Planning of generally balanced nested
row-column designs for near-factorial
experiments
Agnieszka Lacka♠ & Rosemary A. Bailey♣
♠ Department
of Mathematical and Statistical Methods
Poznań University of Life Sciences
♣ School
of Mathematical Sciences, Queen Mary, University of London
School of Mathematics and Statistics, University of St Andrews
1.07.2014
A. Lacka (PULS)
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The Problem
Near–factorial experiment
A. Lacka (PULS)
0
|{z}
1, 2, 3, ...., w
|
{z
}
control
treatment
other
treatments
Nested row-column designs...
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The Problem
Near–factorial experiment
0
|{z}
1, 2, 3, ...., w
|
{z
}
control
treatment
other
treatments
FACTORS:
T occurring on t levels
U occurring on u levels
A. Lacka (PULS)
0
|{z}
11, 12, 13, ...., tu
|
{z
}
control
treatment
other
treatments
Nested row-column designs...
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The Problem
Nested row–column design
|
{z
n1 blocks
}



|
A. Lacka (PULS)
{z
}
n3 columns
Nested row-column designs...
n2 rows
n = n1 n2 n3
v = w + 1 = tu + 1
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Orthogonal block structure
DESIGN
A. Lacka (PULS)
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Orthogonal block structure
DESIGN
Blocks
(n1 levels)
B (1)
A. Lacka (PULS)
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Orthogonal block structure
DESIGN
Blocks
(n1 levels)
B (1)
Rows
(n2 levels)
R[B] (2)
A. Lacka (PULS)
Columns
(n3 levels)
C[B] (3)
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Orthogonal block structure
DESIGN
Blocks
(n1 levels)
B (1)
Rows
(n2 levels)
R[B] (2)
Columns
(n3 levels)
C[B] (3)
R#C[B]
(4)
A. Lacka (PULS)
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Orthogonal block structure
DESIGN
Blocks
(n1 levels)
B (1)
Rows
(n2 levels)
R[B] (2)
Columns
(n3 levels)
C[B] (3)
C(2)
R#C[B]
(4)
A. Lacka (PULS)
C(1)
C(3)
C(4) = C
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Assumption #1
GENERAL BALANCE
A. Lacka (PULS)
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Assumption #1
GENERAL BALANCE
↓
R−1/2 C(k) R−1/2 , k = 1, ..., 4
commute
A. Lacka (PULS)
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Assumption #1
GENERAL BALANCE
↓
R−1/2 C(k) R−1/2 , k = 1, ..., 4
commute
↓
common eigenvectors,
which all correspond to treatment contrasts
A. Lacka (PULS)
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Assumption #1
GENERAL BALANCE
↓
R−1/2 C(k) R−1/2 , k = 1, ..., 4
commute
↓
common eigenvectors,
which all correspond to treatment contrasts
↓
eigenvalues =
canonical efficiency factors λi
A. Lacka (PULS)
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Assumption #2
CONTROL ORTHOGONALITY
A. Lacka (PULS)
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Assumption #2
CONTROL ORTHOGONALITY
↓
n2 m2 = n3 m3
A. Lacka (PULS)
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Assumption #2
CONTROL ORTHOGONALITY
↓
n2 m2 = n3 m3
↓
r0 = n1 n2 m2
A. Lacka (PULS)
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Assumption #2
CONTROL ORTHOGONALITY
↓
n2 m2 = n3 m3
↓
r0 = n1 n2 m2
↓
contrast between the control and the other treatments
has full efficiency in the R#C[B] bottom stratum (λ0 = 1)
A. Lacka (PULS)
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Assumptions #3 and #4
S – an association scheme
on the w non-control
treatments
r – replication of
non-control
treatments
A. Lacka (PULS)
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Assumptions #3 and #4
S – an association scheme
on the w non-control
treatments
r – replication of
non-control
treatments
&
.
C=
A. Lacka (PULS)
wd
−d1w
−d10w
L + wd Jw
Nested row-column designs...
d = r0 r/n
L – Bose–Mesner
algebra A of S
with L1w = 0w
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Assumptions #3 and #4
S – an association scheme
on the w non-control
treatments
r – replication of
non-control
treatments
&
.
C=
wd
−d1w
−d10w
L + wd Jw
d = r0 r/n
L – Bose–Mesner
algebra A of S
with L1w = 0w
↓
Moore–Penrose
gen. inv
A. Lacka (PULS)
C− =
we
−e1w
−e10w
−
L + we Jw
Nested row-column designs...
dev2 = 1
L− – Moore–Penrose
gen. inv. of L,
which is also in A
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Consequences
contrast
A. Lacka (PULS)
x0 τ
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Consequences
x0 τ
contrast
↓
variance
normalized contrast
A. Lacka (PULS)
x0 C− x 2
x0 x σ
Nested row-column designs...
σ 2 − R#C[B]
stratum
variance
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Consequences
x0 τ
contrast
↓
variance
normalized contrast
.
&
x = (1, −w−1 , . . . , −w−1 )0
n
2
r0 rv σ
A. Lacka (PULS)
σ 2 − R#C[B]
stratum
variance
x0 C− x 2
x0 x σ
x = (1, 0, . . . , −1, 0)0
1
2
Nested row-column designs...
1
r0
1
+ rw
+` σ2
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What about the other contrasts?
. . . .
Trivial
R(t, u)
association
schemes
GD(t, u)
EGD(p, q, u)
A. Lacka (PULS)
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What about the other contrasts?
. . . .
Trivial
R(t, u)
association
schemes
D
T
GD(t, u)
EGD(p, q, u)
A. Lacka (PULS)
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What about the other contrasts?
. . . .
Trivial
R(t, u)
D
association
schemes
T
GD(t, u)
L = α(Iw − w−1 Jw )
` = (w − 1)/αw
EGD(p, q, u)
A. Lacka (PULS)
Nested row-column designs...
λT = α/r
(α −1 σ 2 )
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Designs with supplemented balance #1
v = w + 1 (w 6= 2)
n1 arbitrary
n2 = n3 = w
α=
(w−2)(w+1)n1
w
type
df
B
R[B]
C[B]
R#C[B]
with control
1
0
0
0
1
A. Lacka (PULS)
all
w−1
0
1
(w−1)w
1
(w−1)w
2
1 − (w−1)w
Nested row-column designs...
A
C
D
B
D
B
A
C
B C
D A
C B
A D
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Group-divisible association scheme
GD(t, u)
w = tu
D
T
U[T]
A. Lacka (PULS)
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Group-divisible association scheme
GD(t, u)
w = tu
L = wβ ST + u[α + (t − 1)β ]SU[T]
D
ST = u−1 It ⊗ Ju − S0
SU[T] = Iw − ST − S0
S0 = w−1 Jw
T
` = (t − 1)w−2 β −1 + t(u − 1)(wu)−1 [α + (t − 1)β ]−1
λT = wβ /r
λU[T] = u[α + (t − 1)β ]/r
((wβ )−1 σ 2 )
(u−1 [α + (t − 1)β ]−1 σ 2 )
A. Lacka (PULS)
Nested row-column designs...
U[T]
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Supplemented group-divisible design #2
v = tu + 1 treatments
n1 = t blocks
n2 = n3 = tu rows and columns
t = 3, u = 2
Block 2
A. Lacka (PULS)
1
2
3
4
5
6
1
2
3
4
5
6
column
column
column
column
column
column
column
column
column
column
column
column
1
2
3
4
5
6
1
2
3
4
5
6
row
row
row
row
row
row
Block 3
column
column
column
column
column
column
Block 1
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
2
3
4
5
6
1
3
4
5
6
1
2
4
5
6
1
2
3
5
6
1
2
3
4
6
1
2
3
4
5
2
3
4
5
6
1
3
4
5
6
1
2
4
5
6
1
2
3
5
6
1
2
3
4
Nested row-column designs...
6
1
2
3
4
5
2
3
4
5
6
1
3
4
5
6
1
2
4
5
6
1
2
3
5
6
1
2
3
4
6
1
2
3
4
5
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Supplemented group-divisible design #2
v = tu + 1 treatments
n1 = t blocks
n2 = n3 = tu rows and columns
t = 3, u = 2
Block 2
A. Lacka (PULS)
1
2
3
4
5
6
1
2
3
4
5
6
column
column
column
column
column
column
column
column
column
column
column
column
1
2
3
4
5
6
1
2
3
4
5
6
row
row
row
row
row
row
Block 3
column
column
column
column
column
column
Block 1
0
0
3
4
5
6
1
2
0
0
5
6
1
2
3
4
0
0
0
3
4
5
6
0
3
4
5
6
0
0
4
5
6
0
0
3
5
6
0
0
3
4
6
0
0
3
4
5
2
0
0
5
6
1
0
0
5
6
1
2
0
5
6
1
2
0
5
6
1
2
0
0
Nested row-column designs...
6
1
2
0
0
5
2
3
4
0
0
1
3
4
0
0
1
2
4
0
0
1
2
3
0
0
1
2
3
4
0
1
2
3
4
0
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Supplemented group-divisible design #2
v = tu + 1 treatments
n1 = t blocks
n2 = n3 = tu rows and columns
t = 3, u = 2
column
column
column
column
column
column
1
2
3
4
5
6
column
column
column
column
column
column
1
2
3
4
5
6
1
2
3
4
5
6
column
column
column
column
column
column
row
row
row
row
row
row
Block 3
1
2
3
4
5
6
Block 2
Block 1
0
0
21
22
31
32
11
12
0
0
31
32
11
12
21
22
0
0
0
21
22
31
32
0
A. Lacka (PULS)
21
22
31
32
0
0
22
31
32
0
0
21
31
32
0
0
21
22
32
0
0
21
22
31
12
0
0
31
32
11
0
0
31
32
11
12
0
31
32
11
12
0
31
32
11
12
0
0
Nested row-column designs...
32
11
12
0
0
31
12
21
22
0
0
11
21
22
0
0
11
12
22
0
0
11
12
21
0
0
11
12
21
22
1.07.2014
0
11
12
21
22
0
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Supplemented group-divisible design #2
v = tu + 1
n1 = t
n2 = n3 = tu
α=
t2 −1
t
β=
t2 −t−1
t
Canonical efficiency factors in each stratum
for the design obtained with the use of construction 2
type
with control contrasts - l. of T contrasts - l. of U
T
U[T]
number
1
t−1
t (u − 1)
1
B
0
0
t(t−1)
R[B]
0
0
0
C[B]
0
0
0
1
R# C[B]
1
1 − t(t−1)
1
A. Lacka (PULS)
Nested row-column designs...
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Supplemented group-divisible design #2
v = tu + 1 treatments
n1 = t blocks
n2 = n3 = tu rows and columns
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
1
2
3
4
5
6
7
8
R#C[B]
A. Lacka (PULS)
9
10
11
12
13
14
15
16
B
Nested row-column designs...
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Extended GD association scheme
w = tu
t = pq
EGD(p, q, u)
D
P
T[P]
U[T]
A. Lacka (PULS)
Nested row-column designs...
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Extended GD association scheme
EGD(p, q, u)
w = tu
t = pq
D
L = wγSP + qu[β + (p − 1)γ]ST[P] + u[α + (q − 1)β + (p − 1)qγ]SU[T]
ST = u−1 It ⊗ Ju − S0
SU[T] = Iw − ST − S0
S0 = w−1 Jw
SP = (qu)−1 Ip ⊗ Jqu − S0
ST[P] = ST − SP
` = (p − 1)w−2 γ −1 + p(q − 1)(wqu)−1 [β + (p − 1)γ]−1
+t(u − 1)(wu)−1 [α + (q − 1)β + (p − 1)qγ]−1
λP = wγ/r
((wγ)−1 σ 2 )
P
T[P]
λT[P] = qu[β + (p − 1)γ]/r
((qu)−1 [β + (p − 1)γ]−1 σ 2 )
U[T]
u
r [α + (q − 1)β
+ (p − 1)qγ]
λU[T] =
(u−1 [α + (q − 1)β + (p − 1)qγ]−1 σ 2 )
A. Lacka (PULS)
Nested row-column designs...
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Supplemented extended GD design #3
v = tu + 1 treatments
t = pq treatments
n1 = p blocks
n2 = n3 = tu rows and columns
p = q = 2 and u = 3
A0
B
C
D
replace with:
3
1
2
A→ 1
2
3
3 0
A0 → 0 2
0 0
Base block 1
B
C
D
C0
D
A
D
A0
B
A
B
C0
2
3
1
0
0
1
A. Lacka (PULS)
6
B→ 4
5
6
B0 → 0
0
B0
C
D
A
4
5
6
0
5
0
5
6
4
0
0
4
9
C→ 7
8
9
C0 → 0
0
7
8
9
0
8
0
Nested row-column designs...
Base block 2
C
D
A
D0
A
B
A
B0
C
B
C
D0
8
9
7
0
0
7
12
D→ 10
11
12
D0 → 0
0
10
11
12
0
11
0
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11
12
10
0
0
10
19 / 33
Supplemented extended GD design #3
v = tu + 1 treatments
t = pq treatments
n1 = p blocks
n2 = n3 = tu rows and columns
p = q = 2 and u = 3
A0
B
C
D
replace with:
13 11
A→ 11 12
12 13
A0 →
13
0
0
0
12
0
Base block 1
B
C
D
C0
D
A
D
A0
B
A
B
C0
12
13
11
0
0
11
A. Lacka (PULS)
23
B→ 21
22
B0 →
23
0
0
B0
C
D
A
21
22
23
0
22
0
22
23
21
0
0
21
33
C→ 31
32
C0 →
33
0
0
Nested row-column designs...
31
32
33
0
32
0
Base block 2
C
D
A
D0
A
B
A
B0
C
B
C
D0
32
33
31
0
0
31
43
D→ 41
42
D0 →
41
42
43
43
0
0
0
42
0
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42
43
41
0
0
41
20 / 33
Supplemented extended GD design #3
v = tu + 1 treatments
t = pq treatments
n1 = p blocks
n2 = n3 = tu rows and columns
p = 4, q = 1 and u = 3
Base block
A0
B
C
D
A0
B
C
D
A0
B
C
D
replace with:
13
11
12
A→ 11
12
13
A0 →
13
0
0
1
D
C
B
A0
12
13
11
0
12
0
A. Lacka (PULS)
0
0
11
B0
D
C
A
Base block 2
A
C
D
B0
A
C
D
B0
A
C
D
B0
23
B→ 21
22
B0 →
23
0
0
21
22
23
0
22
0
22
23
21
0
0
21
C0
D
A
B
Base block 3
B
A
D
C0
B
A
D
C0
B
A
D
C0
33
C→ 31
32
C0 →
33
0
0
Nested row-column designs...
31
32
33
0
32
0
32
33
31
0
0
31
D0
A
C
B
Base block 4
B
C
A
D0
B
C
A
D0
B
C
A
D0
43
D→ 41
42
D0 →
43
0
0
1.07.2014
41
42
43
0
42
0
42
43
41
0
0
41
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Supplemented extended GD design #3
v = tu + 1 treatments
t = pq treatments
n1 = p blocks
n2 = n3 = tu rows and columns
p = 1, q = 4 and u = 3
A0
C
D
B
replace with:
13 11
A→ 11 12
12 13
A0 →
13
0
0
0
12
0
12
13
11
0
0
11
A. Lacka (PULS)
23
B→ 21
22
B0 →
23
0
0
21
22
23
0
22
0
Base block 1
D
B
C
B0
D
A
A
C0
B
C
A
D0
22
23
21
0
0
21
33
C→ 31
32
C0 →
33
0
0
Nested row-column designs...
31
32
33
0
32
0
32
33
31
0
0
31
43
D→ 41
42
D0 →
41
42
43
43
0
0
0
42
0
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42
43
41
0
0
41
22 / 33
Supplemented extended GD design #3
α=
γ=
w(w+u−3)−(p+1)(u−1)2
u2 pq2
w(w−2u+2)+(u−1)(w+p+1−pu−u)
β=
u2 pq2
2
w(w−u+1)−(u−1)
u2 pq2
type
number
Canonical efficiency factors in each stratum
for the design obtained with the help of construction 3
between t
factor U
with
between p
groups within
within one level
control
groups of T
one p group
of factor T
P
T[P]
U[T]
1
p−1
p (q − 1)
(u − 1)pq
B
0
R[B]
0
C[B]
0
R#C[B]
A. Lacka (PULS)
1
(u−1)2
tu(tu−u+1)
0
0
1−
(u−1)2
tu(tu−u+1)
0
0
(u−1)2
tu(tu−u+1)
(u−1)2
tu(tu−u+1)
2(u−1)2
1 − tu(tu−u+1)
1
tu(tu−u+1)
1
tu(tu−u+1)
2
1 − tu(tu−u+1)
Nested row-column designs...
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Rectangular association scheme R(t, u)
w = tu
D
T
U
T#U
A. Lacka (PULS)
Nested row-column designs...
1.07.2014
24 / 33
Rectangular association scheme R(t, u)
w = tu
L = t[β + (u − 1)γ]ST + u[α + (t − 1)γ]SU
+[uα + tβ + (tu − t − u)γ]STU
ST = u−1 It ⊗ Ju − S0
S0 = w−1 Jw
D
SU = t−1 Jt ⊗ Iu − S0
STU = Iw − ST − SU − S0
T
` = (t − 1)(wt)−1 [β
U
+ (u − 1)γ]−1 + (u − 1)(wu)−1 [α + (t − 1)γ]−1
+(t − 1)(u − 1)w−1 [uα + tβ + (tu − t − u)γ]−1
λT = t[β + (u − 1)γ]/r
(t−1 [β + (u − 1)γ]−1 σ 2 )
λU = u[α + (t − 1)γ]/r
(u−1 [α + (t − 1)γ]−1 σ 2 )
T#U
λTU = [uα + tβ + (tu − t − u)γ]/r
([uα + tβ + (tu − t − u)γ]−1 σ 2 )
A. Lacka (PULS)
Nested row-column designs...
1.07.2014
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Supplemented rectangular design #4
v = tu + 1 treatments
n2 = u rows
n3 = tu columns
t = 3 and u = 4
A. Lacka (PULS)
column 3
column 4
column 5
column 6
column 7
column 8
column 9
column 10
column 11
column 12
1
2
3
4
column 2
row
row
row
row
column 1
Block 1
4
2
1
3
1
3
4
2
3
1
2
4
2
4
3
1
8
6
5
7
5
7
8
6
7
5
6
8
6
8
7
5
12
10
9
11
9
11
12
10
11
9
10
12
10
12
11
9
Nested row-column designs...
1.07.2014
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Supplemented rectangular design #4
v = tu + 1 treatments
n2 = u rows
n3 = tu columns
t = 3 and u = 4
column 3
column 4
column 5
column 6
column 7
column 8
column 9
column 10
column 11
column 12
1
2
3
4
column 2
row
row
row
row
column 1
Block 1
14
12
11
13
11
13
14
12
13
11
12
14
12
14
13
11
24
22
21
23
21
23
24
22
23
21
22
24
22
24
23
21
34
32
31
33
31
33
34
32
33
31
32
34
32
34
33
31
A. Lacka (PULS)
Nested row-column designs...
1.07.2014
26 / 33
Supplemented rectangular design #4
v = tu + 1 treatments
n2 = u rows
n3 = tu columns
t = 3 and u = 4
column 3
column 4
column 5
column 6
column 7
column 8
column 9
column 10
column 11
column 12
1
2
3
4
column 2
row
row
row
row
column 1
Block 1
0
12
11
13
11
0
14
12
13
11
0
14
12
14
13
0
0
22
21
23
21
0
24
22
23
21
0
24
22
24
23
0
0
32
31
33
31
0
34
32
33
31
0
34
32
34
33
0
A. Lacka (PULS)
Nested row-column designs...
1.07.2014
27 / 33
Supplemented rectangular design #4
v = tu + 1 treatments
n2 = u rows
n3 = tu columns
β=
α=
n1 (w+1)(u−2)
tu2
γ=
n1 (u−2)
tu2
2n1 (u−1)
tu2
Canonical efficiency factors in each stratum
for the design obtained with the use of construction 4
type
with control factor T
factor U
interaction
T
U
T#U
nr
1
t−1
u−1
(t − 1) (u − 1)
B
0
0
0
0
1
R[B]
0
0
0
u(u−1)
u−1
1
1
C[B]
0
u
u(u−1)
u(u−1)
1
2
1
R#C[B]
1
1 − u(u−1)
1 − u(u−1)
u
A. Lacka (PULS)
Nested row-column designs...
1.07.2014
28 / 33
Supplemented rectangular design #4
v = tu + 1 treatments
n2 = u rows
n3 = tu columns
1
0,9
0,8
0,7
0,6
0,5
R#C[B]
0,4
0,3
0,2
0,1
0
2
4
6
8
10
T
A. Lacka (PULS)
12
U
14
16
18
20
T#U
Nested row-column designs...
1.07.2014
29 / 33
.... #5
v = t2 + 1 treatments
n2 = r = t + 1 rows
n3 = r0 = t(t + 1)
(t-power of prime)
t = u = 3 ⇒ t2 + t + 1 = 13
Perfect difference set: {1, 2, 5, 7}
A B
B C
E F
G H
0
11
13
21
A. Lacka (PULS)
C D
D E
G H
I J
E
F
I
K
F G
G H
J K
L M
H
I
L
A
I
J
M
B
J
K
A
C
K L
L M
B C
D E
M
A
D
F
11 12 0 13 0 21 22 23 31 32 33
12 0 13 0 21 22 23 31 32 33 0
0 21 22 23 31 32 33 0
0 11 12
22 23 31 32 33 0
0 11 12 0 13
Nested row-column designs...
1.07.2014
30 / 33
.... #5
v = t2 + 1 treatments
n2 = r = t + 1 rows
n3 = r0 = t(t + 1)
`=
(t−1)(t+1)2
n1 t2 (t2 +t+1)
(
(t-power of prime)
t+1
σ2
n1 (t2 +t+1)
)
Canonical efficiency factors in each stratum for the design
obtained from Construction 5
A. Lacka (PULS)
type
df
B
R[B]
C[B]
with control
1
0
0
0
R#C[B]
1
all other
t2 − 1
0
0
t
(1+t)2
1− t 2
(1+t)
Nested row-column designs...
1.07.2014
31 / 33
what to choose?
t = 2, u = 2, r0 = 8, r = 6, n1 = 2, n2 = n3 = 4
construction #3 (p = 2, q = 1)
construction #1
0
4
2
3
3
0
4
1
4
1
0
2
2
3
1
0
λ0 = 1
λT = 0.8333
0
4
2
3
3
0
4
1
4
1
0
2
(0.1333σ 2 )
(0.2σ 2 )
2
3
1
0
1
0
3
4
3
4
1
0
4
3
0
2
λ0 = 1
λT = 0.9167
λU[T] = 0.8333
(0.1583σ 2 )
A. Lacka (PULS)
0
2
4
3
3
0
1
2
0
4
2
1
1
2
3
0
2
1
0
4
(0.1333σ 2 )
(0.1818σ 2 )
(0.2σ 2 )
(0.1560σ 2 )
Nested row-column designs...
1.07.2014
32 / 33
what to choose?
p = 2, q = 1, t = 2, u = 3, n1 = 2, n2 = n3 = 6
construction #3
1
0
0
4
6
5
0
2
0
6
5
4
0
0
3
5
4
6
4
6
5
1
0
0
6
5
4
0
2
0
5
4
6
0
0
3
λ0 = 1
λT = 0.8333
λU[T] = 0.9167
4
0
0
1
3
2
0
5
0
3
2
1
construction #3(mod)
0
0
6
2
1
3
1
3
2
4
0
0
(0.0536σ 2 )
(0.15σ 2 )
(0.1363σ 2 )
3
2
1
0
5
0
2
1
3
0
0
6
0
0
0
4
6
5
0
0
0
6
5
4
λ0 = 1
λT = 0.5
λU[T] = 1
(0.0892σ 2 )
A. Lacka (PULS)
0
0
0
5
4
6
4
6
5
0
0
0
6
5
4
0
0
0
5
4
6
0
0
0
0
0
0
1
3
2
0
0
0
3
2
1
0
0
0
2
1
3
1
3
2
0
0
0
3
2
1
0
0
0
2
1
3
0
0
0
(0.0476σ 2 )
(0.3333σ 2 )
(0.1667σ 2 )
(0.1111σ 2 )
Nested row-column designs...
1.07.2014
33 / 33