aa r X i v : . [ s t a t . O T ] D ec New plans orthogonal through the block factor.
Sunanda Bagchi*1363, 10th cross, Kengeri Satellite Town,Bangalore 560060,IndiaAbstract
In the present paper we construct plans orthogonal through the block factor (POTBs). Wedescribe procedures for adding blocks as well as factors to an initial plan and thus generatea bigger plan. Using these procedures we construct POTBs for symmetrical experiments withfactors having three or more levels. We also construct a series of plans inter-class orthogonalthrough the block factor for two-level factors.
A situation in which a treatment factor is neither orthogonal nor confounded to a nuisance factorwas first explored in Morgan and Uddin (1996) in the context of nested row-column designs.They derived a sufficient condition for a treatment factor, possibly non-orthogonal to the nuisancefactors, to be orthogonal to another treatment factor. They also derived a sufficient conditionfor optimality and constructed several series of orthogonal main effect plans (OMEPs) satisfyingoptimality properties. Mukherjee, Dey and Chatterjee (2002) discussed and constructed maineffect plans (MEPs) on small-sized blocks, not necessarily orthogonal to all treatment factors.Their plans also satisfy optimality properties. Optimal blocked MEPs of similar type are alsoconstructed in Das and Dey (2004). Wang (2004) constructed plans for two-level factors on blocksof size two, estimating interaction effects also.Bose and Bagchi (2007) provided plans satisfying properties similar to those of the plans ofMukherjee, Dey and Chatterjee (2002), but requiring fewer blocks. In Bagchi (2010) the conceptof orthogonality through the block factor [see Definition 2.2] is introduced. In that paper it hasbeen shown that a plan orthogonal through the block factor (POTB) may exist in a set up, wherean OMEP can not exist. Making use of the Hadamard matrices in various way, Jacroux and hisco-authors (20011, ... 2017) have come up with a number of such plans, mostly for two-levelfactors, many of them satisfying optimality properties. Other authors providing POTBs includeChen, Lin, Yang, and Wang (2015) and Saharay and Dutta (2016).Preece (1966) constructed ‘BIBDs for two sets of treatments’. Subsequently several authorsconstructed similar combinatorial objects. Among these, the ones relevant to the present paperare ‘balanced Graeco-Latin block designs’ of Seberry (1979), ‘Graeco-Latin designs of type 1’ ofStreet (1981) and ‘Perfect Graeco-Latin balanced incomplete block designs (PERGOLAs)’ of Reesand Preece (1999). We note that all these combinatorial designs are, in fact, two-factor POTBssatifying certain additional properties. We discuss these interesting combinatorial designs brieflyin Section 3.In the present paper our main objective is to provide plans in those set ups where no OMEPis available, accommodating as many factors as possible and deviating “as little as possible”1rom orthogonality. We construct a few series of POTBs for symmetrical experiment with factorshaving three or more levels. We also define plans inter-class orthogonal through the block factor(PIOTBs) [see Definition 6.1] and construct a series of such plans.In Section 2 we present the definition of a POTB along with its attractive features. Thelater sections are devoted to construction. In Section 3 we obtain a few infinite series of POTBsfor symmetric experiments with four or less factors, each with five or more levels [see Theorems3.1, 3.2 and 3.3]. In Section 4 we describe methods of recursive construction. In Section 5 weuse these methods and construct two series of POTBs for three-level factors on blocks of sizefour [see Theorems 5.1 and 5.3]. Finally, in Section 6 we construct an infinite series of PIOTBswith orthogonal classes of small size for two-level factors [see Theorem 6.1]. Many of the plansconstructed are saturated.
We shall consider main effect plans for a symmetrical experiment with m factors, laid out onblocks of constant size. Notation 2.1. (a) P will denote a main effect plan for a s m experiment consisting of b blockseach of size k . n will denote the total number of runs. Thus, n = bk .(b) The set of levels for each factor is denoted by S , the set of integers modulo s , unless statedotherwise. S m will denote the following set of m × vectors. S m = { ( x , · · · x m ) ′ : x i ∈ S } .(c) A i denotes the i th factor, i = 1 , , · · · m . The vector x = ( x , x , · · · x m ) ′ ∈ S m representsa level combination or run, in which A i is at level x i , i = 1 , , · · · m .(d) B = { B j , j = 1 , · · · b } will denote the set of all blocks of P , Thus, B j ⊂ S m , | B j | = k, ≤ j ≤ b . Sometimes we describe a plan in terms of its blocks.(e) The replication vector of A i is denoted by the s × vector r i , the p th entry of which isthe number of runs x of P such that x i = p, p ∈ S . R i denotes a diagonal matrix with diagonalentries same as those of r i in the same order, ≤ i ≤ m .(f ) For ≤ i, j ≤ m , the A i versus A j incidence matrix is the s i × s j matrix N ij . The ( p, q ) thentry of this matrix is N ij ( p, q ) , which is the number of runs x of P such that x i = p, x j = q, p ∈ S i , q ∈ S j . When j = i, N ij = R i .(g) L i will denote the A i -versus block incidence matrix, ≤ i ≤ m . Thus, the ( p, j ) th entryof the L i is L i ( p, j ) = | x ∈ B j : x i = p | , p ∈ S, ≤ j ≤ b, ≤ i ≤ m. (h) The s × vector α i will denote the vector of unknown effects of A i , ≤ i ≤ m . Consider the normal equations for a plan P as described above. If we eliminate the generaleffects and the vector of block effects from this system of equations, we get the reduced normalequation for the vectors of all (unknown) effects of all the treatment factors. This is a system of ms equations, but it is convenient to view it as m systems of s equations each, the i th systemequations is of the form m X j =1 C ij ; B c α j = Q i ; B . (2.1)2ere C ij ; B , ≤ j ≤ m are the coefficient matrices and Q i ; B is the vector of adjusted (for theblocks) totals for A i .For a fixed i , we can eliminate c α j , j = i from ( 2.1 ) and getthe reduced normal equation for b α i as C i ;¯ i b α i = Q i ;¯ i . (2.2)We omit the expressions for the quantities C ij ; B , C i ;¯ i , Q i ; B and Q i ;¯ i above. Those are notnecessary here and are available in Bagchi and Bagchi (2020), for instance. With this backgroundwe present a few definitions. Definition 2.1.
An m-factor MEP is said to be ‘connected’ if
Rank ( C i ;¯ i ) = s − , for every i = 1 , , · · · m . Definition 2.2. [ Bagchi (2010)] Fix i = j ≤ i, j ≤ m . The factors A i and A j are said to be orthogonal through the block factor (OTB) if kN ij = L i ( L j ) ′ . (2.3) We denote this by A i ⊥ bl A j .A plan P is said to be a plan orthogonal through the block factor (POTB) if A i ⊥ bl A j for every pair ( i, j ) , i = j, i, j = 1 , · · · m . Remark 2.1:
Condition ( 2.3 ) is equivalent to equation (7) of Morgan and Uddin (1996) inthe context of nested row-column designs.Let us try to see the implications of orthogonality through the block factor. Let SS i ; all (respectively SS i ; B ) denote sum of squares for A i , adjusted for all other factors (respectively theblock factor). The following results are known. Theorem 2.1.
Consider a plan P . Fix i ∈ { , · · · m } .(a) [Bagchi(2010)] If for j = i A i ⊥ bl A j , then(i) C ij ; B = 0 and (ii) Cov ( l ′ b α i , m ′ c α j ) = 0 , for l ′ s = 0 = m ′ s .(b)[Bagchi (2020)] Further, A i ⊥ bl A j , ∀ j = i is necessary and sufficient for the following.(i) C i ;¯ i = C ii ; B and (ii) SS i ;¯ i = SS i ;0 with probability 1. Discussion :
Theorem 2.1 says the following about the inference on the factors of a connectedmain effect plan. The inference on a factor A i depends only on the relationship between A i andthe block factor if and only if A i is orthogonal to every other treatment factor through the blockfactor. Moreover, the data analysis of a POTB is very similar to the data analysis of a blockdesign with s treatments.It is well-known that the orthogonal MEP obtained from an orthogonal array is the bestpossible MEP in the sense that the estimates have the maximum precision among all MEPs inthe same set up. The same cannot be said about an POTB since its performance also depends onthe relationships of the treatment factors with the block factor. In the next theorem a guidelinefor the search for a ‘good’ POTB is provided. We omit the proof which can be obtained bygoing along the same lines as in the proofs of Lemma 1 and Theorem 1 of Mukherjee, Dey andChatterjee (2002). [See Shah and Sinha (1989) for definitions, results and other details aboutstandard optimality criteria] 3 heorem 2.2. Suppose a connected POTB ρ ∗ satisfies the following condition. For a factor A i and a non-increasing optimality criterion φ , L i is the incidence matrix of a block design d whichis φ -optimal in a certain class of connected block designs with s treatments and b blocks of sizek each. Then, ρ ∗ is φ -optimal in a similar class of connected m-factor MEPs in the same set-upas ρ ∗ for the inference on A i . In particular, using the well-known optimality results of Kiefer (1975) and Takeuchi (1961)we get the following result.
Corollary 2.1.
Suppose ρ ∗ is a connected POTB. Fix i ∈ { , · · · m } .(a) If L i is the incidence matrix of a BIBD, then, for the inference on A i , ρ ∗ is universallyoptimal in the class of all m-factor connected MEP containing ρ ∗ .(b) If L i is the incidence matrix of a group divisible design satisfying λ = λ + 1 , then ρ ∗ isE-optimal in the class of all m-factor connected MEP containing ρ ∗ , for the inference on A i . In view of the above result, we introduce the following term.
Definition 2.3.
A connected POTB is said to be balanced if each of its factors form a BIBDwith the block factor , that is L i is the incidence matrix of a BIBD for each i, ≤ i ≤ m We now present a small example of a balanced POTB on six blocks of size two each. It hastwo factors, each with four levels 0,1,2,3.
Example 1 [Bagchi and Bose (2007)] :Blocks → B B B B B B Factors ↓ A A We shall now proceed to construct POTBs for a symmetric experiment. Most of the constructionsare of recursive type, in the sense that from a given initial plan we generate a plan by addingblocks and/or factors.
Definition 3.1.
Consider an initial plan P for an s m experiment as described in Notation 2.1.For B ∈ B and v ∈ S m , B + v will denote the following set of k runs. B + v = { x + v, x ∈ B } .Here x + v = [ x i + v i : 1 ≤ i ≤ m ] ′ , where the addition in each co-ordinate is modulo s .By the plan generated from P by adding S we shall mean the plan (for the same exper-iment) having the set of blocks { B + u m : u ∈ S, B ∈ B} . The new plan P will be denoted by P ⊕ S . We shall now proceed to construction. We begin with plans with a small set of factors. Let S + denote S ∪ {∞} . The following rule will define addition in S + . u + ∞ = ∞ = ∞ + u, u ∈ S. (3.4)4 heorem 3.1. Suppose s is an integer ≥ . Then POTBs with block size two exists for thefollowing experiments.(a) For an s experiment a POTB P on s blocks exists. In the case s = 5 , P is balanced.(b) (i) For an s experiment a POTB P on s blocks exists. If s = 10 , then P is E-optimalfor the inference on each factor.(ii) Moreover, if s ≥ , there exists a POTB P with the same parameters as P , but non-isomorphic to the same. If s = 9 , P is balanced.(c) A POTB P for a ( s + 1) experiment with s blocks exists, whenever n ≥ . Proof :
In each case, we present the blocks of an initial plan P . The required plan is P ⊕ S [see Definition 3.1]. Here a, b, c, d are distinct members of S \ { } . That the final plan is aPOTB can be verified by straightforward computation. Proofs for the optimality properties arepresented.(a) The blocks of P are given below.Blocks → B B Factors ↓ A a -a b -b A b -b -a a .If s = 5, taking a = 1 , b = 2 we get a balanced POTB.(b) (i) The blocks B l , l = 1 , · · · P are as follows.Blocks → B B B B Factors ↓ A A a -a 0 -a -b b 0 b A A b -b 0 -b a -a a 0 .If n = 10, we take a = 1 and b = 3. Then for every i = 1 , · · · L i is the incidence matrix of agroup divisible design with five groups, the jth group being the pair of levels { j, j + 5 } j = 0 , · · · λ = 0 and λ = 1. This plan is, therefore, E-optimal for the inference on all the fourfactors by the result of Corollary 2.1 (b).(b) (ii) The blocks B l , l = 1 , · · · P are as follows.Blocks → B B B B Factors ↓ A a -a b -b c -c -d d A b -b -a a -d d -c c A c -c d -d -a a b -b A d -d -c c b -b a -aBy taking a = 1 , b = 2 , c = 3 and d = 4 in the case s = 9, we get a balanced POTB.(c) The set of levels for each factor is S + . The blocks B l , l = 1 , · · · → B B B B B B Factors ↓ A ∞ a -a b -b c -c a -a a -a A a -a 0 ∞ c -c -b b a -a -a a A b -b c -c 0 ∞ a -a -c c -c c A c -c b -b a -a 0 ∞ -c c c -c . (cid:3) We now list a few combinatorial structures in the literature which are actually balancedPOTBs (for symmetrical or asymmetrical experiments).(a)
Balanced Graco-Latin block design defined and constructed in Seberry (1979) hevetwo factors.(b)
Graco-Latin block design of type 1 of Street (1981) are also two-factor balancedPOTBs satisfying N = J. (c) Perfect Graeco-Latin balanced incomplete block designs (PERGOLAs) definedand discussed extensively in Rees and Preece (1999) are two-factor balanced POTBs satisfying N N ′ = N ′ N = f I s + gJ s , where f, g are integers . (3.5)Here I n is the identity matrix and J n is the all-one matrix of order n .(d) Mutually orthogonal BIBDs defined and constructed by Morgan and Uddin (1996)are multi-factor balanced POTBs.
Remark 3.1:
The definition of neither balanced Graco-Latin block designs nor of mutuallyorthogonal BIBDs include condition ( 3.5 ). However, it is interesting to note that all thesedesigns constructed so far do satisfy this condition. One would, therefore, suspect that thiscondition is implicit in the definition. We have, however, found a balanced POTB which doesnot satisfy this condition, as is shown below.
Theorem 3.2.
Let s be a positive integer ≥ . Then(a) there exists a symmetric POTB P with three factors each having s + 1 levels on b = 6 s blocks of size two.(b) In the case s = 5 , we get a Balanced POTB. The restriction to any two of the factorsreduces it to a PERGOLA, except that condition ( 3.5 ) is not satisfied. Proof : (a) Let S + be the set of levels for each factor. Consider an initial plan P with theset of factors { A , A , A } and B = { B ij , i = 1 , , j = 0 , , } , where B ij ’s are as shown in thetable below. The required plan P = P ⊕ S .Blocks → B B B B B B Factors ↓ A ∞ ∞ A ∞ ∞ A -1 1 0 1 ∞ ∞ P satisfies ( 2.3 ) follows by straightforward verification.(b) Let s = 5. One can verify that the incidence matrices satisfy the following.6 ij = , i, j = 0 , , . (3.6)Moreover, L i ( L i ) ′ = 8 I + 2 J , i = 0 , , . (3.7)We see that each L i is the incidence matrix of a BIBD with parameters ( v = 6 , b = 30 , r =10 , k = 2 , λ = 2). Thus, by Definition 2.3 P is a balanced POTB. However, N ij does not satisfy( 3.5 ), i = j, i, j = 0 , , (cid:3) Next we construct a series of balanced POTBs using finite fields. We first introduce thefollowing notation.
Notation 3.1. (i) F denotes an union counting multiplicity.(ii) For a set A and an integer n , nA denotes the multiset in which every member of A occurs n times.(iii) For subsets A and B of a group ( G, +) , A − B = { a − b : a ∈ A, b ∈ B } . Notation 3.2. (i) s is an odd prime power. t = ( s − / . F denotes the Galois field of order s . Further, F ∗ = F \ { } and F + = F ∪ {∞} .(ii) α denotes a primitive element of F .(iii) C denotes the subgroup of order t of the multiplicative group of F and C the coset of C .Thus, C is the set of all non-zero squares of F , while C is the set of all non-zero non-squaresof F .(iv) ( i, j ) denotes the number of ordered pairs of integers (k,l) such that the following equationis satisfied in F . [ This notation is borrowed from the theory of cyclotomy] α k = α l , k ≡ i, l ≡ j (mod 2) . We present the following well-known result for ready reference. [See equations (11.6.30),(11.6.40) and (11.6.43) of Hall (1986)].
Lemma 3.1.
The difference between the cosets of F ∗ can be expressed in terms of the cyclotomynumbers as follows. C − C = [ k =0 ( k, C k . The following cyclotomy numbers are known.
Case 1: t odd. (0,0) = (1,1) = (1,0) = (t-1)/2, (0,1) = (t+1)/2.
Case 2: t even. (0,0) = t/2 -1, (0,1) = (1,0) = (1,1) = t/2.
A series of two-factor balanced POTBs : heorem 3.3. Suppose s is an odd prime or a prime power. Then there exists a balanced POTB P ∗ for a ( s + 1) experiment on b = 2 s blocks of size ( s + 1) / . Proof :
The set of levels of each factor is F + . We shall present the initial plan P consistingof a pair of blocks. The required POTB is P ∗ = P ⊕ F .Let δ ∈ C . Consider three 2 × ( t + 1) arrays R , R and R , the rows of which are indexedby { , } and the columns by C ∪ { } . The entries of the arrays are as given below. R (1 ,
0) = R (0 ,
0) = R (0 ,
0) = 0 and R (0 ,
0) = R (1 ,
0) = R (1 ,
0) = ∞ . (3.8)For x = 0 , , y ∈ C , R ( x, y ) = δ x y, R ( x, y ) = δ − x y and R ( x, y ) = δ x − y. (3.9)For i = 0 , ,
2, let B i be the block, the runs of which are the columns of R i . When t is even, B and B constitute P , while B and B constitute P when t is odd.Clearly block size is t + 1 = ( s + 1) /
2. To show that P ∗ satisfies the required property, wehave to show that(a) P ∗ is a POTB and (b) each factor forms a BIBD with the block factor.Condition (b) follows from the construction in view of Lemma 3.1. So, we prove (a). Let uswrite N for N . The rows and columns of N are indexed by F + . From ( 3.8 ) and (3.9), we seethat N ( ii ) = 0 , i ∈ F + and N ( ∞ , i ) = N ( i, ∞ ) = 1 , i ∈ F. (3.10)So, we assume i = j ∈ F . Let u = j − i . Then, N ( ij ) is the number of times u appears inthe multiset (cid:26) ( δ − C F ( δ − − C if t is even( δ − C F (1 − δ − ) C if t is oddSince − ∈ C if and only if t is even, δ − − δ − t is odd.Therefore, the relations above together with ( 3.10 ) above imply that N = J s +1 − I s +1 . (3.11)Now we take up L L ′ = H (say). From ( 3.8 ) and (3.9), we see that H ( ii ) = 0 , i ∈ F + . (3.12)Further, for every i ∈ F , H ( ∞ , i ) is the replication number of i in the block design generated bythe initial block { } ∪ C . Similarly, H ( i, ∞ ) is the replication number of i in the block designgenerated by the initial block { } ∪ C if t is odd and { } ∪ C otherwise. Thus, H ( ∞ , i ) = H ( i, ∞ ) = t + 1 , i ∈ F. (3.13)We, therefore, assume i = j, i, j ∈ F . Let u = j − i . Then, H ( ij ) is the number of times u appears in the multiset (cid:26) (( { } ∪ C ) − C ) F ( C − ( { } ∪ C )) if t is even(( { } ∪ C ) − C ) F ( C − ( { } ∪ C )) if t is oddThese relations, together with ( 3.12 ), ( 3.13 ) and Lemma 3.1 imply that H = ( t + 1)( J s +1 − I s +1 ). Therefore, in view of ( 3.11 ), ( 2.3 ) follows and we are done. (cid:3) More on recursive construction
In this section we describe procedures for adding factors as well as blocks to an initial plan.
Notation 4.1.
Consider a subset V of S m .For every i, ≤ i ≤ m , V i will denote the following multiset of | V | members of S . V i = { v i : v = ( v , · · · v m ) ′ ∈ V } . Similarly, V ij will denote the following multiset of | V | members of S × S . V ij = { ( v i , v j ) : v = ( v , · · · v m ) ′ ∈ V } . Definition 4.1.
Consider an initial plan P for an s m experiment as described in Notation 2.1.Let V be as in Notation 4.1. By the plan P + V generated from P along V we shall meanthe plan (for the same experiment) having the set of blocks B + V = { B + v : v ∈ V, B ∈ B} ,where B + v is as in Definition 3.1. Usually, V will contain the -vector, so that the blocks of P will also be blocks of P + V . The next lemma provides a few sufficient conditions on P and V so that a given pair offactors are orthogonal through the block factor in P + V . The proof is by direct verification. Remark 4.1:
In an initial plan, say P , one or more levels of one or more factors may beabsent. P may still be a POTB if ( 2.3 ) holds (with one or more row/column of N ij ’s beingnull vectors) for every unordered pair of ( i, j ). In such cases one has to choose V such that alllevels of all factors do appear in P + V . Lemma 4.1.
Consider an initial plan P for an s experiment. For V ⊂ F × F , consider P + V .The following conditions on P and V are sufficient for P + V to be a POTB.(a) In P all the levels of the first factor appear and V = { (0 , i ) , i ∈ S } .(b) P is arbitrary and V = { ( i, j ) , i, j ∈ S } .(c) P has a pair of blocks B , B each of size 2, as described below. Let i = j, k = l ∈ S .Let x = ( i, i ) ′ , y = ( j, j ) , x = ( k, l ) ′ and y = ( l, k ) ′ . B i consists of runs x i and y i , i = 0 , . V = ( u, u ) , u ∈ S .(d) P is a POTB in which with one or more levels of one or both factors may be absent. V is such that every member of S appears at least once in each V i , i = 1 , . Our next procedure enlarges the set of factors of a given plan, while keeping the number ofblocks fixed.
Definition 4.2. (a) Consider a plan P as in Notation 2.1. Suppose there is another plan P ′ having b blocks of size k each. We shall combine these two plans to get another one with a largerset of factors.Let x ij (respectively ˜ x ij ) denote the j th run in the ith block of P (respectively P ′ ) , ≤ j ≤ k, ≤ i ≤ b . Let y ij = (cid:2) x ij ˜ x ij (cid:3) ′ , ≤ j ≤ k, ≤ i ≤ b . Then, the plan on b blocks of size k with y ij as the j th run in the ith block, ≤ j ≤ k, ≤ i ≤ b is said to obtained by joining thefactors of P and P ′ together. The new plan will be denoted by (cid:2) P P ′ (cid:3) .(b) In case P ′ is a copy of P then (cid:2) P P ′ (cid:3) is denoted by P . For t ≥ , the plan P t is defined in the same way. In this case we name the factors of P and its power P t as in thenotation below. otation 4.2. Consider a plan P having a set of m factors F = { A, · · · M } . The set of factorsof P t will be named as F = t [ i =1 F i , where F i = { A i , · · · M i } . Combining Definitions 4.1 and 4.2 we get a recursive construction described below.
Definition 4.3.
Consider an initial plan P for an s m experiment laid on b blocks of size k each.Consider a p × q array H = (( h ij )) ≤ i ≤ p, ≤ j ≤ q . We now obtain a plan for an s mq experiment on bp blocks of size k using the array H as follows. We first obtain P q following Definition 4.2.Let v i = (cid:2) h i . ′ t h i . ′ t · · · h iq . ′ t (cid:3) ′ , ≤ i ≤ p and V H = { v i , ≤ i ≤ p } .Our required plan P is P q + V H and it will be denoted by H ♦ P . Symbolically, P = H ♦ P = P q + V H . (4.14)Our task is to find a suitable array H so that the plan H ♦ P satisfies certain desirable prop-erties. A natural choice for H is an orthogonal array of strength 2. We shall use a modificationof an orthogonal array so as to accommodate a few more factors. Definition 4.4 (Rao(1946)) . Let m, N, t ≥ be integers and s is an integer ≥ . Then anorthogonal array of strength t is an m × N array, with the entries from a set S of s symbolssatisfying the following. All the s t t -tuples with symbols from S appear equally often as columnsin every t × N subarray. Such an array is denoted by OA ( N, m, s, t ) . Notation 4.3. (a) The set of symbols of an OA ( N, m, s, is assumed to be the set of integersmodulo s .(b) The array obtained by adding a column of all zeros (in the th position, say) to an OA ( N, m − , s, will be denoted by Q ( N, m, s ) . Exploring the properties of an orthogonal array of strength 2, we get the following result fromthe recursive construction described in Definition 4.3.
Theorem 4.1.
Consider a plan P for an s t experiment on b blocks of size k each. If an OA ( N, m − , s, exists, then ∃ a plan P with a set of s mt factors on bN blocks of size k eachwith the following properties. Here the factors of P q as well as P are named according to Notation4.2.(a) For P = Q, P, Q ∈ F , P i ⊥ bl Q i for every i, ≤ i ≤ m − , if and only if P ⊥ bl Q in P .(b) P i ⊥ bl Q j , P, Q ∈ F , i = j, ≤ i, j ≤ m − , . Proof :
By assumption Q = Q ( N, m, s ) exists. The required plan P is Q ♦ P . Property (a)follows from the construction while (b) follows from (b) of Lemma 4.1. Remark 4.2 :
Table 1 of Rees and Preece (1999) presents a number of examples of PER-GOLAs [see the statement proceeding ( 3.5 )]. An Application of Theorem 4.1 on each of themwould yield a balanced POTB for a larger set of factors.Finally, we describe a procedure of modifying the sets of levels of factors. Specifically, givena pair of plans with the same number of factors and the same block size, we obtain a plan bymerging the sets of levels of the corresponding factors of the given plans.10 efinition 4.5.
Consider a pair of plans P and P each having t factors and blocks of size k .Let S i denote the set of levels of each factor of P i , s i = | S i | , i = 1 , . We assume that S = S .Let U = S ∪ S and u = | U | . The plan consisting of all the blocks of P and P taken togetherwill be viewed as a plan, say P ∪ P , for an u t experiment in the following sense.(a) Each factor of P ∪ P will have U as the set of levels.(b) Fix p ∈ U . Let R ijp denote the set of runs of P j , in which the level p of the i th factorappears, j = 1 , , ≤ i ≤ t . [Needless to mention that R ijp = φ if p is not in S j .] Then, the level p of the i th factor of P ∪ P appears in exactly the runs in R i p ⊔ R i p , ≤ i ≤ t . Remark 4.3:
From Definition 4.5 we see that for p ∈ U , the replication number of level p of the i th factor of P ∪ P is r i ( p ) + r i ( p ), where r ij ( p ) is the replication number of level p ofthe i th factor of P j .For instance, in Theorem 5.1 below, Definition 4.5 is used to construct P h by merging thecorresponding factors of P h and P h . There, S = { , } , while S = { , } . Thus, while both P h and P h are equireplicate, the replication number of level 0 of each factor of P h is double ofthe levels 1 and 2 of the same factor.The following result is an immediate consequence of Definition 4.5 . Lemma 4.2.
Consider a pair of connected plans P and P , as in Definition 4.5 (recall Definition2.1). Then, we can say the following about the plan P = P ∪ P .(a) If both P and P are POTB, then so is P .(b) P is connected, if and only if S ∩ S = φ . In this section we make use of the tools described in Section 4 to generate plans for three-levelfactors. The factors of the initial and final plans are named in accordance with Notation 4.2.
Theorem 5.1. If h is the order of a Hadamard matrix, then there exists a connected and saturatedPOTB P h for a h experiment in h blocks of size each. Proof :
Let O = OA (4 , , ,
2) with S = { , } . Let P be the plan consisting of a singleblock consisting of the four columns of O as runs. Thus, P is an OMEP for a 2 experiment.By hypothesis Q = Q ( h, h,
2) exists. Let P h = Q ♦ P and P h be obtained from P h byreplacing level 1 of every factor by the level 2. Next we construct our required plan P h = P h ∪P h by using Definition 4.5. By construction P h has 2 h blocks of size 4 each.We now show that P h is a POTB. We note that by Theorem 4.1, each of P h and P h is aPOTB for a 2 h experiment on h blocks of size 4 each. The sets of levels of each factor of themare { , } and { , } respectively. It follows from Lemma 4.2 that P h is a connected POTB foran experiment with 3 h factors, the set of levels of each factor being { , , } . Since the availabledegrees of freedom for the treatment factors is 2 h (4 −
1) which is the same as the required degreesof freedom, the plan is saturated. (cid:3)
We now take h = 2 and present the plan P for a 3 experiment on four blocks of size foureach. 11 able 5.1 : The plan P Blocks → B B B B Factors ↓ A
00 11 00 11 00 22 00 22 B
01 01 01 01 02 02 02 02 C
01 10 01 10 02 20 02 20 A
00 11 11 00 00 22 22 00 B
01 01 10 10 02 02 20 20 C
01 10 10 01 02 20 20 02For the next construction we need some more notations.
Notation 5.1. O is as in the proof of Theorem 5.1. T will denote the array obtained from O by replacing each 1 by 2 and ˜ T the array obtained from T by interchanging 0 and 2. Theorem 5.2.
A POTB for a experiment on two blocks of size four exists. Proof :
Let B = O , B = T and B = ˜ T . The set of columns of each of themconstitutes an OMEP for a 2 experiment, the set of levels of factors being { , } for B , while { , } for the other two.Let ρ (respectively ρ ) denote the plan consisting of the pair of blocks B , B (respectively B , B ). By Lemma 4.2, each of ρ and ρ is a POTB for a 3 experiment. (cid:3) Using the pair of plans constructed above, we generate a bigger plan.
Theorem 5.3. (a) If there exists an OA ( N, m, , , then there exists a connected POTB P m fora m +1) experiment in N blocks of size each.In particular P m is saturated whenever N = 3 n and m = (3 n − − / , for an integer n ≥ .(b) There exists a connected POTB for a experiment in blocks of size each. Proof of (a):
Let the factors of ρ and ρ be named as A, B, C and ˜ A, ˜ B, ˜ C respectively.Let O = OA ( N, m, ,
2) and Q = Q ( N, m, P m and P m as follows. P m = Q ♦ ρ and P m = O ♦ ρ . Clearly, P m and P m are plans for 3 m +1) and 3 m experiments respectively, each on 2 N blocks of size 4. Following Notation 4.2, we name of the factors of these plans as follows.The factors of P m are A , B , C , A , B , C , · · · A m , B m , C m and the factors of P m are ˜ A , ˜ B , ˜ C , · · · ˜ A m , ˜ B m , ˜ C m . Now we combine the factors of P m and P m following Definition 4.2 (a) and thus obtain ourrequired plan P m . Symbolically, P m = (cid:2) P m P m (cid:3) . By construction, P m is a plan for 2 m + 1 three-level factors on 2 N blocks of size 4 each. Weshall now show that it is a POTB. 12heorems 4.1 and 5.2 imply that each one of P m and P m is a POTB. Therefore, if we showthe following relation, then we are done. P i ⊥ bl ˜ Q j , P, Q ∈ { A, B, C } , i ∈ I ∪ { } , j ∈ I, where I = { , · · · m } . (5.15)To show this relation, we fix P i and ˜ Q j as above. Case 1. i, j ∈ I : Since ρ and ρ are POTBs, ( 5.15 ) follows from Lemma 4.1 (d), whenever Q = P . Again, (c) of the same Lemma proves ( 5.15 ) for the case Q = P . Case 2. i = 0 , j ∈ I : We take P as the first and ˜ Q j as the second factor. Then applyingLemma 4.1 (a) we get ( 5.15 ).Hence the proof of the first part is complete.To prove the second part, we see that P m is saturated when N = 2 m + 1. Now Rao (1946)has shown that an OA ( s n , ( s n − / ( s − , s,
2) exists whenever n ≥
2. (see Theorem 3.20 ofHedayat, Sloane and Stufken (1999) for instance). Putting s = 3, we get the result. Proof of (b) :
Let O = (cid:20) (cid:21) . and Q = . Now the construction for theplan, say P , is just like that in Case (a). The verification is also exactly like the same in Case(a) with I = { } . (cid:3) We now present P . Table 5.2 : The plan P Blocks → B B B B B B Factors ↓ A
00 11 00 22 00 11 00 22 00 11 00 22 B
01 01 02 02 01 01 02 02 01 01 02 02 C
01 10 02 20 01 10 02 20 01 10 02 20 A
00 11 00 22 11 22 11 00 22 00 22 11 B
01 01 02 02 12 12 10 10 20 20 21 21 C
01 10 02 20 12 21 10 01 20 02 21 12˜ A
00 11 22 00 11 22 00 11 22 00 11 22˜ B
01 01 20 20 12 12 01 01 20 20 12 12˜ C
01 10 20 02 12 21 01 10 20 02 12 21
Inter-class orthogonal plans are defined in Bagchi (2019) in the context of plans without anyblocking factor. Here we extend the definition to the present context - the orthogonality beingthrough the block factor.
Definition 6.1.
Let us consider a plan ρ . Suppose the set of all factors of ρ can be dividedinto several classes in such a way that if two factors belong to different classes, then they areorthogonal through the block factor. Such a plan ρ is called a “Plan Inter-class Orthogonalthrough the Blocks (PIOTB)” and the classes will be referred to as “orthogonal classes”.
13e shall now proceed towards the construction of a series of PIOTBs. Using the relationbetween orthogonal arrays of strength two and Hadamard matrices, [see Theorem 7.5 in Hedayat,Sloane and Stuffken (1999), for instance], we see that a Q ( n, n,
2) exists whenever n is the orderof a Hadamard matrix. Theorem 6.1.
Suppose Hadamard matrices of orders m and n exist. Then, there exists asaturated PIOTB P ( m,n ) for a mn experiment on n blocks of size m + 1 each. There are n orthogonal classes of size m each. Proof :
By hypothesis Q m = Q ( m, m,
2) exists. Let R be the m × m + 1 array obtainedby juxtaposing a column of all-ones to Q m . Let P be the plan for a 2 m experiment on a singleblock consisting of m + 1 runs, which are the columns of R . Let us name the factors of P as A, B, · · · M } . Note that the column added to Q m saves A from being confounded with the block.By hypothesis, Q n = Q ( n, n,
2) exists. Let P ( m,n ) = Q n ♦ P . Clearly, P ( m,n ) is an main effectplan for a 2 mn experiment with parameters as in the statement. By construction, no factor isconfounded with the block factor. Using Theorem 4.1 and the property of P , we see that P n is interclass orthogonal with orthogonal classes { A i , B i , · · · M i } , ≤ i ≤ n (recall Notation 4.2).Hence the result. (cid:3) We now present the plans P (4 , . Table 6.1 : The plan P , Blocks → B B B B Factors ↓ A
00 00 1 00 00 1 00 00 1 00 00 1 B
00 11 1 00 11 1 00 11 1 00 11 1 C
01 01 1 01 01 1 01 01 1 01 01 1 D
01 10 1 01 10 1 01 10 1 01 10 1 A
00 00 1 00 00 1 11 11 0 11 11 0 B
00 11 1 00 11 1 11 00 0 11 00 0 C
01 01 1 01 01 1 10 10 0 10 10 0 D
01 10 1 01 10 1 10 01 0 10 01 0 A
00 00 1 11 11 0 00 00 1 11 11 0 B
00 11 1 11 00 0 00 11 1 11 11 0 C
01 01 1 10 10 0 01 01 1 10 10 0 D
01 10 1 10 01 0 01 10 1 10 010 A
00 00 1 11 11 0 11 11 0 00 00 1 B
00 11 1 11 00 0 11 00 0 00 11 1 C
01 01 1 10 10 0 10 10 0 01 01 1 D
01 10 1 10 01 0 10 01 0 01 10 1There are four orthogonal classes, which are { A i , B i , C i , D i } , i = 1 , , , Theorem 6.2.
A saturated PIOTB exists for a experiment on four blocks of size four each. Proof :
Consider the following plan P . It is easy to see that it is a PIOTB with non-orthgonalclasses { P , P } , P = A, B, C . 14 able 6.2 : Plan P Blocks → B B B B Factors ↓ A
00 12 00 21 00 12 00 21 B
01 02 02 01 10 20 20 10 C
01 20 02 10 02 10 01 20 A
01 01 02 02 01 01 02 02 B
01 10 02 20 10 01 20 02 C
00 11 00 22 11 00 22 00
Remark 6.1:
A POTB for a 4 experiment on 4 blocks of size 4 is well-known [can beobtained by treating a row of OA(16,5,4,2) as the block factor]. By collapsing two of the levelsof each factor to one level one gets a POTB for a 3 experiment on the same set up. Allowingnon-orthogonality we have been able to accommodate two more three-level factors, making itsaturated.
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