Levi graphs and concurrence graphs as tools to evaluate designs R. A. Bailey [email protected] The Norman Biggs Lecture, May 2012 1/35 What makes a block design good for experiments? I have v treatments that I want to compare. I have b blocks. Each block has space for k treatments (not necessarily distinct). How should I choose a block design? 2/35 Two designs with v = 5, b = 7, k = 3: which is better? Conventions: columns are blocks; order of treatments within each block is irrelevant; order of blocks is irrelevant. 1 1 1 1 2 2 2 1 1 1 1 2 2 2 2 3 3 4 3 3 4 1 3 3 4 3 3 4 3 4 5 5 4 5 5 2 4 5 5 4 5 5 binary non-binary A design is binary if no treatment occurs more than once in any block. 3/35 Two designs with v = 15, b = 7, k = 3: which is better? 1 1 2 3 4 5 6 1 1 1 1 1 1 1 2 4 5 6 10 11 12 2 4 6 8 10 12 14 3 7 8 9 13 14 15 3 5 7 9 11 13 15 replications differ by ≤ 1 queen-bee design The replication of a treatment is its number of occurrences. A design is a queen-bee design if there is a treatment that occurs in every block. 4/35 Two designs with v = 7, b = 7, k = 3: which is better? 1 2 3 4 5 6 7 1 2 3 4 5 6 7 2 3 4 5 6 7 1 2 3 4 5 6 7 1 4 5 6 7 1 2 3 3 4 5 6 7 1 2 balanced (2-design) non-balanced A binary design is balanced if every pair of distinct treaments occurs together in the same number of blocks. 5/35 If w is an experimental unit, put f (w) = treatment on w g(w) = block containing w. For i = 1, . , v and j = 1, . , b, let nij = jfw : f (w) = i and g(w) = jgj = number of experimental units in block j which have treatment i. The v × b incidence matrix N has entries nij. Experimental units and incidence matrix There are bk experimental units. 6/35 For i = 1, . , v and j = 1, . , b, let nij = jfw : f (w) = i and g(w) = jgj = number of experimental units in block j which have treatment i. The v × b incidence matrix N has entries nij. Experimental units and incidence matrix There are bk experimental units. If w is an experimental unit, put f (w) = treatment on w g(w) = block containing w. 6/35 The v × b incidence matrix N has entries nij. Experimental units and incidence matrix There are bk experimental units. If w is an experimental unit, put f (w) = treatment on w g(w) = block containing w. For i = 1, . , v and j = 1, . , b, let nij = jfw : f (w) = i and g(w) = jgj = number of experimental units in block j which have treatment i. 6/35 Experimental units and incidence matrix There are bk experimental units. If w is an experimental unit, put f (w) = treatment on w g(w) = block containing w. For i = 1, . , v and j = 1, . , b, let nij = jfw : f (w) = i and g(w) = jgj = number of experimental units in block j which have treatment i. The v × b incidence matrix N has entries nij. 6/35 I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge w joining vertex f (w) to vertex g(w). It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j. Levi graph The Levi graph G˜ of a block design D has 7/35 I one vertex for each block, I one edge for each experimental unit, with edge w joining vertex f (w) to vertex g(w). It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j. Levi graph The Levi graph G˜ of a block design D has I one vertex for each treatment, 7/35 I one edge for each experimental unit, with edge w joining vertex f (w) to vertex g(w). It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j. Levi graph The Levi graph G˜ of a block design D has I one vertex for each treatment, I one vertex for each block, 7/35 It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j. Levi graph The Levi graph G˜ of a block design D has I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge w joining vertex f (w) to vertex g(w). 7/35 It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j. Levi graph The Levi graph G˜ of a block design D has I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge w joining vertex f (w) to vertex g(w). 7/35 Levi graph The Levi graph G˜ of a block design D has I one vertex for each treatment, I one vertex for each block, I one edge for each experimental unit, with edge w joining vertex f (w) to vertex g(w). It is a bipartite graph, with nij edges between treatment-vertex i and block-vertex j. 7/35 ¨¨HH ¨¨ HH ¨¨ HH HH ¨¨ H ¨ HH ¨¨ H¨ 4 x 1x2 x 3x Example 1: v = 4, b = k = 3 1 2 1 3 3 2 4 4 2 8/35 ¨¨HH ¨¨ HH ¨¨ HH HH ¨¨ H ¨ HH ¨¨ H¨ Example 1: v = 4, b = k = 3 1 2 1 3 3 2 4 4 2 4 x 1x2 x 3x 8/35 HH HH HH ¨¨ ¨ ¨¨ ¨ Example 1: v = 4, b = k = 3 1 2 1 3 3 2 4 4 2 4 ¨¨ ¨¨ x ¨¨ HH H1 2 HxH x H 3x 8/35 Example 1: v = 4, b = k = 3 1 2 1 3 3 2 4 4 2 4 ¨¨HH ¨¨ x HH ¨¨ HH HH ¨¨ H1 2¨ HxH ¨¨ x H¨ 3x 8/35 Example 1: v = 4, b = k = 3 1 2 1 3 3 2 4 4 2 4 ¨¨HH ¨¨ x HH ¨¨ HH HH ¨¨ H1 2¨ HxH ¨¨ x H¨ 3x 8/35 5 8 1 x @ @ x@ x @ 2 x 4 @ @ x@ @ x63 x 7 x Example 2: v = 8, b = 4, k = 3 1 2 3 4 2 3 4 1 5 6 7 8 9/35 Example 2: v = 8, b = 4, k = 3 1 2 3 4 2 3 4 1 5 6 7 8 5 8 1 x @ @ x@ x @ 2 x 4 @ @ x@ @ x63 x 7 x 9/35 I one vertex for each treatment, I one edge for each unordered pair a, w, with a 6= w, g(a) = g(w) and f (a) 6= f (w): this edge joins vertices f (a) and f (w). There are no loops. If i 6= j then the number of edges between vertices i and j is b lij = ∑ nisnjs; s=1 this is called the concurrence of i and j, and is the (i, j)-entry of L = NN>. Concurrence graph The concurrence graph G of a block design D has 10/35 I one edge for each unordered pair a, w, with a 6= w, g(a) = g(w) and f (a) 6= f (w): this edge joins vertices f (a) and f (w). There are no loops. If i 6= j then the number of edges between vertices i and j is b lij = ∑ nisnjs; s=1 this is called the concurrence of i and j, and is the (i, j)-entry of L = NN>. Concurrence graph The concurrence graph G of a block design D has I one vertex for each treatment, 10/35 There are no loops. If i 6= j then the number of edges between vertices i and j is b lij = ∑ nisnjs; s=1 this is called the concurrence of i and j, and is the (i, j)-entry of L = NN>. Concurrence graph The concurrence graph G of a block design D has I one vertex for each treatment, I one edge for each unordered pair a, w, with a 6= w, g(a) = g(w) and f (a) 6= f (w): this edge joins vertices f (a) and f (w). 10/35 There are no loops. If i 6= j then the number of edges between vertices i and j is b lij = ∑ nisnjs; s=1 this is called the concurrence of i and j, and is the (i, j)-entry of L = NN>. Concurrence graph The concurrence graph G of a block design D has I one vertex for each treatment, I one edge for each unordered pair a, w, with a 6= w, g(a) = g(w) and f (a) 6= f (w): this edge joins vertices f (a) and f (w). 10/35 If i 6= j then the number of edges between vertices i and j is b lij = ∑ nisnjs; s=1 this is called the concurrence of i and j, and is the (i, j)-entry of L = NN>. Concurrence graph The concurrence graph G of a block design D has I one vertex for each treatment, I one edge for each unordered pair a, w, with a 6= w, g(a) = g(w) and f (a) 6= f (w): this edge joins vertices f (a) and f (w).