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Association Schemes in Designed Experiments Association Schemes Arise in Designed Experiments in Many Ways

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Association Schemes in Designed Experiments Association Schemes Arise in Designed Experiments in Many Ways Abstract Association schemes in designed experiments Association schemes arise in designed experiments in many ways. R. A. Bailey They were first used in incomplete-block designs, but University of St Andrews QMUL (emerita) they are implicit in the treatment structure of factorial designs and in many common block structures, such as row-column designs or nested blocks. What is nice about them is the link between the matrices which show the patterns and the matrices which project onto the common eigenspaces. In recent work, Agnieszka Łacka and I have considered designs 25th International Workshop on Matrices and Statistics where the treatments consist of all combinations of levels of (IWMS 2016), two treatment factors and one additional control treatment. Funchal, June 2016 We construct nested row-column designs which have what we call control orthogonality and supplemented partial balance. 1/36 2/36 An example of a designed experiment Outline Example I want to compare 6 new varieties of wheat (3 Dutch, 3 British). I can use 30 plots, which form a 5 6 rectangle in a single field. × How do I decide which variety to plant in which plot? I Association schemes In general, II Incomplete-block designs I Ω is a set of experimental units, III Blocking factors with random effects I is a set of treatments, IV Near-factorial treatments T V I a design is a function f : Ω such that Nested row-column designs for near-factorial treatments → T if ω Ω then f (ω) is the treatment allocated to unit ω. ∈ Ω is given: usually it has some sort of structure. is given: usually it has some sort of structure. T My job is to choose f , possibly subject to some constraints. 3/36 4/36 I. What is an association scheme? First example of an association scheme (s = 2) consists of 6 new varieties of wheat (3 Dutch, 3 British). T The structure on the set Ω of experimental units is often an Dutch British association scheme. 1 2 3 4 5 6 The structure on the set of treatments is often another T association scheme. D 1 But what is an association scheme? D 2 An association scheme on Ω with s associate classes D 3 is a partition of the unordered pairs of distinct elements of Ω B 4 into s classes such that . B 5 (a technical condition that I’ll tell you about later). B 6 And similarly for . T Class 1, shown in red: different varieties from the same country. Class 2, shown in blue: varieties from different countries. Class 0, shown in white: same variety. 5/36 6/36 Second example of an association scheme (s = 3) Third example of an association scheme (s = 2) Row 1 1 ... 1 2 2 ... 5 Diallel cross from 5 parental lines with no self-crosses Column 1 2 ... 6 1 2 ... 6 12 13 14 15 23 24 25 34 35 45 R1 C1 ... ... 1, 2 { } R1 C2 ... ... 1, 3 { } . .. .. 1, 4 . { } Ω is a 5 6 rectangle. 1, 5 × R1 C6 ... .. { } 2, 3 R2 C1 ... ... { } 2, 4 R2 C2 ... ... { } 2, 5 . .. .. { } . 3, 4 { } R5 C6 ... ... 3, 5 { } 4, 5 Class 1, shown in red: different plots in the same row. { } Class 2, shown in blue: different plots in the same column. Class 1, in red: crosses with one parental line in common. Class 3, shown in yellow: plots in different rows and columns. Class 2, in blue: crosses with no parental line in common. Class 0, shown in white: same plot. Class 0, in white: same cross. 7/36 8/36 Fourth example of an association scheme (s = 4) Adjacency matrices So where do matrices come in? Given an association scheme on a set of size N, 3 blocks, each consisting of 4 rows 6 columns the adjacency matrix for class i is the N N matrix × × 1 if (α, β) is in class i whose (α, β) entry is equal to (0 otherwise. Dutch British A0 = I 1 2 3 4 5 6 Class 0: same plot. D 1 0 1 1 0 0 0 0 0 0 1 1 1 Class 1: different plots in the same row. D 2 1 0 1 0 0 0 0 0 0 1 1 1 Class 2: different plots in the same column. 1 1 0 0 0 0 0 0 0 1 1 1 Class 3: plots in the same block but different rows and columns. D 3 A = A = 1 2 Class 4: plots in different blocks. 0 0 0 0 1 1 1 1 1 0 0 0 B 4 0 0 0 1 0 1 1 1 1 0 0 0 B 5 0 0 0 1 1 0 1 1 1 0 0 0 B 6 9/36 10/36 Definition of association scheme First example (N = 6, s = 2) Definition 0 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1 1 An association scheme with s classes on a set of size N consists 1 1 0 0 0 0 0 0 0 1 1 1 of N N adjacency matrices A0, A1, A2,..., As satisfying A = IA = A = × 0 1 0 0 0 0 1 1 2 1 1 1 0 0 0 (i) A0 = I (the identity matrix); 0 0 0 1 0 1 1 1 1 0 0 0 (ii) A0 + A1 + A2 + + As = J (the all-1 matrix); 0 0 0 1 1 0 1 1 1 0 0 0 ··· (iii) A is symmetric for i = 0, 1, 2, . , s; i a0 = 1 a1 = 2 a2 = 3 (iv) For 0 i s and 0 j s, ≤ ≤ ≤ ≤ AiAj is a linear combination of A0, A1, A2,..., As. A0Ai = AiA0 = Ai for i = 0, 1, 2. 2 The diagonal entries of Ai are equal to the row sums of Ai, so an easy consequence of this definition is that AiJ = JAi = aiJ for i = 0, 1, 2. there are integers a = 1, a , a ,..., a such that 2 0 1 2 s So it is enough to check A1. Ai has ai non-zero entries in every row and in every column. 2 A1 = 2I + A1, so this is an association scheme. 11/36 12/36 Bose–Mesner algebra Eigenspaces in our examples Given an association scheme with s classes on a set of size N, let consists of 6 new varieties of wheat (3 Dutch, 3 British). be the set of all real linear combinations of A0, A1, A2,..., As. T A N = 6 and s = 2; a = 1, a = 2 and a = 3. Conditions (iii) and (iv) show that is a commutative algebra. 0 1 2 A It is called the Bose–Mesner algebra. subspace description dimension W constant vectors 1 Theorem 0 N W1 differences between countries 1 There are subspaces W0,W1,...,Ws of R such that W2 differences within countries 4 1. if j = k then W is orthogonal to W ; 6 j k 2. every vector v in RN can be expressed uniquely Ω is a 5 6 rectangle. × as v = ∑ w with w W ; N = 30 and s = 3; a0 = 1, a1 = 5, a2 = 4 and a3 = 20. j j j ∈ j 3. Wj is contained in an eigenspace of Ai, for all i and all j; subspace description dimension 4. the idempotent matrix Pj of orthogonal projection onto Wj W0 constant vectors 1 is in , so it is a linear combination of A ,...,A ; W1 differences between rows 4 A 0 s W2 differences between columns 5 5. if λij is the the eigenvalue of Ai on Wj then Ai = ∑j λijPj; W3 everything else 20 6. W0 is the 1-dimensional space of constant vectors. 13/36 14/36 Eigenspaces in more of our examples II. Incomplete-block designs Diallel cross from 5 parental lines with no self-crosses. N = 10 and s = 2; a0 = 1, a1 = 6 and a2 = 3. Ω is a set of experimental units; is a set of treatments; subspace description dimension T f (ω) is the treatment allocated to unit ω. W constant vectors 1 0 Suppose that Ω is partitioned into blocks of equal size, W main effects of parental lines 4 1 and g(ω) is the block containing unit ω. W2 everything else 5 The usual linear model for the response Yω is 3 blocks, each consisting of 4 rows 6 columns. × 2 N = 72 and s = 4; a0 = 1, a1 = 5, a2 = 3, a3 = 15 and a4 = 48. E(Yω) = τf (ω) + βg(ω) and Var(Y) = σ I. subspace description dimension In vector form W constant vectors 1 0 Y = X1τ + X2β + e, W1 differences between blocks 2 where the ω-row of matrix X picks out the treatment f (ω) W2 differences between rows within blocks 9 1 and the ω-row of matrix X2 picks out the block g(ω). W3 differences between columns within blocks 15 W4 everything else 45 15/36 16/36 Concurrence matrix and information matrix Partial balance 1 2 N12 = X10 X2 and C = X10 X1 k− N12N120 ; also Var(τˆ ) = σ C−. Y = X1τ + X2β + e. − Definition Put N12 = X10 X2, whose entries show how often each treatment occurs in each block (we assume at most once). An incomplete-block design is partially balanced if there is an association scheme on such that The entries in the concurrence matrix N12N120 show how often T the information matrix C is in its Bose–Mesner algebra .
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