference [3] gave a good summary. Based on Theorem 1 and the MA designs given in Chapter 3 of [3], the W-MA and S-MA FFSP 2n1+n2−k1+0 designs can be constructed by simply adding n2 factors independent of the n1 WP factors to represent the SP factors.
Fork2≠0, we have the necessary condition for a 2n1+n2−k1+k2 design to have W-MA or S-MA as follows.Lemma 1.
If a2n1+n2−k1+k2 design d has W-MA or S-MA, then each of the n1+n2 factors is involved in some defining word of d.Proof.
Supposed is a 2n1+n2−k1+k2 design determined by k1+k2 independent defining words w1,…,wk1,s1,…,sk2. Without loss of generality, suppose factor F is not involved in any defining word. By adding factor F to one of the SP type defining word, say s1, we can obtain a 2n1+n2−k1+k2 design d′. According to whether or not factor F is involved, the defining words of d′ can be classified into two groups. Let G1 denote the group containing the defining words involving factor F and G2 denote the other group. Each defining word in G1 has one more factor (F) than the corresponding defining word of d. The defining words in the other group are the same as those of the design d. Thus, the design d′ has less aberration of type W and S than d, and d cannot have W-MA or S-MA. □
By Lemma1, we obtain the following result directly, which can help us construct some W-MA and S-MA designs with the special case of k2=1.Corollary 1.
If a2n1+n2−k1+1 design d has W-MA or S-MA, the generating matrix B of d must have a submatrix B˜ with n2 columns, such that B˜ has k1 row vectors with all 0’s and one row vector with all 1’s.Recall that the two-step randomization in an FFSP design results in two sources of errors in ANOVA, the WP and SP error terms, and the former is larger than the latter in general. If a full design is considered (i.e.,k1=k2=0), this would entail lower estimation efficiency of WP effects compared to SP effects. In the same manner, in an FFSP design, estimation from a WP alias set has a lower efficiency than that from an SP alias set ([8, 10]). So, in a good FFSP design, the main effects of the SP factors should not appear in a WP alias set. From this point, we obtain the following two lemmas.Lemma 2.
If a2n−k design d can be split into a 2n1+n2−k1+k2 design with n2−k2=1, the generating matrix B of d must have a column with k1 0’s and k2 1’s. Especially, B must have a column with all 1’s for k1=0.Lemma 3.
If a2n−k design d can be split into a 2n1+n2−k1+k2 design, the generating matrix B of d must have the following form up to permutation of columns:(7)B=B110B21B22,where B11 is a k1×n1 matrix with rankB11=k1 and B22 is a k2×n2 matrix with rankB22=k2, each row vector of B11 has at least three 1’s and each row vector of B22 has at least two 1’s.Theorem 2.
Regarding an FFSPRegarding an FFSP2n1+n2−0+k2 design d0 as an FF 2n−k2 design d, where n=n1+n2, if the 2n−k2 design d has MA, then d0 has W-MA and S-MA.Proof.
Note that whenk1=0, we have Ai,0=0 and Ai=Ai,1 for any i=3,…,n1+n2. When regarding an FFSP 2n1+n2−0+k2 design d0 as an FF 2n−k2 design d, both (5) and (6) of d0 reduce to A3,1,…,An1+n2,1, the wordlength pattern of d. Theorem 2 follows directly from Definition 2 and the definition of MA 2n−k2 design.Theorem 2 implies that we can construct W-MA and S-MA2n1+n2−0+k2 designs via 2n−k2 designs through decomposing n into suitable n1 and n2 with n=n1+n2. Then, we propose the following Algorithm 1 to construct W-MA and S-MA 2n1+n2−0+k2 designs via 2n−k2 designs.Reference [3] gave the 16-run MA 2n1+n2−0+k2 designs. By Algorithm 1, we obtain all the 32-run MA 2n1+n2−0+k2 designs with n1+n2≤12 in Table 1.Table 1
Catalogue of W-MA and S-MA2n1+n2−0+k2designs with 32 runs.
n1n2k2si1511234562411234563311234564211234561621236 124572521236 124573422356 124574322356 124571731236 1247 134582631236 1247 134583531246 2347 134584432356 2457 134581841236 1247 1258 134592741236 1247 1258 134593641256 2357 2458 134594541256 2357 2458 134591951236 1247 1258 13459 2345t02851236 1247 1258 13459 2345t03751246 2347 2458 12359 1345t04651356 1457 2358 2459 12345t011061236 1247 1348 1259 135t0145t12961236 1247 1348 1259 135t0145t13861246 1347 2348 1459 245t0345t14761256 1357 1458 2359 245t0345t111171236 1247 1348 2349 125t0135t1145t221071236 1247 1348 2349 125t0135t1145t23972346 1257 1358 1459 235t0245t1345t24871256 1357 2358 1459 245t0345t012345t2si denotes the independent SP-type defining words and t0, t1, t2, denote the factors 10, 11, 12, respectively.The WP settings of a2n1+n2−k1+k2 design constitute a 2n1−k1 design. It is clear that the resolution of a 2n1+n2−k1+k2 design is no larger than that of the corresponding 2n1−k1 design. Note that the resolution of a 2n1−k1 design is at most n1. Thus, if a 2Rn−k design d can be split into a 2n1+n2−k1+k2 design with k1>0, then n1≥R. Therefore, for given n1, n2, k1, and k2, to construct the W-MA and S-MA 2n1+n2−k1+k2 designs, we narrow the search to 2n−k designs with resolution R≤n1. In particular, if k1=1, we select the defining word with length no more than n1 as the first row of the generating matrixB.Algorithm 1: Construction algorithm to the W-MA and S-MA 2n1+n2−0+k2 designs.
Step 1
For givenn1, n2, and k2, rank 2n−k2 designs according to the MA criterion, say d1,d2,…, where n=n1+n2.Step 2
Write down the generating matrixB of the design d1 according to its defining contrast subgroup, where B is a k2×n matrix with full row rank.Step 3
Try to decomposeB into B1B2 by permuting the columns such that B1 and B2 have n1 and n2 columns, respectively, and B2 has full row rank. It also requires that B2 has a column with all 1’s if n2−k2=1.Ifd1 can be split as above, then it is the W-MA and S-MA 2n1+n2−0+k2 design. Otherwise, delete it from the list, go to d2 and repeat Steps 2 and 3.From [3], we know that the maximum resolution of 2n−2 designs is Rmax=2n/3, here x denotes the integer part of x.Remark 1.
Consider a2Rmaxn−2 design d whose defining words have the same length, i.e., ARmaxd=3. Then, d has MA and 2n=3Rmax. For any Rmax≤n1≤n−2, taking any defining word as the first row of generating matrix B of d, design d can be split into a 2n1+n2−1+1 design, which simultaneously has MA, W-MA, and S-MA.Remark 2.
Consider a2Rn−2 design d whose defining words have different lengths. Suppose that taking each defining word as the first row of generating matrix B can generate a 2n1+n2−1+1 design. Then, n1>R. Denote the 2n1+n2−1+1 designs with the shortest word and longest word putting in the first row as d1 and d3, respectively, and the other design d2. Then, under the W-MA criterion, d3 is superior to d2, and d2 is superior to d1, while under the S-MA criterion, d1 is superior to d2, and d2 is superior to d3.The following example illustrates the use of Algorithm 2 and Remarks1-2.Algorithm 2: Construction algorithm to the W-MA and S-MA 2n1+n2−1+k2.
Step 1
For givenn and k, rank 2n−k designs according to the MA criterion.Step 2
For givenn1 and n2 satisfying n=n1+n2, select the 2n−k designs with resolution R≤n1 according to the original order, say d1,d2,….Step 3
Ford1, pick k defining words from its defining contrast subgroup and generate the generating matrix B.Step 4
Try to decomposeB into B110B21B22 by permuting the columns, such that B11 is an n1-dimensional non null row vector with at least three 1’s, and B22 is a k−1×n2 matrix with rankB22=k−1 and each row of B22 has at least two 1’s.(a) IfB can be decomposed as above, then d1 corresponds to the MA 2n1+n2−1+k−1 design d1′. If B11 corresponds to any of the shortest defining words, then d1′ is the unique S-MA 2n1+n2−1+k−1 design. If B11 corresponds to any of the longest defining words, d1′ is the unique W-MA 2n1+n2−1+k−1 design.(b) Otherwise, delete it from the list, go tod2 and repeat Steps 3 and 4.Example 1.
Consider the construction of the W-MA and S-MA2n1+n2−1+1 designs with n1+n2=9. The MA 29−2 design d0 has resolution 6 and the defining contrast subgroup is(8)G=I,123457,123689,456789.Since A6d0=3, pick any defining word as the first row of the generating matrix B, d0 can be split into the 27+2−1+1 and 26+3−1+1 designs. Without loss of generality, take w1=123457 and s1=123689 to generate(9)B=111110100111001011.
Then,B satisfies the decomposition condition of Algorithm 2 for the 27+2−1+1 (d0′) design and 26+3−1+1 (d0′′, by permuting columns 6 and 7) design. By Remark 1, d0′ is the unique MA, W-MA, and S-MA 27+2−1+1 design, with WP factors 1,2,3,4,5,6 and 7=12345 and SP factors 8,9=12368. The independent defining words of d0′ are w1 (WP type) and s1 (or s2=456789, SP type). Similarly, the 26+3−1+1 design d0′′ simultaneously has MA, W-MA, and S-MA. d0′′ has WP factors 1,2,3,4,5,6=12345 and SP factors 7,8,9=12378. The independent defining words of d0′′ are changed to w1=123456 (WP type) and s1=123789 (SP type) after the permutation of columns. Obviously, they have Wwd0′=Wwd0′′=06,1,2, where 0s denotes s successive zeros. To construct 25+4−1+1 design, consider the optimal 2V9−2 design d1 under the MA criterion, which has Wd1=0,0,1,1,1. The defining contrast subgroup of d1 is(10)G=I,12345,126789,3456789.
Denoteb1′=1,1,1,1,1,0,0,0,0, b2′=1,1,0,0,0,1,1,1,1, and b3′=0,0,1,1,1,1,1,1,1. Pick b1′, b2′, and b3′ as the first row of B, respectively. Without loss of generality, denote(11)B1=b1′b2′,B2=b2′b1′,B3=b3′b2′.
Then,B1 corresponds to 25+4−1+1 (d̃), 26+3−1+1, and 27+2−1+1 (d1′) designs; B2 corresponds to 26+3−1+1 and 27+2−1+1 (d1′′) designs; B3 corresponds to a 27+2−1+1 (d1′′′) design. Among these FFSP designs split by d1, d̃ is the unique MA and hence W-MA and S-MA 25+4−1+1 design, the three 27+2−1+1 designs are inferior to d0′, and the two 26+3−1+1 designs are inferior to d0′′. Ww’s for the three 27+2−1+1 designs d1′,d1′′,d1′′′ are, respectively, Wwd1′=04,1,0,0,1,0,1, Wwd1′′=04,0,1,1,0,0,1, and Wwd1′′′=04,0,1,0,1,1,0. Obviously, under the W-MA criterion, d1′′′ is superior to d1′′, and d1′′ is superior to d1′, while under the S-MA criterion, d1′ is superior to d1′′, and d1′′ is superior to d1′′′. Then, we go to the optimal 2IV9−2 design d2 under the MA criterion, which has A4d2=A6d2=A8d2=1 and the other Aid2=0. Pick the shortest word as the first row of B, we obtain the unique MA 24+5−1+1 design, which also uniquely has W-MA and S-MA. At last we construct the unique MA 23+6−1+1 design from the optimal 2III9−2 design (see Table 1). Here, all 128-run W-MA and S-MA 2n1+n2−1+1 designs are constructed according to Algorithm 2.Now, we give three special cases of optimal2n1+n2−1+1 designs.Remark 3.
For givenn2≥2, 23+n2−1+1 design with WP factors 1,2,3=12 and SP factors 4,5,…,n2+2,145⋯n2+2 is the unique MA 23+n2−1+1 design, which has A3,0=An2+1,1=An2+2,1=1.Remark 4.
For givenn2≥2, 24+n2−1+1 design with WP factors 1,2,3,4=123 and SP factors 5,…,n2+3,125⋯n2+3 is the unique MA 24+n2−1+1 design, which has A4,0=1 and An2+2,1=2.Remark 5.
For givenn2≥3, 25+n2−1+1 design with WP factors 1,2,3,4, 5=1234 and SP factors 6,…,n2+4,126⋯n2+4 is the unique MA 25+n2−1+1 design, which has A5,0=1 and An2+2,1=An2+3,1=1.From Algorithm 2 and Remarks1–5, we obtain all the W-MA and S-MA 2n1+n2−1+1 designs with 8, 16, 32, 64, and 128 runs, and all the W-MA, S-MA 2n1+n2−1+2, and 2n1+n2−1+3 designs with 8, 16, 32, and 64 runs, see Tables 2–4.Table 2
Catalogue of W-MA and S-MA designs withk1=k2=1.
n1n2wisiWwdMAproperty321231451,1,1,0∗421234125602,1,2∗3312314561,0,1,1∗4312341256702,1,0,2∗5212345126703,1,1,1W-MA12351246702,1,02,2S-MA34123145671,02,1,1∗53123451267804,1,1,1∗621234561237804,0,2,1,0W-MA123461257804,1,1,0,1S-MA44123412567802,1,02,2∗351231456781,03,1,1∗6312345612378906,1,2∗7212345712368906,1,2∗541234512678904,1,0,1,1∗451234125678902,1,03,2∗3612314567891,04,1,1∗wi and si denote the independent WP- and SP-type defining words, respectively. ∗MA, W-MA, and S-MA design. 0s denotes s successive zeros.Table 3
Catalogue of W-MA and S-MA designs withk1=1andk2=2.
n1n2wisiWwdMA property33123145 246(1, 3, 3)∗4312341256 135702,1,6∗34123146 2457(1, 1, 3, 2)14412341257 1356802,1,2,4∗53123451267 136803,3,1,3W-MA12352367 1246802,1,2,0,4S-MA351231457 24681,0,2,3,1∗45123412568 1357902,1,0,4,2∗54123451268 1467903,1,1,3,2W-MA123512468 1347902,1,02,4,2S-MA6312345612378 1457903,1,0,4,1,1W-MA123612578 1347902,1,02,4,0,2S-MA3612314568 245791,02,3,3∗wi and si denote the independent WP- and SP-type defining words, respectively. ∗MA, W-MA, and S-MA design.Table 4
Catalogue of W-MA and S-MA designs withk1=1andk2=3.
n1n2wisiWwdMA property34123145 246 12471,6,7,02,1∗4412341256 1357 235802,1,13∗35123146 157 24581,2,7∗4512341257 1268 1356902,1,5,8∗54123451467 2468 346903,6,1,7∗361231457 1468 25691,0,5,6∗46123412568 12579 1367t002,1,1,8,4∗55123451268 12369 1467t003,2,1,7,4W-MA123512468 12479 1367t002,1,1,0,8,4S-MA641234561278 13479 13567t003,2,0,8,1,3,0,1W-MA123461278 13579 1457t003,2,1,7,0,4S-MA3712314568 24579 12467t01,0,1,6,6,1∗wi and si denote the independent WP- and SP-type defining words, respectively. ∗MA, W-MA, and S-MA design. t0 denotes the factor 10.Now, we give a general algorithm to construct the W-MA and S-MA2n1+n2−k1+k2 designs. By using this algorithm, we obtain some 2n1+n2−2+1 and 2n1+n2−2+2designs.Algorithm 3: Construction algorithm to the W-MA and S-MA 2n1+n2−k1+k2.
Step 1
For givenn1 and n2, rank 2n−k designs according to the MA criterion, say d1,d2,…, where n=n1+n2,k=k1+k2.Step 2
Write down the generating matrixB of design d1 according to its defining contrast subgroup, where B is a k×n matrix with rank(B) = k.Step 3
Try to decomposeB into B110B21B22, such that up to permutation of columns, B11 is a k1×n1 matrix with rankB11=k1, and B22 is a k2×n2 matrix with rankB22=k2. Note that each row of B11 has at least three 1’s and each row of B22 has at least two 1’s.(a) IfB can be decomposed as above, then it is the MA 2n1+n2−k1+k2 design. If there is only one decomposition, it corresponds to the unique MA design, otherwise determine the W-MA and S-MA 2n1+n2−k1+k2 designs by comparing their wordlength patterns.(b) Otherwise, delete it from the list and go tod2 and Steps 2 and 3.


## 4. Tables of Optimal Designs

Some catalogues of W-MA and S-MA2n1+n2−k1+k2 FFSP designs are tabulated in Tables 1–6. In the tables, we use the compact notation to denote the factorial effects and defining words. For a 2n1+n2−k1+k2 FFSP design, let n1+n2=n, q1=n1−k1, q2=n2−k2, and q=q1+q2. Denote the WP factors as 1,…,q1,q1+1,…,n1, where the last k1 factors are some interactions of 1,…,q1, which generate the independent WP type defining words. The SP factors are denoted as n1+1,…,n1+q2,n1+q2+1,…,n, where the last k2 factors correspond to the independent SP type defining words.Table 5
Catalogue of W-MA and S-MA designs withk1=2andk2=1.
n1n2wisiWwd52124 1352367(2, 0, 1, 2, 0, 2)53124 135236782,0,1,02,2,2621235 12461347802,3,02,454124 1352367892,0,1,03,2,2631235 124613478902,3,04,4721236 124571348902,1,0,2,2,0,255124 135236789t02,0,1,03,2,2641235 1246134789t002,3,05,4731236 1245713489t002,1,0,2,02,2,0,2wi and si denote the independent WP and SP type defining words, respectively. Each design is MA, W-MA, and S-MA design and t0 denotes the factor 10.Table 6
Catalogue of W-MA and S-MA designs withk1=2andk2=2.
n1n2wisiWwd53124 135167 2368(2, 1, 1, 6, 0, 4)54124 135168 23679(2, 1, 1, 2, 0, 4, 4, 1)631235 12461278 1347902,3,3,0,855124 13523679 12368t02,0,1,02,5,6,02,1641235 12461379 1278t002,3,02,7,0,4731236 1245712489 1358t002,1,1,2,6,0,4wi and si denote the independent WP and SP type defining words, respectively. Each design is MA, W-MA, and S-MA design and t0 denotes the factor 10.Table1 lists all the 32-run W-MA and S-MA 2n1+n2−0+k2 designs with n1+n2≤12. All the designs have MA and are split from the MA 2n−k2 designs, where n=n1+n2. All the W-MA and S-MA 2n1+n2−1+1 designs with 8, 16, 32, 64, and 128 runs are listed in Table 2, where ∗ in the last column denotes the design simultaneously has MA, W-MA, and S-MA. Tables 3 and 4, respectively, list all the W-MA and S-MA 2n1+n2−1+2 and 2n1+n2−1+3 designs with 8, 16, 32, and 64 runs. For given n1,n2,k1, and k2 in Tables 2–4, both the W-MA and S-MA designs are MA designs, though not isomorphic. Tables 5 and 6 give some 2n1+n2−2+1 and 2n1+n2−2+2 designs, which are obtained from Algorithm 3 and each which are obtained from Algorithm 3 and design simultaneously has MA, W-MA, and S-MA.Use of the design tables is illustrated in the following example.Example 2.
Consider the W-MA and S-MA25+2−1+1 designs in Table 2. Here, n=7, q1=4, and q2=1. For the W-MA design d1, we have the WP type defining word w1d1=12345 and the SP type defining word s1d1=1267, which means that 1,2,3,4 and 5=1234 are the WP factors and 6,7=126 are the SP factors. The two words w1d1 and s1d1 generate another SP type defining word, say s2d1=w1d1s1d1=34567. Thus, we obtain A50d1=1, A41d1=A51d1=1, and the wordlength pattern Wwd1=03,1,1,1, here we omit two 0’s after the last 1. Similarly, the S-MA 25+2−1+1 design d2 has two independent defining words: w1d2=1235 (WP type) and s1d2=12467 (SP type). Thus, the WP factors for d2 are 1,2,3,4 and 5=123, and SP factors are 6,7=1246. The wordlength pattern is Wwd2=02,1,02,2. Both d1 and d2 are MA 25+2−1+1 designs and they have wordlength pattern Wd1=Wd2=0,1,2. Obviously, d1 and d2 are nonisomorphic.


## 5. Conclusions

FFSP designs are useful in many scientific, agricultural, industrial, and engineering experiments. In an FFSP design, the factors whose levels are difficult to change are called whole plot (WP) factors, and the rest are called subplot (SP) factors. An FFSP design includes a two-phase randomization, which leads to two sources of errors in ANOVA. Therefore, the SP factors and the WP factors have different status.MA criterion treats the WP and SP factors with the same importance, which is inconsistent with the physical truth. In addition, the MA FFSP designs are not always unique up to isomorphism. All the criteria for the five scenarios in [6], the WP-MA criterion proposed in [4], and the SP-MA criterion proposed in [14] violate the effect hierarchy principle somewhat.According to the effect hierarchy principle and some actual circumstances and needs, this paper proposes the W-MA and S-MA criteria, which, respectively, correspond to the cases of WP factors and SP factors being more important. By decomposing the generating matrix of2n−k FF designs, we give some algorithms for splitting a 2n−k FF design into a W-MA or S-MA 2n1+n2−k1+k2 design. By using these algorithms, we obtain some W-MA and S-MA 2n1+n2−k1+k2 designs for small k1 and k2.



---

*Source: 9417813-2022-09-27.xml*