Using graphs and Laplacian eigenvaluesto evaluate block designs

R. A. BaileyUniversity of St Andrews / QMUL (emerita)

Ongoing joint work with Peter J. Cameron

Modern Trends in Algebraic Graph Theory,Villanova, June 2014

1/52

Abstract

Consider an experiment to compare v treatments in b blocks ofsize k.

Statisticians use various criteria to decide which design is best.

It turns out that most of these criteria are defined by theLaplacian eigenvalues of one of the two graphs defined by theblock design:I the Levi graph, which has v + b vertices,I or the concurrence graph, which has v vertices.

The algebraic approach shows that sometimes all the criteriaprefer highly symmetric designs but sometimes they favourvery different ones.

2/52

Abstract

Consider an experiment to compare v treatments in b blocks ofsize k.

Statisticians use various criteria to decide which design is best.

It turns out that most of these criteria are defined by theLaplacian eigenvalues of one of the two graphs defined by theblock design:I the Levi graph, which has v + b vertices,I or the concurrence graph, which has v vertices.

The algebraic approach shows that sometimes all the criteriaprefer highly symmetric designs but sometimes they favourvery different ones.

2/52

Abstract

Consider an experiment to compare v treatments in b blocks ofsize k.

Statisticians use various criteria to decide which design is best.

It turns out that most of these criteria are defined by theLaplacian eigenvalues of one of the two graphs defined by theblock design:I the Levi graph, which has v + b vertices,I or the concurrence graph, which has v vertices.

The algebraic approach shows that sometimes all the criteriaprefer highly symmetric designs but sometimes they favourvery different ones.

2/52

Abstract

Consider an experiment to compare v treatments in b blocks ofsize k.

Statisticians use various criteria to decide which design is best.

It turns out that most of these criteria are defined by theLaplacian eigenvalues of one of the two graphs defined by theblock design:I the Levi graph, which has v + b vertices,I or the concurrence graph, which has v vertices.

The algebraic approach shows that sometimes all the criteriaprefer highly symmetric designs but sometimes they favourvery different ones.

2/52

An experiment on detergents

In a consumer experiment, twelve housewives volunteer to testnew detergents. There are 16 new detergents to compare, but itis not realistic to ask any one volunteer to compare this manydetergents.

Each housewife tests one detergent per washload for each offour washloads, and assesses the cleanliness of each washload.

The experimental units are the 48 washloads.The housewives form 12 blocks of size 4.

The treatments are the 16 new detergents.

3/52

An experiment on detergents

In a consumer experiment, twelve housewives volunteer to testnew detergents. There are 16 new detergents to compare, but itis not realistic to ask any one volunteer to compare this manydetergents.

Each housewife tests one detergent per washload for each offour washloads, and assesses the cleanliness of each washload.

The experimental units are the 48 washloads.The housewives form 12 blocks of size 4.

The treatments are the 16 new detergents.

3/52

An experiment on detergents

In a consumer experiment, twelve housewives volunteer to testnew detergents. There are 16 new detergents to compare, but itis not realistic to ask any one volunteer to compare this manydetergents.

Each housewife tests one detergent per washload for each offour washloads, and assesses the cleanliness of each washload.

The experimental units are the 48 washloads.The housewives form 12 blocks of size 4.

The treatments are the 16 new detergents.

3/52

Experiments in blocks

I have v treatments that I want to compare.I have b blocks, with k plots in each block.

blocks b k treatments vhousewives 12 4 detergents 16

How should I choose a block designfor these values of b, v and k?

What makes a block design good?

4/52

Experiments in blocks

I have v treatments that I want to compare.I have b blocks, with k plots in each block.

blocks b k treatments vhousewives 12 4 detergents 16

How should I choose a block designfor these values of b, v and k?

What makes a block design good?

4/52

Experiments in blocks

I have v treatments that I want to compare.I have b blocks, with k plots in each block.

blocks b k treatments vhousewives 12 4 detergents 16

How should I choose a block designfor these values of b, v and k?

What makes a block design good?

4/52

Experiments in blocks

I have v treatments that I want to compare.I have b blocks, with k plots in each block.

blocks b k treatments vhousewives 12 4 detergents 16

How should I choose a block designfor these values of b, v and k?

What makes a block design good?

4/52

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 22 3 3 4 3 3 43 4 5 5 4 5 5

1 1 1 1 2 2 21 3 3 4 3 3 42 4 5 5 4 5 5

binary non-binary

A design is binary if no treatment occurs more than once in anyblock.

5/52

Two designs with v = 15, b = 7, k = 3: which is better?

1 1 2 3 4 5 62 4 5 6 10 11 123 7 8 9 13 14 15

1 1 1 1 1 1 12 4 6 8 10 12 143 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 thatoccurs in every block.

6/52

Two designs with v = 7, b = 7, k = 3: which is better?

1 2 3 4 5 6 72 3 4 5 6 7 14 5 6 7 1 2 3

1 2 3 4 5 6 72 3 4 5 6 7 13 4 5 6 7 1 2

balanced (2-design) non-balanced

A binary design is balanced if every pair of distinct treamentsoccurs together in the same number of blocks.

7/52

Experimental units and incidence matrix

There are bk experimental units.

If ω is an experimental unit, put

f (ω) = treatment on ω

g(ω) = block containing ω.

For i = 1, . . . , v put

ri = |{ω : f (ω) = i}| = replication of treatment i.

For i = 1, . . . , v and j = 1, . . . , b, let

nij = |{ω : f (ω) = i and g(ω) = j}|

= number of experimental units in block j which havetreatment i.

The v× b incidence matrix N has entries nij.

8/52

Experimental units and incidence matrix

There are bk experimental units.

If ω is an experimental unit, put

f (ω) = treatment on ω

g(ω) = block containing ω.

For i = 1, . . . , v put

ri = |{ω : f (ω) = i}| = replication of treatment i.

For i = 1, . . . , v and j = 1, . . . , b, let

nij = |{ω : f (ω) = i and g(ω) = j}|

= number of experimental units in block j which havetreatment i.

The v× b incidence matrix N has entries nij.

8/52

Experimental units and incidence matrix

There are bk experimental units.

If ω is an experimental unit, put

f (ω) = treatment on ω

g(ω) = block containing ω.

For i = 1, . . . , v put

ri = |{ω : f (ω) = i}| = replication of treatment i.

For i = 1, . . . , v and j = 1, . . . , b, let

nij = |{ω : f (ω) = i and g(ω) = j}|

= number of experimental units in block j which havetreatment i.

The v× b incidence matrix N has entries nij.

8/52

Experimental units and incidence matrix

There are bk experimental units.

If ω is an experimental unit, put

f (ω) = treatment on ω

g(ω) = block containing ω.

For i = 1, . . . , v put

ri = |{ω : f (ω) = i}| = replication of treatment i.

For i = 1, . . . , v and j = 1, . . . , b, let

nij = |{ω : f (ω) = i and g(ω) = j}|

= number of experimental units in block j which havetreatment i.

The v× b incidence matrix N has entries nij.

8/52

Experimental units and incidence matrix

There are bk experimental units.

If ω is an experimental unit, put

f (ω) = treatment on ω

g(ω) = block containing ω.

For i = 1, . . . , v put

ri = |{ω : f (ω) = i}| = replication of treatment i.

For i = 1, . . . , v and j = 1, . . . , b, let

nij = |{ω : f (ω) = i and g(ω) = j}|

= number of experimental units in block j which havetreatment i.

The v× b incidence matrix N has entries nij.8/52

Levi graph

The Levi graph G̃ of a block design ∆ has

I one vertex for each treatment,I one vertex for each block,I one edge for each experimental unit, with edge ω joining

vertex f (ω) (the treatment on ω)to vertex g(ω) (the block containing ω).

It is a bipartite graph,with nij edges between treatment-vertex i and block-vertex j.

9/52

Levi graph

The Levi graph G̃ of a block design ∆ has

I one vertex for each treatment,

I one vertex for each block,I one edge for each experimental unit, with edge ω joining

vertex f (ω) (the treatment on ω)to vertex g(ω) (the block containing ω).

It is a bipartite graph,with nij edges between treatment-vertex i and block-vertex j.

9/52

Levi graph

The Levi graph G̃ of a block design ∆ has

I one vertex for each treatment,I one vertex for each block,

I one edge for each experimental unit, with edge ω joiningvertex f (ω) (the treatment on ω)to vertex g(ω) (the block containing ω).

It is a bipartite graph,with nij edges between treatment-vertex i and block-vertex j.

9/52

Levi graph

The Levi graph G̃ of a block design ∆ has

I one vertex for each treatment,I one vertex for each block,I one edge for each experimental unit, with edge ω joining

vertex f (ω) (the treatment on ω)to vertex g(ω) (the block containing ω).

It is a bipartite graph,with nij edges between treatment-vertex i and block-vertex j.

9/52

Levi graph

The Levi graph G̃ of a block design ∆ has

I one vertex for each treatment,I one vertex for each block,I one edge for each experimental unit, with edge ω joining

vertex f (ω) (the treatment on ω)to vertex g(ω) (the block containing ω).

It is a bipartite graph,with nij edges between treatment-vertex i and block-vertex j.

9/52

Levi graph

The Levi graph G̃ of a block design ∆ has

I one vertex for each treatment,I one vertex for each block,I one edge for each experimental unit, with edge ω joining

vertex f (ω) (the treatment on ω)to vertex g(ω) (the block containing ω).

It is a bipartite graph,with nij edges between treatment-vertex i and block-vertex j.

9/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2

HHH

HHH

������

���

���

HHHHHH

10/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2

HHH

HHH

������

���

���

HHHHHH

10/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2H

HHH

HH

������

���

���

HHHHHH

10/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2H

HHH

HH

������

���

���

HHHHHH

10/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2H

HHH

HH

������

���

���

HHHHHH

10/52

Example 2: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

11/52

Example 2: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

11/52

Concurrence graph

The concurrence graph G of a block design ∆ has

I one vertex for each treatment,I one edge for each unordered pair α, ω, with α 6= ω,

g(α) = g(ω) (in the same block) and f (α) 6= f (ω):this edge joins vertices f (α) and f (ω).

There are no loops.

If i 6= j then the number of edges between vertices i and j is

λij =b

∑s=1

nisnjs;

this is called the concurrence of i and j,and is the (i, j)-entry of Λ = NN>.

12/52

Concurrence graph

The concurrence graph G of a block design ∆ has

I one vertex for each treatment,

I one edge for each unordered pair α, ω, with α 6= ω,g(α) = g(ω) (in the same block) and f (α) 6= f (ω):this edge joins vertices f (α) and f (ω).

There are no loops.

If i 6= j then the number of edges between vertices i and j is

λij =b

∑s=1

nisnjs;

this is called the concurrence of i and j,and is the (i, j)-entry of Λ = NN>.

12/52

Concurrence graph

The concurrence graph G of a block design ∆ has

I one vertex for each treatment,I one edge for each unordered pair α, ω, with α 6= ω,

g(α) = g(ω) (in the same block) and f (α) 6= f (ω):this edge joins vertices f (α) and f (ω).

There are no loops.

If i 6= j then the number of edges between vertices i and j is

λij =b

∑s=1

nisnjs;

this is called the concurrence of i and j,and is the (i, j)-entry of Λ = NN>.

12/52

Concurrence graph

The concurrence graph G of a block design ∆ has

I one vertex for each treatment,I one edge for each unordered pair α, ω, with α 6= ω,

g(α) = g(ω) (in the same block) and f (α) 6= f (ω):this edge joins vertices f (α) and f (ω).

There are no loops.

If i 6= j then the number of edges between vertices i and j is

λij =b

∑s=1

nisnjs;

this is called the concurrence of i and j,and is the (i, j)-entry of Λ = NN>.

12/52

Concurrence graph

The concurrence graph G of a block design ∆ has

I one vertex for each treatment,I one edge for each unordered pair α, ω, with α 6= ω,

g(α) = g(ω) (in the same block) and f (α) 6= f (ω):this edge joins vertices f (α) and f (ω).

There are no loops.

If i 6= j then the number of edges between vertices i and j is

λij =b

∑s=1

nisnjs;

this is called the concurrence of i and j,and is the (i, j)-entry of Λ = NN>.

12/52

Concurrence graph

The concurrence graph G of a block design ∆ has

I one vertex for each treatment,I one edge for each unordered pair α, ω, with α 6= ω,

g(α) = g(ω) (in the same block) and f (α) 6= f (ω):this edge joins vertices f (α) and f (ω).

There are no loops.

If i 6= j then the number of edges between vertices i and j is

λij =b

∑s=1

nisnjs;

this is called the concurrence of i and j,and is the (i, j)-entry of Λ = NN>.

12/52

Concurrence graph

The concurrence graph G of a block design ∆ has

I one vertex for each treatment,I one edge for each unordered pair α, ω, with α 6= ω,

g(α) = g(ω) (in the same block) and f (α) 6= f (ω):this edge joins vertices f (α) and f (ω).

There are no loops.

If i 6= j then the number of edges between vertices i and j is

λij =b

∑s=1

nisnjs;

this is called the concurrence of i and j,and is the (i, j)-entry of Λ = NN>.

12/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2HH

HHHH

����

��

������

HHHH

HH

x

x

x

x

������

@@

@@

@@

3 4

1 2

Levi graph concurrence graph

can recover design may have more symmetrymore vertices

more edges if k = 2 more edges if k ≥ 4

13/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2HH

HHHH

����

��

������

HHHH

HH

x

x

x

x

������

@@

@@

@@

3 4

1 2

Levi graph concurrence graph

can recover design may have more symmetrymore vertices

more edges if k = 2 more edges if k ≥ 4

13/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2HH

HHHH

����

��

������

HHHH

HH

x

x

x

x

������

@@

@@

@@

3 4

1 2

Levi graph concurrence graphcan recover design may have more symmetry

more verticesmore edges if k = 2 more edges if k ≥ 4

13/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2HH

HHHH

����

��

������

HHHH

HH

x

x

x

x

������

@@

@@

@@

3 4

1 2

Levi graph concurrence graphcan recover design may have more symmetry

more vertices

more edges if k = 2 more edges if k ≥ 4

13/52

Example 1: v = 4, b = k = 3

1 2 13 3 24 4 2

x� � �

xx x

3

4

1 2HH

HHHH

����

��

������

HHHH

HH

x

x

x

x

������

@@

@@

@@

3 4

1 2

Levi graph concurrence graphcan recover design may have more symmetry

more verticesmore edges if k = 2 more edges if k ≥ 4

13/52

Example 2: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

14/52

Example 2: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

14/52

Example 3: v = 15, b = 7, k = 3

1 1 2 3 4 5 62 4 5 6 10 11 123 7 8 9 13 14 15

1 1 1 1 1 1 12 4 6 8 10 12 143 5 7 9 11 13 15

x13 x7 x8 x11

x10

x1

x2

x14

x12 x6

x9x15

x4 x5x3

�����

TTT

�����

TTTTT

bb

bbb

"""""

bbb

bb

"""""

x1

�����

������

`````̀

QQ

QQQ

JJJJJ

DDDDDD

��

���

`````̀QQQQQ

JJJJJ

x12

x11

x10x9x5

x6

x7 x8

x14x15

x13x4 x3 x2

��

@@

@@

��

15/52

Laplacian matrices

The Laplacian matrix L of the concurrence graph G is av× v matrix with (i, j)-entry as follows:

I if i 6= j thenLij = −(number of edges between i and j) = −λij;

I Lii = valency of i = ∑j 6=i

λij.

The Laplacian matrix L̃ of the Levi graph G̃ is a(v + b)× (v + b) matrix with (i, j)-entry as follows:

I L̃ii = valency of i

=

{k if i is a blockreplication ri of i if i is a treatment

I if i 6= j then Lij = −(number of edges between i and j)

=

0 if i and j are both treatments0 if i and j are both blocks−nij if i is a treatment and j is a block, or vice versa.

16/52

Laplacian matrices

The Laplacian matrix L of the concurrence graph G is av× v matrix with (i, j)-entry as follows:I if i 6= j then

Lij = −(number of edges between i and j) = −λij;

I Lii = valency of i = ∑j 6=i

λij.

The Laplacian matrix L̃ of the Levi graph G̃ is a(v + b)× (v + b) matrix with (i, j)-entry as follows:

I L̃ii = valency of i

=

{k if i is a blockreplication ri of i if i is a treatment

I if i 6= j then Lij = −(number of edges between i and j)

=

0 if i and j are both treatments0 if i and j are both blocks−nij if i is a treatment and j is a block, or vice versa.

16/52

Laplacian matrices

The Laplacian matrix L of the concurrence graph G is av× v matrix with (i, j)-entry as follows:I if i 6= j then

Lij = −(number of edges between i and j) = −λij;I Lii = valency of i = ∑

j 6=iλij.

The Laplacian matrix L̃ of the Levi graph G̃ is a(v + b)× (v + b) matrix with (i, j)-entry as follows:

I L̃ii = valency of i

=

{k if i is a blockreplication ri of i if i is a treatment

I if i 6= j then Lij = −(number of edges between i and j)

=

0 if i and j are both treatments0 if i and j are both blocks−nij if i is a treatment and j is a block, or vice versa.

16/52

Laplacian matrices

The Laplacian matrix L of the concurrence graph G is av× v matrix with (i, j)-entry as follows:I if i 6= j then

Lij = −(number of edges between i and j) = −λij;I Lii = valency of i = ∑

j 6=iλij.

The Laplacian matrix L̃ of the Levi graph G̃ is a(v + b)× (v + b) matrix with (i, j)-entry as follows:

I L̃ii = valency of i

=

{k if i is a blockreplication ri of i if i is a treatment

I if i 6= j then Lij = −(number of edges between i and j)

=

0 if i and j are both treatments0 if i and j are both blocks−nij if i is a treatment and j is a block, or vice versa.

16/52

Laplacian matrices

The Laplacian matrix L of the concurrence graph G is av× v matrix with (i, j)-entry as follows:I if i 6= j then

Lij = −(number of edges between i and j) = −λij;I Lii = valency of i = ∑

j 6=iλij.

The Laplacian matrix L̃ of the Levi graph G̃ is a(v + b)× (v + b) matrix with (i, j)-entry as follows:

I L̃ii = valency of i

=

{k if i is a blockreplication ri of i if i is a treatment

I if i 6= j then Lij = −(number of edges between i and j)

=

0 if i and j are both treatments0 if i and j are both blocks−nij if i is a treatment and j is a block, or vice versa.

16/52

Laplacian matrices

The Laplacian matrix L of the concurrence graph G is av× v matrix with (i, j)-entry as follows:I if i 6= j then

Lij = −(number of edges between i and j) = −λij;I Lii = valency of i = ∑

j 6=iλij.

The Laplacian matrix L̃ of the Levi graph G̃ is a(v + b)× (v + b) matrix with (i, j)-entry as follows:I L̃ii = valency of i

=

{k if i is a blockreplication ri of i if i is a treatment

I if i 6= j then Lij = −(number of edges between i and j)

=

0 if i and j are both treatments0 if i and j are both blocks−nij if i is a treatment and j is a block, or vice versa.

16/52

Laplacian matrices

The Laplacian matrix L of the concurrence graph G is av× v matrix with (i, j)-entry as follows:I if i 6= j then

Lij = −(number of edges between i and j) = −λij;I Lii = valency of i = ∑

j 6=iλij.

The Laplacian matrix L̃ of the Levi graph G̃ is a(v + b)× (v + b) matrix with (i, j)-entry as follows:I L̃ii = valency of i

=

{k if i is a blockreplication ri of i if i is a treatment

I if i 6= j then Lij = −(number of edges between i and j)

=

0 if i and j are both treatments0 if i and j are both blocks−nij if i is a treatment and j is a block, or vice versa.

16/52

Connectivity

All row-sums of L and of L̃ are zero,so both matrices have 0 as eigenvalueon the appropriate all-1 vector.

TheoremThe following are equivalent.

1. 0 is a simple eigenvalue of L;2. G is a connected graph;3. G̃ is a connected graph;4. 0 is a simple eigenvalue of L̃;5. the design ∆ is connected in the sense that all differences between

treatments can be estimated.

From now on, assume connectivity.

Call the remaining eigenvalues non-trivial.They are all non-negative.

17/52

Connectivity

All row-sums of L and of L̃ are zero,so both matrices have 0 as eigenvalueon the appropriate all-1 vector.

TheoremThe following are equivalent.

1. 0 is a simple eigenvalue of L;2. G is a connected graph;3. G̃ is a connected graph;4. 0 is a simple eigenvalue of L̃;5. the design ∆ is connected in the sense that all differences between

treatments can be estimated.

From now on, assume connectivity.

Call the remaining eigenvalues non-trivial.They are all non-negative.

17/52

Connectivity

All row-sums of L and of L̃ are zero,so both matrices have 0 as eigenvalueon the appropriate all-1 vector.

TheoremThe following are equivalent.

1. 0 is a simple eigenvalue of L;2. G is a connected graph;3. G̃ is a connected graph;4. 0 is a simple eigenvalue of L̃;5. the design ∆ is connected in the sense that all differences between

treatments can be estimated.

From now on, assume connectivity.

Call the remaining eigenvalues non-trivial.They are all non-negative.

17/52

Connectivity

All row-sums of L and of L̃ are zero,so both matrices have 0 as eigenvalueon the appropriate all-1 vector.

TheoremThe following are equivalent.

1. 0 is a simple eigenvalue of L;2. G is a connected graph;3. G̃ is a connected graph;4. 0 is a simple eigenvalue of L̃;5. the design ∆ is connected in the sense that all differences between

treatments can be estimated.

From now on, assume connectivity.

Call the remaining eigenvalues non-trivial.They are all non-negative.

17/52

Generalized inverse

Under the assumption of connectivity,the Moore–Penrose generalized inverse L− of L is defined by

L− =(

L +1v

Jv

)−1

− 1v

Jv,

where Jv is the v× v all-1 matrix.

(The matrix1v

Jv is the orthogonal projector onto the null spaceof L.)

The Moore–Penrose generalized inverse L̃− of L̃ is definedsimilarly.

18/52

Generalized inverse

Under the assumption of connectivity,the Moore–Penrose generalized inverse L− of L is defined by

L− =(

L +1v

Jv

)−1

− 1v

Jv,

where Jv is the v× v all-1 matrix.

(The matrix1v

Jv is the orthogonal projector onto the null spaceof L.)

The Moore–Penrose generalized inverse L̃− of L̃ is definedsimilarly.

18/52

Estimation

We measure the response Yω on each experimenal unit ω.

If experimental unit ω has treatment i and is in block m(f (ω) = i and g(ω) = m), then we assume that

Yω = τi + βm + random noise.

We will do an experiment, collect data yω on each experimentalunit ω, then want to estimate certain functions of the treatmentparameters using functions of the data.

We want to estimate contrasts ∑i xiτi with ∑i xi = 0.

In particular, we want to estimate all the simple differencesτi − τj.

19/52

Estimation

We measure the response Yω on each experimenal unit ω.

If experimental unit ω has treatment i and is in block m(f (ω) = i and g(ω) = m), then we assume that

Yω = τi + βm + random noise.

We will do an experiment, collect data yω on each experimentalunit ω, then want to estimate certain functions of the treatmentparameters using functions of the data.

We want to estimate contrasts ∑i xiτi with ∑i xi = 0.

In particular, we want to estimate all the simple differencesτi − τj.

19/52

Estimation

We measure the response Yω on each experimenal unit ω.

If experimental unit ω has treatment i and is in block m(f (ω) = i and g(ω) = m), then we assume that

Yω = τi + βm + random noise.

We will do an experiment, collect data yω on each experimentalunit ω, then want to estimate certain functions of the treatmentparameters using functions of the data.

We want to estimate contrasts ∑i xiτi with ∑i xi = 0.

In particular, we want to estimate all the simple differencesτi − τj.

19/52

Estimation

We measure the response Yω on each experimenal unit ω.

If experimental unit ω has treatment i and is in block m(f (ω) = i and g(ω) = m), then we assume that

Yω = τi + βm + random noise.

We will do an experiment, collect data yω on each experimentalunit ω, then want to estimate certain functions of the treatmentparameters using functions of the data.

We want to estimate contrasts ∑i xiτi with ∑i xi = 0.

In particular, we want to estimate all the simple differencesτi − τj.

19/52

Estimation

We measure the response Yω on each experimenal unit ω.

If experimental unit ω has treatment i and is in block m(f (ω) = i and g(ω) = m), then we assume that

Yω = τi + βm + random noise.

We will do an experiment, collect data yω on each experimentalunit ω, then want to estimate certain functions of the treatmentparameters using functions of the data.

We want to estimate contrasts ∑i xiτi with ∑i xi = 0.

In particular, we want to estimate all the simple differencesτi − τj.

19/52

Variance: why does it matter?

We want to estimate all the simple differences τi − τj.

Put Vij = variance of the best linear unbiased estimator forτi − τj.

The length of the 95% confidence interval for τi − τj isproportional to

√Vij.

(If we always present results using a95% confidence interval, then our interval will contain the truevalue in 19 cases out of 20.)

The smaller the value of Vij, the smaller is the confidenceinterval, the closer is the estimate to the true value(on average), and the more likely are we to detect correctlywhich of τi and τj is bigger.

We can make better decisions about new drugs, about newvarieties of wheat, about new engineering materials . . . if wemake all the Vij small.

20/52

Variance: why does it matter?

We want to estimate all the simple differences τi − τj.

Put Vij = variance of the best linear unbiased estimator forτi − τj.

The length of the 95% confidence interval for τi − τj isproportional to

√Vij. (If we always present results using a

95% confidence interval, then our interval will contain the truevalue in 19 cases out of 20.)

The smaller the value of Vij, the smaller is the confidenceinterval, the closer is the estimate to the true value(on average), and the more likely are we to detect correctlywhich of τi and τj is bigger.

We can make better decisions about new drugs, about newvarieties of wheat, about new engineering materials . . . if wemake all the Vij small.

20/52

Variance: why does it matter?

We want to estimate all the simple differences τi − τj.

Put Vij = variance of the best linear unbiased estimator forτi − τj.

The length of the 95% confidence interval for τi − τj isproportional to

√Vij. (If we always present results using a

95% confidence interval, then our interval will contain the truevalue in 19 cases out of 20.)

The smaller the value of Vij, the smaller is the confidenceinterval, the closer is the estimate to the true value(on average), and the more likely are we to detect correctlywhich of τi and τj is bigger.

We can make better decisions about new drugs, about newvarieties of wheat, about new engineering materials . . . if wemake all the Vij small.

20/52

Variance: why does it matter?

We want to estimate all the simple differences τi − τj.

Put Vij = variance of the best linear unbiased estimator forτi − τj.

The length of the 95% confidence interval for τi − τj isproportional to

√Vij. (If we always present results using a

95% confidence interval, then our interval will contain the truevalue in 19 cases out of 20.)

The smaller the value of Vij, the smaller is the confidenceinterval, the closer is the estimate to the true value(on average), and the more likely are we to detect correctlywhich of τi and τj is bigger.

We can make better decisions about new drugs, about newvarieties of wheat, about new engineering materials . . . if wemake all the Vij small.

20/52

How do we calculate variance?

TheoremAssume that all the noise is independent, with variance σ2.If ∑i xi = 0, then the variance of the best linear unbiased estimator of∑i xiτi is equal to

(x>L−x)kσ2.

In particular, the variance of the best linear unbiased estimator of thesimple difference τi − τj is

Vij =(

L−ii + L−jj − 2L−ij)

kσ2.

(This follows from assumption

Yω = τi + βm + random noise.

by using standard theory of linear models.)

21/52

How do we calculate variance?

TheoremAssume that all the noise is independent, with variance σ2.If ∑i xi = 0, then the variance of the best linear unbiased estimator of∑i xiτi is equal to

(x>L−x)kσ2.

In particular, the variance of the best linear unbiased estimator of thesimple difference τi − τj is

Vij =(

L−ii + L−jj − 2L−ij)

kσ2.

(This follows from assumption

Yω = τi + βm + random noise.

by using standard theory of linear models.)21/52

. . . Or we can use the Levi graph

TheoremThe variance of the best linear unbiased estimator of the simpledifference τi − τj is

Vij =(

L̃−ii + L̃−jj − 2L̃−ij)

σ2.

(Or βi − βj, appropriately labelled.)

(This follows from assumption

Yω = τi − β̃m + random noise.

by using standard theory of linear models.)

22/52

. . . Or we can use the Levi graph

TheoremThe variance of the best linear unbiased estimator of the simpledifference τi − τj is

Vij =(

L̃−ii + L̃−jj − 2L̃−ij)

σ2.

(Or βi − βj, appropriately labelled.)

(This follows from assumption

Yω = τi − β̃m + random noise.

by using standard theory of linear models.)

22/52

. . . Or we can use the Levi graph

TheoremThe variance of the best linear unbiased estimator of the simpledifference τi − τj is

Vij =(

L̃−ii + L̃−jj − 2L̃−ij)

σ2.

(Or βi − βj, appropriately labelled.)

(This follows from assumption

Yω = τi − β̃m + random noise.

by using standard theory of linear models.)

22/52

How do we calculate these generalized inverses?

We need L− or L̃−.

I Add a suitable multiple of J,use GAP to find the inverse with exact rational coefficients,subtract that multiple of J.

I If the matrix is highly patterned,guess the eigenspaces,then invert each non-zero eigenvalue.

I Direct use of the graph: coming up.

Not all of these methods are suitable for generic designs with avariable number of treatments.

23/52

How do we calculate these generalized inverses?

We need L− or L̃−.

I Add a suitable multiple of J,use GAP to find the inverse with exact rational coefficients,subtract that multiple of J.

I If the matrix is highly patterned,guess the eigenspaces,then invert each non-zero eigenvalue.

I Direct use of the graph: coming up.

Not all of these methods are suitable for generic designs with avariable number of treatments.

23/52

How do we calculate these generalized inverses?

We need L− or L̃−.

I Add a suitable multiple of J,use GAP to find the inverse with exact rational coefficients,subtract that multiple of J.

I If the matrix is highly patterned,guess the eigenspaces,then invert each non-zero eigenvalue.

I Direct use of the graph: coming up.

Not all of these methods are suitable for generic designs with avariable number of treatments.

23/52

How do we calculate these generalized inverses?

We need L− or L̃−.

I Add a suitable multiple of J,use GAP to find the inverse with exact rational coefficients,subtract that multiple of J.

I If the matrix is highly patterned,guess the eigenspaces,then invert each non-zero eigenvalue.

I Direct use of the graph: coming up.

Not all of these methods are suitable for generic designs with avariable number of treatments.

23/52

How do we calculate these generalized inverses?

We need L− or L̃−.

I Add a suitable multiple of J,use GAP to find the inverse with exact rational coefficients,subtract that multiple of J.

I If the matrix is highly patterned,guess the eigenspaces,then invert each non-zero eigenvalue.

I Direct use of the graph: coming up.

Not all of these methods are suitable for generic designs with avariable number of treatments.

23/52

How do we calculate these generalized inverses?

We need L− or L̃−.

I Add a suitable multiple of J,use GAP to find the inverse with exact rational coefficients,subtract that multiple of J.

I If the matrix is highly patterned,guess the eigenspaces,then invert each non-zero eigenvalue.

I Direct use of the graph: coming up.

Not all of these methods are suitable for generic designs with avariable number of treatments.

23/52

Electrical networks

We can consider the concurrence graph G as an electricalnetwork with a 1-ohm resistance in each edge.Connect a 1-volt battery between vertices i and j.Current flows in the network, according to these rules.

1. Ohm’s Law:In every edge, voltage drop = current × resistance =current.

2. Kirchhoff’s Voltage Law:The total voltage drop from one vertex to any other vertexis the same no matter which path we take from one to theother.

3. Kirchhoff’s Current Law:At every vertex which is not connected to the battery, thetotal current coming in is equal to the total current goingout.

Find the total current I from i to j, then use Ohm’s Law todefine the effective resistance Rij between i and j as 1/I.

24/52

Electrical networks: variance

TheoremThe effective resistance Rij between vertices i and j in G is

Rij =(

L−ii + L−jj − 2L−ij)

.

SoVij = Rij × kσ2.

Effective resistances are easy to calculate withoutmatrix inversion if the graph is sparse.

25/52

Electrical networks: variance

TheoremThe effective resistance Rij between vertices i and j in G is

Rij =(

L−ii + L−jj − 2L−ij)

.

SoVij = Rij × kσ2.

Effective resistances are easy to calculate withoutmatrix inversion if the graph is sparse.

25/52

Electrical networks: variance

TheoremThe effective resistance Rij between vertices i and j in G is

Rij =(

L−ii + L−jj − 2L−ij)

.

SoVij = Rij × kσ2.

Effective resistances are easy to calculate withoutmatrix inversion if the graph is sparse.

25/52

Example 2 calculation: v = 8, b = 4, k = 3

V = 23 I = 24 R =2324

z

z

z

zz z

z

z

����

��

HHHH

HH

HHHH

HH

����

��

������

AAAAAA

AAAAAA

������

5

7

8 6

4

1 2

3

[0]U5

�5

[10]-

10

[5]

Y3

[3]

*

3[6]

66

�3

[9]

U 3[12]

-6

62*13

[23]

j11

26/52

Example 2 calculation: v = 8, b = 4, k = 3

V = 23 I = 24 R =2324

z

z

z

zz z

z

z

����

��

HHHH

HH

HHHH

HH

����

��

������

AAAAAA

AAAAAA

������

5

7

8 6

4

1 2

3

[0]U5

�5

[10]-

10

[5]

Y3

[3]

*

3[6]

66

�3

[9]

U 3[12]

-6

62*13

[23]

j11

26/52

Example 2 calculation: v = 8, b = 4, k = 3

V = 23 I = 24 R =2324

z

z

z

zz z

z

z

����

��

HHHH

HH

HHHH

HH

����

��

������

AAAAAA

AAAAAA

������

5

7

8 6

4

1 2

3

[0]U5

�5

[10]-

10

[5]

Y3

[3]

*

3[6]

66

�3

[9]

U 3[12]

-6

62*13

[23]

j11

26/52

Example 2 calculation: v = 8, b = 4, k = 3

V = 23 I = 24 R =2324

z

z

z

zz z

z

z

����

��

HHHH

HH

HHHH

HH

����

��

������

AAAAAA

AAAAAA

������

5

7

8 6

4

1 2

3

[0]U5

�5

[10]-

10

[5]

Y3

[3]

*

3[6]

66

�3

[9]

U 3[12]

-6

62*13

[23]

j11

26/52

Example 2 calculation: v = 8, b = 4, k = 3

V = 23 I = 24 R =2324

z

z

z

zz z

z

z

����

��

HHHH

HH

HHHH

HH

����

��

������

AAAAAA

AAAAAA

������

5

7

8 6

4

1 2

3

[0]U5

�5

[10]-

10

[5]

Y3

[3]

*

3[6]

66

�3

[9]

U 3[12]

-6

62*13

[23]

j11

26/52

. . . Or we can use the Levi graph

If i and j are treatment vertices in the Levi graph G̃and R̃ij is the effective resistance between them in G̃ then

Vij = R̃ij × σ2.

27/52

Example 2 yet again: v = 8, b = 4, k = 3

V = 23 I = 8 R̃ =238

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

6

�

-R

?3

33

3

3[15][0]

R- �

55

5 R8 [23]

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

28/52

Example 2 yet again: v = 8, b = 4, k = 3

V = 23 I = 8 R̃ =238

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

6

�

-R

?

3

33

3

3[15][0]

R- �

55

5 R8 [23]

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

28/52

Example 2 yet again: v = 8, b = 4, k = 3

V = 23 I = 8 R̃ =238

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

6

�

-R

?3

33

3

3[15][0]

R- �

55

5 R8 [23]

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

28/52

Example 2 yet again: v = 8, b = 4, k = 3

V = 23 I = 8 R̃ =238

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

6

�

-R

?3

33

3

3[15][0]

R- �

55

5

R8 [23]

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

28/52

Example 2 yet again: v = 8, b = 4, k = 3

V = 23 I = 8 R̃ =238

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

6

�

-R

?3

33

3

3[15][0]

R- �

55

5 R8 [23]

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

28/52

Example 2 yet again: v = 8, b = 4, k = 3

V = 23 I = 8 R̃ =238

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

6

�

-R

?3

33

3

3[15][0]

R- �

55

5 R8 [23]

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

28/52

Optimality: Average pairwise variance

The variance of the best linear unbiased estimator of the simpledifference τi − τj is

Vij =(

L−ii + L−jj − 2L−ij)

kσ2 = Rijkσ2.

We want all of the Vij to be small.

Put V̄ = average value of the Vij. Then

V̄ =2kσ2 Tr(L−)

v− 1= 2kσ2 × 1

harmonic mean of θ1, . . . , θv−1,

where θ1, . . . , θv−1 are the nontrivial eigenvalues of L.

29/52

Optimality: Average pairwise variance

The variance of the best linear unbiased estimator of the simpledifference τi − τj is

Vij =(

L−ii + L−jj − 2L−ij)

kσ2 = Rijkσ2.

We want all of the Vij to be small.

Put V̄ = average value of the Vij. Then

V̄ =2kσ2 Tr(L−)

v− 1= 2kσ2 × 1

harmonic mean of θ1, . . . , θv−1,

where θ1, . . . , θv−1 are the nontrivial eigenvalues of L.

29/52

Optimality: Average pairwise variance

The variance of the best linear unbiased estimator of the simpledifference τi − τj is

Vij =(

L−ii + L−jj − 2L−ij)

kσ2 = Rijkσ2.

We want all of the Vij to be small.

Put V̄ = average value of the Vij. Then

V̄ =2kσ2 Tr(L−)

v− 1= 2kσ2 × 1

harmonic mean of θ1, . . . , θv−1,

where θ1, . . . , θv−1 are the nontrivial eigenvalues of L.

29/52

A-Optimality

A block design is called A-optimal if it minimizes the averageof the variances Vij;

—equivalently, it maximizes the harmonic mean of thenon-trivial eigenvalues of the Laplacian matrix L;over all block designs with block size k and the given v and b.

30/52

A-Optimality

A block design is called A-optimal if it minimizes the averageof the variances Vij;—equivalently, it maximizes the harmonic mean of thenon-trivial eigenvalues of the Laplacian matrix L;

over all block designs with block size k and the given v and b.

30/52

A-Optimality

A block design is called A-optimal if it minimizes the averageof the variances Vij;—equivalently, it maximizes the harmonic mean of thenon-trivial eigenvalues of the Laplacian matrix L;over all block designs with block size k and the given v and b.

30/52

Optimality: Confidence region

When v > 2 the generalization of confidence interval is theconfidence ellipsoid around the point (τ̂1, . . . , τ̂v) in thehyperplane in Rv with ∑i τi = 0. The volume of this confidenceellipsoid is proportional to√√√√v−1

∏i=1

1θi

= (geometric mean of θ1, . . . , θv−1)−(v−1)/2

=1√

v× number of spanning trees for G.

31/52

Optimality: Confidence region

When v > 2 the generalization of confidence interval is theconfidence ellipsoid around the point (τ̂1, . . . , τ̂v) in thehyperplane in Rv with ∑i τi = 0. The volume of this confidenceellipsoid is proportional to√√√√v−1

∏i=1

1θi

= (geometric mean of θ1, . . . , θv−1)−(v−1)/2

=1√

v× number of spanning trees for G.

31/52

Optimality: Confidence region

When v > 2 the generalization of confidence interval is theconfidence ellipsoid around the point (τ̂1, . . . , τ̂v) in thehyperplane in Rv with ∑i τi = 0. The volume of this confidenceellipsoid is proportional to√√√√v−1

∏i=1

1θi

= (geometric mean of θ1, . . . , θv−1)−(v−1)/2

=1√

v× number of spanning trees for G.

31/52

D-Optimality

A block design is called D-optimal if it minimizes the volumeof the confidence ellipsoid for (τ̂1, . . . , τ̂v) ;

—equivalently, it maximizes the geometric mean of thenon-trivial eigenvalues of the Laplacian matrix L;—equivalently, it maximizes the number of spanning trees forthe concurrence graph G;—equivalently, it maximizes the number of spanning trees forthe Levi graph G̃;over all block designs with block size k and the given v and b.

32/52

D-Optimality

A block design is called D-optimal if it minimizes the volumeof the confidence ellipsoid for (τ̂1, . . . , τ̂v) ;—equivalently, it maximizes the geometric mean of thenon-trivial eigenvalues of the Laplacian matrix L;

—equivalently, it maximizes the number of spanning trees forthe concurrence graph G;—equivalently, it maximizes the number of spanning trees forthe Levi graph G̃;over all block designs with block size k and the given v and b.

32/52

D-Optimality

A block design is called D-optimal if it minimizes the volumeof the confidence ellipsoid for (τ̂1, . . . , τ̂v) ;—equivalently, it maximizes the geometric mean of thenon-trivial eigenvalues of the Laplacian matrix L;—equivalently, it maximizes the number of spanning trees forthe concurrence graph G;

—equivalently, it maximizes the number of spanning trees forthe Levi graph G̃;over all block designs with block size k and the given v and b.

32/52

D-Optimality

A block design is called D-optimal if it minimizes the volumeof the confidence ellipsoid for (τ̂1, . . . , τ̂v) ;—equivalently, it maximizes the geometric mean of thenon-trivial eigenvalues of the Laplacian matrix L;—equivalently, it maximizes the number of spanning trees forthe concurrence graph G;—equivalently, it maximizes the number of spanning trees forthe Levi graph G̃;

over all block designs with block size k and the given v and b.

32/52

D-Optimality

A block design is called D-optimal if it minimizes the volumeof the confidence ellipsoid for (τ̂1, . . . , τ̂v) ;—equivalently, it maximizes the geometric mean of thenon-trivial eigenvalues of the Laplacian matrix L;—equivalently, it maximizes the number of spanning trees forthe concurrence graph G;—equivalently, it maximizes the number of spanning trees forthe Levi graph G̃;over all block designs with block size k and the given v and b.

32/52

Optimality: Worst case

If x is a contrast in Rv then the variance of the estimator of x>τis (x>L−x)kσ2.

If we multiply every entry in x by a constant c then thisvariance is multiplied by c2; and so is x>x.

The worst case is for contrasts x giving the maximum value of

x>L−xx>x

.

These are precisely the eigenvectors corresponding to θ1,where θ1 is the smallest non-trivial eigenvalue of L.

33/52

Optimality: Worst case

If x is a contrast in Rv then the variance of the estimator of x>τis (x>L−x)kσ2.

If we multiply every entry in x by a constant c then thisvariance is multiplied by c2;

and so is x>x.

The worst case is for contrasts x giving the maximum value of

x>L−xx>x

.

These are precisely the eigenvectors corresponding to θ1,where θ1 is the smallest non-trivial eigenvalue of L.

33/52

Optimality: Worst case

If x is a contrast in Rv then the variance of the estimator of x>τis (x>L−x)kσ2.

If we multiply every entry in x by a constant c then thisvariance is multiplied by c2; and so is x>x.

The worst case is for contrasts x giving the maximum value of

x>L−xx>x

.

These are precisely the eigenvectors corresponding to θ1,where θ1 is the smallest non-trivial eigenvalue of L.

33/52

Optimality: Worst case

If x is a contrast in Rv then the variance of the estimator of x>τis (x>L−x)kσ2.

If we multiply every entry in x by a constant c then thisvariance is multiplied by c2; and so is x>x.

The worst case is for contrasts x giving the maximum value of

x>L−xx>x

.

These are precisely the eigenvectors corresponding to θ1,where θ1 is the smallest non-trivial eigenvalue of L.

33/52

Optimality: Worst case

If x is a contrast in Rv then the variance of the estimator of x>τis (x>L−x)kσ2.

If we multiply every entry in x by a constant c then thisvariance is multiplied by c2; and so is x>x.

The worst case is for contrasts x giving the maximum value of

x>L−xx>x

.

These are precisely the eigenvectors corresponding to θ1,where θ1 is the smallest non-trivial eigenvalue of L.

33/52

E-Optimality

A block design is called E-optimal if it maximizes thesmallest non-trivial eigenvalue of the Laplacian matrix L;

over all block designs with block size k and the given v and b.

34/52

E-Optimality

A block design is called E-optimal if it maximizes thesmallest non-trivial eigenvalue of the Laplacian matrix L;over all block designs with block size k and the given v and b.

34/52

BIBDs are optimal

Theorem (Kshirsagar, 1958; Kiefer, 1975)

If there is a balanced incomplete-block design (BIBD) (2-design)for v treatments in b blocks of size k,then it is A-, D- and E-optimal.Moreover, no non-BIBD is A-, D- or E-optimal.

35/52

D-optimality: spanning trees

A spanning tree for the graph is a collection of edges of thegraph which form a tree (connected graph with no cycles)and which include every vertex.

Cheng (1981), after Gaffke (1978), after Kirchhoff (1847):

product of non-trivial eigenvalues of L= v× number of spanning trees.

So a design is D-optimal if and only if its concurrence graph Ghas the maximal number of spanning trees.

This is easy to calculate by hand when the graph is sparse.

36/52

D-optimality: spanning trees

A spanning tree for the graph is a collection of edges of thegraph which form a tree (connected graph with no cycles)and which include every vertex.

Cheng (1981), after Gaffke (1978), after Kirchhoff (1847):

product of non-trivial eigenvalues of L= v× number of spanning trees.

So a design is D-optimal if and only if its concurrence graph Ghas the maximal number of spanning trees.

This is easy to calculate by hand when the graph is sparse.

36/52

D-optimality: spanning trees

A spanning tree for the graph is a collection of edges of thegraph which form a tree (connected graph with no cycles)and which include every vertex.

Cheng (1981), after Gaffke (1978), after Kirchhoff (1847):

product of non-trivial eigenvalues of L= v× number of spanning trees.

So a design is D-optimal if and only if its concurrence graph Ghas the maximal number of spanning trees.

This is easy to calculate by hand when the graph is sparse.

36/52

What about the Levi graph?

Theorem (Gaffke, 1982)

Let G and G̃ be the concurrence graph and Levi graph for a connectedincomplete-block design for v treatments in b blocks of size k.Then the number of spanning trees for G̃ is equal tokb−v+1 times the number of spanning trees for G.

So a block design is D-optimal if and only ifits Levi graph maximizes the number of spanning trees.

If v ≥ b + 2 it is easier to count spanning trees in the Levi graphthan in the concurrence graph.

37/52

What about the Levi graph?

Theorem (Gaffke, 1982)

Let G and G̃ be the concurrence graph and Levi graph for a connectedincomplete-block design for v treatments in b blocks of size k.Then the number of spanning trees for G̃ is equal tokb−v+1 times the number of spanning trees for G.

So a block design is D-optimal if and only ifits Levi graph maximizes the number of spanning trees.

If v ≥ b + 2 it is easier to count spanning trees in the Levi graphthan in the concurrence graph.

37/52

What about the Levi graph?

Theorem (Gaffke, 1982)

Let G and G̃ be the concurrence graph and Levi graph for a connectedincomplete-block design for v treatments in b blocks of size k.Then the number of spanning trees for G̃ is equal tokb−v+1 times the number of spanning trees for G.

So a block design is D-optimal if and only ifits Levi graph maximizes the number of spanning trees.

If v ≥ b + 2 it is easier to count spanning trees in the Levi graphthan in the concurrence graph.

37/52

Example 2 one last time: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

8 spanning trees 216 spanning trees

38/52

Example 2 one last time: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph

8 spanning trees 216 spanning trees

38/52

Example 2 one last time: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph8 spanning trees

216 spanning trees

38/52

Example 2 one last time: v = 8, b = 4, k = 3

1 2 3 42 3 4 15 6 7 8

x x xxx x x

x

�

�

�

�

��

��

@@

@@

��

��

@@

@@

15

2

637

4

8

xx

xxx x

x

x���

HHH

HHH

���

���

AAA

AAA

���

5

7

8 6

4

1 2

3

Levi graph concurrence graph8 spanning trees 216 spanning trees

38/52

E-optimality: the edge-cutset lemma

A design is E-optimal if it maximizes the smallest non-trivialeigenvalue θ1 of the Laplacian L of the concurrence graph G.

LemmaLet G have an edge-cutset of size c(set of c edges whose removal disconnects the graph)whose removal separates the graph into components of sizes m and n.Then

θ1 ≤ c(

1m

+1n

).

If c is small but m and n are both large, then θ1 is small.

39/52

E-optimality: the edge-cutset lemma

A design is E-optimal if it maximizes the smallest non-trivialeigenvalue θ1 of the Laplacian L of the concurrence graph G.

LemmaLet G have an edge-cutset of size c(set of c edges whose removal disconnects the graph)whose removal separates the graph into components of sizes m and n.Then

θ1 ≤ c(

1m

+1n

).

If c is small but m and n are both large, then θ1 is small.

39/52

E-optimality: the edge-cutset lemma

A design is E-optimal if it maximizes the smallest non-trivialeigenvalue θ1 of the Laplacian L of the concurrence graph G.

LemmaLet G have an edge-cutset of size c(set of c edges whose removal disconnects the graph)whose removal separates the graph into components of sizes m and n.Then

θ1 ≤ c(

1m

+1n

).

If c is small but m and n are both large, then θ1 is small.

39/52

E-optimality: the vertex-cutset lemma

A design is E-optimal if it maximizes the smallest non-trivialeigenvalue θ1 of the Laplacian L of the concurrence graph G.

LemmaLet G have a vertex-cutset of size c(set of c vertices whose removal disconnects the graph)whose removal separates the graph into components of sizes m and n,with m′ and n′ edges between them and the vertices in the cutset.Then

θ1 ≤m′n2 + n′m2

mn(n + m),

which is at most c if no multiple edges are involved.

If m′ << m and n′ << n then θ1 is small.

40/52

E-optimality: the vertex-cutset lemma

A design is E-optimal if it maximizes the smallest non-trivialeigenvalue θ1 of the Laplacian L of the concurrence graph G.

LemmaLet G have a vertex-cutset of size c(set of c vertices whose removal disconnects the graph)whose removal separates the graph into components of sizes m and n,with m′ and n′ edges between them and the vertices in the cutset.Then

θ1 ≤m′n2 + n′m2

mn(n + m),

which is at most c if no multiple edges are involved.

If m′ << m and n′ << n then θ1 is small.

40/52

E-optimality: the vertex-cutset lemma

A design is E-optimal if it maximizes the smallest non-trivialeigenvalue θ1 of the Laplacian L of the concurrence graph G.

LemmaLet G have a vertex-cutset of size c(set of c vertices whose removal disconnects the graph)whose removal separates the graph into components of sizes m and n,with m′ and n′ edges between them and the vertices in the cutset.Then

θ1 ≤m′n2 + n′m2

mn(n + m),

which is at most c if no multiple edges are involved.

If m′ << m and n′ << n then θ1 is small.

40/52

Minimal connectivity

If the block design is connected then bk ≥ b + v− 1.

If the block design is connected and b(k− 1) = v− 1 thenthe Levi graph is a tree andthe concurrence graph is a b-tree of k-cliques.

x x x xx x x x

x x xx

x xx

�����

TTT

���

TTTTTT

bbb

bb

"""""

bb

bbb

"""""

x �����

������

`````̀

QQQ

QQ

JJJJJ

DDDDDD

���

��

`````̀QQQQQ

JJJJJ

xxxx

xx x x

xxxx

x x ��

@@

@@

��

41/52

Minimal connectivity

If the block design is connected then bk ≥ b + v− 1.

If the block design is connected and b(k− 1) = v− 1 thenthe Levi graph is a tree andthe concurrence graph is a b-tree of k-cliques.

x x x xx x x x

x x xx

x xx

�����

TTT

���

TTTTTT

bbb

bb

"""""

bb

bbb

"""""

x �����

������

`````̀

QQQ

QQ

JJJJJ

DDDDDD

���

��

`````̀QQQQQ

JJJJJ

xxxx

xx x x

xxxx

x x ��

@@

@@

��

41/52

Minimal connectivity

If the block design is connected then bk ≥ b + v− 1.

If the block design is connected and b(k− 1) = v− 1 thenthe Levi graph is a tree andthe concurrence graph is a b-tree of k-cliques.

x x x xx x x x

x x xx

x xx

�����

TTT

���

TTTTTT

bbb

bb

"""""

bb

bbb

"""""

x �����

������

`````̀

QQQ

QQ

JJJJJ

DDDDDD

���

��

`````̀QQQQQ

JJJJJ

xxxx

xx x x

xxxx

x x ��

@@

@@

��

41/52

Optimality of minimally connected designs

The Levi graph is a tree,so all connected designs are equally good under the D-criterion.

The Levi graph is a tree,so effective resistance = graph distance,so the only A-optimal designs are the queen-bee designs.

The concurrence graph is a b-tree of k-cliques,so the Cutset Lemmas show thatthe only E-optimal designs are the queen-bee designs.

42/52

Optimality of minimally connected designs

The Levi graph is a tree,so all connected designs are equally good under the D-criterion.

The Levi graph is a tree,so effective resistance = graph distance,so the only A-optimal designs are the queen-bee designs.

The concurrence graph is a b-tree of k-cliques,so the Cutset Lemmas show thatthe only E-optimal designs are the queen-bee designs.

42/52

Optimality of minimally connected designs

The Levi graph is a tree,so all connected designs are equally good under the D-criterion.

The Levi graph is a tree,so effective resistance = graph distance,so the only A-optimal designs are the queen-bee designs.

The concurrence graph is a b-tree of k-cliques,so the Cutset Lemmas show thatthe only E-optimal designs are the queen-bee designs.

42/52

Can we use the Levi graph to find E-optimal designs?

For binary designs with equal replication,θ1(L) is a monotonic increasing function of θ1(L̃).

But, queen-bee designs are E-optimal under minimalconnectivity,and some non-binary designs are E-optimal.

For general block designs, we do not know if we can use theLevi graph to investigate E-optimality.

43/52

Can we use the Levi graph to find E-optimal designs?

For binary designs with equal replication,θ1(L) is a monotonic increasing function of θ1(L̃).

But, queen-bee designs are E-optimal under minimalconnectivity,

and some non-binary designs are E-optimal.

For general block designs, we do not know if we can use theLevi graph to investigate E-optimality.

43/52

Can we use the Levi graph to find E-optimal designs?

For binary designs with equal replication,θ1(L) is a monotonic increasing function of θ1(L̃).

But, queen-bee designs are E-optimal under minimalconnectivity,and some non-binary designs are E-optimal.

For general block designs, we do not know if we can use theLevi graph to investigate E-optimality.

43/52

Can we use the Levi graph to find E-optimal designs?

For binary designs with equal replication,θ1(L) is a monotonic increasing function of θ1(L̃).

But, queen-bee designs are E-optimal under minimalconnectivity,and some non-binary designs are E-optimal.

For general block designs, we do not know if we can use theLevi graph to investigate E-optimality.

43/52

Large blocks; many unreplicated treatments

Suppose that r̄ = ∑i ri

v< 2.

New conventions: blocks are rows, and block size = k + n.

b blocks

k plots n plots

......

v treatments bn treatmentsall single replication

whole design ∆

Whole design ∆ has v + bn treatments in b blocks of size k + n;the subdesign Γ has v core treatments in b blocks of size k;call the remaining treatments orphans.

44/52

Large blocks; many unreplicated treatments

Suppose that r̄ = ∑i ri

v< 2.

New conventions: blocks are rows, and block size = k + n.

b blocks

k plots n plots

......

v treatments bn treatmentsall single replication

whole design ∆

Whole design ∆ has v + bn treatments in b blocks of size k + n;

the subdesign Γ has v core treatments in b blocks of size k;call the remaining treatments orphans.

44/52

Large blocks; many unreplicated treatments

Suppose that r̄ = ∑i ri

v< 2.

New conventions: blocks are rows, and block size = k + n.

b blocks

k plots n plots

......

v treatments bn treatmentsall single replication

whole design ∆

Whole design ∆ has v + bn treatments in b blocks of size k + n;the subdesign Γ has v core treatments in b blocks of size k;

call the remaining treatments orphans.

44/52

Large blocks; many unreplicated treatments

Suppose that r̄ = ∑i ri

v< 2.

New conventions: blocks are rows, and block size = k + n.

b blocks

k plots n plots

......

v treatments bn treatmentsall single replication

whole design ∆

Whole design ∆ has v + bn treatments in b blocks of size k + n;the subdesign Γ has v core treatments in b blocks of size k;call the remaining treatments orphans.

44/52

Levi graph: 10 + 5n treatments in 5 blocks of 4 + n plots

1 2 3 4 A1 · · · An

1 5 6 7 B1 · · · Bn

2 5 8 9 C1 · · · Cn

3 6 8 0 D1 · · · Dn

4 7 9 0 E1 · · · En

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����x1

HHH

H x5

�������x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

45/52

Pairwise resistance

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����x1

HHHH x5

�������x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

Resistance(A1, A2) = 2Resistance(A1, B1) = 2 + Resistance(block A, block B) in Γ

Resistance(A1, 8) = 1 + Resistance(block A, 8) in ΓResistance(1, 8) = Resistance(1, 8) in Γ

46/52

Pairwise resistance

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����x1

HHHH x5

�������x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

Resistance(A1, A2) = 2

Resistance(A1, B1) = 2 + Resistance(block A, block B) in ΓResistance(A1, 8) = 1 + Resistance(block A, 8) in Γ

Resistance(1, 8) = Resistance(1, 8) in Γ

46/52

Pairwise resistance

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����x1

HHHH x5

�������x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

Resistance(A1, A2) = 2Resistance(A1, B1) = 2 + Resistance(block A, block B) in Γ

Resistance(A1, 8) = 1 + Resistance(block A, 8) in ΓResistance(1, 8) = Resistance(1, 8) in Γ

46/52

Pairwise resistance

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����x1

HHHH x5

�������x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

Resistance(A1, A2) = 2Resistance(A1, B1) = 2 + Resistance(block A, block B) in Γ

Resistance(A1, 8) = 1 + Resistance(block A, 8) in Γ

Resistance(1, 8) = Resistance(1, 8) in Γ

46/52

Pairwise resistance

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����x1

HHHH x5

�������x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

Resistance(A1, A2) = 2Resistance(A1, B1) = 2 + Resistance(block A, block B) in Γ

Resistance(A1, 8) = 1 + Resistance(block A, 8) in ΓResistance(1, 8) = Resistance(1, 8) in Γ

46/52

Sum of the pairwise variances

Theorem (cf Herzberg and Jarrett, 2007)

The sum of the variances of treatment differences in ∆

= constant + V1 + nV3 + n2V2,where

V1 = the sum of the variances of treatment differences in ΓV2 = the sum of the variances of block differences in ΓV3 = the sum of the variances of sums of

one treatment and one block in Γ.

(If Γ is equi-replicate then V2 and V3 are both increasingfunctions of V1.)

Consequence

For a given choice of k, make Γ as efficient as possible.

47/52

Sum of the pairwise variances

Theorem (cf Herzberg and Jarrett, 2007)

The sum of the variances of treatment differences in ∆

= constant + V1 + nV3 + n2V2,where

V1 = the sum of the variances of treatment differences in ΓV2 = the sum of the variances of block differences in ΓV3 = the sum of the variances of sums of

one treatment and one block in Γ.

(If Γ is equi-replicate then V2 and V3 are both increasingfunctions of V1.)

Consequence

For a given choice of k, make Γ as efficient as possible.47/52

A less obvious consequence

Consequence

If n or b is large,and we want an A-optimal design,it may be best to make Γ a complete block design for k′ controls,even if there is no interest incomparisons between new treatments and controls,or between controls.

48/52

Spanning trees

A spanning tree for the Levi graph is a collection edges whichprovides a unique path between every pair of vertices.

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����

��x1

HHHH

HH x5

�������

����x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

The orphans make no difference to the number of spanningtrees for the Levi graph.

49/52

Spanning trees

A spanning tree for the Levi graph is a collection edges whichprovides a unique path between every pair of vertices.

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

������x1

HHHHHH x5

�������

����x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

The orphans make no difference to the number of spanningtrees for the Levi graph.

49/52

Spanning trees

A spanning tree for the Levi graph is a collection edges whichprovides a unique path between every pair of vertices.

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����

��x1

HHHH

HH x5

�������

����x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

The orphans make no difference to the number of spanningtrees for the Levi graph.

49/52

Spanning trees

A spanning tree for the Levi graph is a collection edges whichprovides a unique path between every pair of vertices.

� �

� �

�

x0

x2�������x7

AAAA x6

BBBBBBB

x4����x8

����

��x1

HHHH

HH x5

�������

����x

9Q

QQ

QQ

QQ x3

xA1

xAn ... @@ xCn

xC1...��

xD1xDn

...@@

xEn xE1

... ��

xB1HH

xBn��· · ·

The orphans make no difference to the number of spanningtrees for the Levi graph.

49/52

D-optimality under very low replication

Consequence

The whole design ∆ is D-optimalfor v + bn treatments in b blocks of size k + nif and only if the core design Γ is D-optimalfor v treatments in b blocks of size k.

Consequence

Even when n or b is large,D-optimal designs do not include uninteresting controls.

50/52

D-optimality under very low replication

Consequence

The whole design ∆ is D-optimalfor v + bn treatments in b blocks of size k + nif and only if the core design Γ is D-optimalfor v treatments in b blocks of size k.

Consequence

Even when n or b is large,D-optimal designs do not include uninteresting controls.

50/52

Conjectures

Conjecture (Underpinned by theoretical work by C.-S. Cheng)

If the A-optimal design is very different from the D-optimal design,then the E-optimal design is (almost) the same as the A-optimaldesign.

Conjecture (Underpinned by theoretical work by C.-S. Cheng)

If the connectivity is more than minimal, then all D-optimal designshave (almost) equal replication.

Conjecture (Underpinned by theoretical work by J. R. Johnsonand M. Walters)

If r̄ > 3.5 then designs optimal under one criterion are (almost)optimal under the other criteria.

51/52

Conjectures

Conjecture (Underpinned by theoretical work by C.-S. Cheng)

If the A-optimal design is very different from the D-optimal design,then the E-optimal design is (almost) the same as the A-optimaldesign.

Conjecture (Underpinned by theoretical work by C.-S. Cheng)

If the connectivity is more than minimal, then all D-optimal designshave (almost) equal replication.

Conjecture (Underpinned by theoretical work by J. R. Johnsonand M. Walters)

If r̄ > 3.5 then designs optimal under one criterion are (almost)optimal under the other criteria.

51/52

Conjectures

Conjecture (Underpinned by theoretical work by C.-S. Cheng)

If the A-optimal design is very different from the D-optimal design,then the E-optimal design is (almost) the same as the A-optimaldesign.

Conjecture (Underpinned by theoretical work by C.-S. Cheng)

If the connectivity is more than minimal, then all D-optimal designshave (almost) equal replication.

Conjecture (Underpinned by theoretical work by J. R. Johnsonand M. Walters)

If r̄ > 3.5 then designs optimal under one criterion are (almost)optimal under the other criteria.

51/52

Main References

I R. A. Bailey and Peter J. Cameron:Combinatorics of optimal designs.In Surveys in Combinatorics 2009(eds. S. Huczynska, J. D. Mitchell andC. M. Roney-Dougal),London Mathematical Society Lecture Note Series, 365,Cambridge University Press, Cambridge, 2009, pp. 19–73.

I R. A. Bailey and Peter J. Cameron:Using graphs to find the best block designs.In Topics in Structural Graph Theory(eds. L. W. Beineke and R. J. Wilson),Cambridge University Press, Cambridge, 2013,pp. 282–317.

52/52