Chapter 13
Complete Block Designs
Randomized Block Design (RBD)
• g > 2 Treatments (groups) to be compared
• r Blocks of homogeneous units are sampled. Blocks can
be individual subjects. Blocks are made up of t subunits
• Subunits within a block receive one treatment. When
subjects are blocks, receive treatments in random order.
• Outcome when Treatment i is assigned to Block j is
labeled Yij
• Effect of Trt i is labeled ai (Typically Fixed)
• Effect of Block j is labeled bj (Typically Random)
• Random error term is labeled eij
• Efficiency gain from removing block-to-block
variability from experimental error
Randomized Complete Block Designs
• Model:
Yij    a i  b j  e ij  i  b j  e ij
2
2




a

0
b
~
N
0
,

e
~
N
0
,

 i
j
b
ij
g
b   e 
i 1
• Test for differences among treatment effects:
• H0: a1  ...  ag  0
• HA: Not all ai = 0
(1  ...  g )
(Not all g are equal)
Typically not interested in measuring block effects (although
sometimes wish to estimate their variance in the population of
blocks). Using Block designs increases efficiency in making
inferences on treatment effects
RBD - ANOVA F-Test (Normal Data)
• Data Structure: (g Treatments, r Subjects (Blocks))
• Mean for Treatment i:
y i.
• Mean for Subject (Block) j:
• Overall Mean:
y. j
y ..
• Overall sample size: N = rg
• ANOVA:Treatment, Block, and Error Sums of Squares

TSS  i 1  j 1 yij  y 
g
r


SSB  g  y  y 
SSE   y  y  y
SST  r i 1 y i   y 
g
dfT  g  1
2
j
j 1
df B  r  1

ij
i

2
r
i 1 j 1
df Total  rg  1
2
r
g

2
j
 y 
 TSS  SST  SSB
df E  ( r  1)( g  1)
RBD - ANOVA F-Test (Normal Data)
• ANOVA Table:
Source
Treatments
Blocks
Error
Total
SS
SST
SSB
SSE
TSS
df
g-1
r-1
(r-1)(g-1)
rg-1
MS
MST = SST/(g-1)
MSB = SSB/(r-1)
MSE = SSE/[(r-1)(g-1)]
•H0: a1  ...  ag  0 (1  ...  g )
• HA: Not all ai = 0
T .S . : Fobs
R.R. : Fobs
(Not all i are equal)
MST

MSE
 Fa , g 1,( r 1)( g 1)
P  val : P ( F  Fobs )
F
F = MST/MSE
Pairwise Comparison of Treatment Means
• Tukey’s Method- with n = (r-1)(g-1)
MSE
Wij  qa ( g , v)
r
Conclude i   j if y i  y j   Wij


Tukey' s Confidence Interval : y i  y j   Wij
• Bonferroni’s Method - with n = (r-1)(g-1), C=g(g-1)/2
Bij  ta / 2,C ,v
2MSE
r
Conclude i   j if y i  y j   Bij


Bonferroni ' s Confidence Interval : y i  y j   Bij
Expected Mean Squares / Relative Efficiency
• Expected Mean Squares: As with CRD, the Expected Mean
Squares for Treatment and Error are functions of the sample
sizes (r, the number of blocks), the true treatment effects
(a1,…,ag) and the variance of the random error terms (2)
• By assigning all treatments to units within blocks, error
variance is (much) smaller for RBD than CRD (which
combines block variation&random error into error term)
• Relative Efficiency of RBD to CRD (how many times as
many replicates would be needed for CRD to have as
precise of estimates of treatment means as RBD does):
MSECR (r  1) MSB  r ( g  1) MSE
RE ( RCB , CR) 

MSE RCB
(rg  1) MSE
Example - Caffeine and Endurance
•
•
•
•
Treatments: g=4 Doses of Caffeine: 0, 5, 9, 13 mg
Blocks: r=9 Well-conditioned cyclists
Response: yij=Minutes to exhaustion for cyclist j @ dose i
Data:
Dose \ Subject
0
5
9
13
1
36.05
42.47
51.50
37.55
2
52.47
85.15
65.00
59.30
3
56.55
63.20
73.10
79.12
4
45.20
52.10
64.40
58.33
5
35.25
66.20
57.45
70.54
6
66.38
73.25
76.49
69.47
7
40.57
44.50
40.55
46.48
8
57.15
57.17
66.47
66.35
9
28.34
35.05
33.17
36.20
Plot of Y versus Subject by Dose
90.00
80.00
70.00
Time to Exhaustion
60.00
0 mg
50.00
5 mg
9mg
40.00
13 mg
30.00
20.00
10.00
0.00
0
1
2
3
4
5
Cyclist
6
7
8
9
10
Example - Caffeine and Endurance
Subject\Dose
1
2
3
4
5
6
7
8
9
Dose Mean
Dose Dev
Squared Dev
TSS
0mg
36.05
52.47
56.55
45.20
35.25
66.38
40.57
57.15
28.34
46.44
-8.80
77.38
5mg
42.47
85.15
63.20
52.10
66.20
73.25
44.50
57.17
35.05
57.68
2.44
5.95
9mg
51.50
65.00
73.10
64.40
57.45
76.49
40.55
66.47
33.17
58.68
3.44
11.86
13mg
37.55
59.30
79.12
58.33
70.54
69.47
46.48
66.35
36.20
58.15
2.91
8.48
Subj MeanSubj Dev Sqr Dev
41.89
-13.34
178.07
65.48
10.24
104.93
67.99
12.76
162.71
55.01
-0.23
0.05
57.36
2.12
4.51
71.40
16.16
261.17
43.03
-12.21
149.12
61.79
6.55
42.88
33.19
-22.05
486.06
55.24
1389.50
103.68
7752.773
TSS  (36.05  55.24) 2    (36.20  55.24) 2  7752.773 dfTotal  4(9)  1  35

SSB  4(41.89  55.24)

   (33.19  55.24)   4(1389.50)  5558.00
SST  9 (46.44  55.24) 2    (58.15  55.24) 2  9(103.68)  933.12 dfT  4  1  3
2
2
df B  9  1  8
SSE  (36.05  41.89  46.44  55.24) 2    (36.20  33.19  58.15  55.24) 2 
 TSS  SST  SSB  7752.773  933.12  5558  1261.653 df E  (4  1)(9  1)  24
Example - Caffeine and Endurance
Source
Dose
Cyclist
Error
Total
df
3
8
24
35
SS
933.12
5558.00
1261.65
7752.77
MS
311.04
694.75
52.57
H 0 : No Caffeine Dose Effect (a1    a 4  0)
H A : Difference s Exist Among Doses
MST 311.04
T .S . : Fobs 

 5.92
MSE 52.57
R.R.(a  0.05) : Fobs  F.05,3, 24  3.01
P  value : P( F  5.92)  .0036 (From EXCEL)
Conclude that true means are not all equal
F
5.92
Example - Caffeine and Endurance
Tukey' s W : q.05, 4, 24
1
 3.90 W  3.90 52.57   9.43
9
Bonferroni ' s B : t.05 / 2, 6, 24
Doses
5mg vs 0mg
9mg vs 0mg
13mg vs 0mg
9mg vs 5mg
13mg vs 5mg
13mg vs 9mg
2
 2.875 B  2.875 52.57   9.83
9
High Mean
57.6767
58.6811
58.1489
58.6811
58.1489
58.1489
Low Mean Difference Conclude
46.4400
11.2367
5>0
46.4400
12.2411
9>0
46.4400
11.7089
13>0
57.6767
1.0044
NSD
57.6767
0.4722
NSD
58.6811
-0.5322
NSD
Example - Caffeine and Endurance
Relative Efficiency of Randomized Block to Completely Randomized Design :
g  4 r  9 MSB  694.75 MSE  52.57
(r  1) MSB  r ( g  1) MSE 8(694.75)  9(3)(52.57) 6977.39
RE ( RCB , CR) 


 3.79
(rg  1) MSE
(9(4)  1)(52.57)
1839.95
Would have needed 3.79 times as many cyclists per dose to have the
same precision on the estimates of mean endurance time.
• 9(3.79)  35 cyclists per dose
• 4(35) = 140 total cyclists
Latin Square Design
• Design used to compare g treatments when there are
two sources of extraneous variation (types of blocks),
each observed at g levels
• Best suited for analyses when g  10
• Classic Example: Car Tire Comparison
– Treatments: 4 Brands of tires (A,B,C,D)
– Extraneous Source 1: Car (1,2,3,4)
– Extraneous Source 2: Position (Driver Front, Passenger
Front, Driver Rear, Passenger Rear)
Car\Position
1
2
3
4
DF
A
B
C
D
PF
B
C
D
A
DR
C
D
A
B
PR
D
A
B
C
Latin Square Design - Model
• Model (g treatments, rows, columns, N=g2) :
yijk    a i  b j   k  e ijk
  Overall Mean
^
  y 
^
a i  Effect of Treatment i a i  y i  y 
bi
 Effect due to row j
^
b j  y  j   y  
^
 k  Effect due to Column k  k  y  k  y 
e ijk  Random Error Term
Latin Square Design - ANOVA & F-Test
g

g
Total Sum of Squares : TSS   yijk  y 

2
df  g 2  1
j 1 k 1
g

Treatment Sum of Squares SST  g  y i  y 
i 1
g

Row Sum of Squares SSR  g  y  j   y 


Column Sum of Squares SSC  g  y  k  y 
k 1
dfT  g  1
2
j 1
g

2
df R  g  1

2
df C  g  1
Error Sum of Squares SSE  TSS  SST  SSR  SSC df E  ( g 2  1)  3( g  1)  ( g  1)( g  2)
• H0: a1 = … = ag = 0
Ha: Not all ai = 0
• TS: Fobs = MST/MSE = (SST/(g-1))/(SSE/((g-1)(g-2)))
• RR: Fobs  Fa, g-1, (g-1)(g-2)
Pairwise Comparison of Treatment Means
• Tukey’s Method- with n = (g-1)(g-2)
Wij  qa ( g , v)
MSE
g
Conclude i   j if y i  y j   Wij


Tukey' s Confidence Interval : y i  y j   Wij
• Bonferroni’s Method - with n = (g-1)(g-2), C=g(g-1)/2
Bij  ta / 2,C ,v
2MSE
g
Conclude i   j if y i  y j   Bij


Bonferroni ' s Confidence Interval : y i  y j   Bij
Expected Mean Squares / Relative Efficiency
• Expected Mean Squares: As with CRD, the Expected Mean
Squares for Treatment and Error are functions of the sample
sizes (g, the number of blocks), the true treatment effects
(a1,…,ag) and the variance of the random error terms (2)
• By assigning all treatments to units within blocks, error
variance is (much) smaller for LS than CRD (which
combines block variation&random error into error term)
• Relative Efficiency of LS to CRD (how many times as
many replicates would be needed for CRD to have as
precise of estimates of treatment means as LS does):
MSECR MSR  MSC  ( g  1) MSE
RE ( LS , CR) 

MSE LS
( g  1) MSE
Replicated Latin Squares
• To Increase Power (and Error degrees of freedom),
experimenters often will use multiple (m>1) gxg latin
squares for their design. There are 3 possible model
structures:
• Model 1: Separate Row and Column blocks used in each
square
• Model 2: Common Row, but separate Column blocks used in
each square
• Model 3: Common Row and Column blocks used in each
square
Model 1 – Separate Row and Column Blocks
yilkl    a i  b j (l )   k (l )   l  e ijkl
Square1
Row(1)
…
Row(g)
Col(1)
…
Col(g)
ANOVA
Source
Treatments
Rows
Columns
Squares
Error
Total
Square2 Col(g+1)
Row(g+1)
…
Row(2g)
i  1,..., g ; j  1,..., g ; k  1,..., g ; l  1,..., m
…
Col(2g)
…
…
…
…
Square m
Col((m-1)g+1)
Row((m-1)g+1)
…
Row(mg)
df
g-1
m(g-1)
m(g-1)
m-1
(mg-m-1)(g-1)
mgg-1
…
Col(mg)
Model 2 – Common Row, Separate Column Blocks
yilkl    a i  b j   k (l )   l  e ijkl
Square1
Row(1)
…
Row(g)
Col(1)
…
Col(g)
ANOVA
Source
Treatments
Rows
Columns
Squares
Error
Total
Square2 Col(g+1)
Row(1)
…
Row(g)
i  1,..., g ; j  1,..., g ; k  1,..., g ; l  1,..., m
…
Col(2g)
…
…
…
…
df
g-1
g-1
m(g-1)
m-1
(mg-2)(g-1)
mgg-1
Square m
Row(1)
…
Row(g)
Col((m-1)g+1)
…
Col(mg)
Model 3 – Common Row and Column Blocks
yilkl    a i  b j   k   l  e ijkl
Square1
Row(1)
…
Row(g)
Col(1)
…
Col(g)
ANOVA
Source
Treatments
Rows
Columns
Squares
Error
Total
Square2
Row(1)
…
Row(g)
Col(1)
i  1,..., g ; j  1,..., g ; k  1,..., g ; l  1,..., m
…
Col(g)
…
…
…
…
df
g-1
g-1
g-1
m-1
(mg+m-3)(g-1)
mgg-1
Square m
Row(1)
…
Row(g)
Col(1)
…
Col(g)
Designs Balanced for Carry-Over Effects
• Subjects receive g treatments, one in each of g time
periods
• Treatments are balanced with equal number of replicates
per time period (across subjects)
• Design balanced such that each treatment follows each
other treatment equal number of times and appears in the
first position equal number of times.
• Carryover effect that observation in Period 1 receives is 0
yijkl    a i  b j   k   l  e ijkl
a i  Direct of effect of Trt i b j  Carry - Over Effect whe n Trt j was in period l  1
 k  Effect of Subject k
 l  Effect of Period l
Example 13.12 – Milk Yield
Period\Cow
1
2
3
Period\Cow
1
2
3
Period\Cow
1
2
3
1
A
B
C
7
A
B
C
13
A
B
C
2
B
C
A
8
B
C
A
14
B
C
A
3
C
A
B
9
C
A
B
15
C
A
B
4
A
C
B
10
A
C
B
16
A
C
B
5
B
A
C
11
B
A
C
17
B
A
C
6
C
B
A
12
C
B
A
18
C
B
A
Example 13.2 – Factor/Carryover Coding
Cow
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
9
9
9
Period
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Trt
1
2
3
2
3
1
3
1
2
1
3
2
2
1
3
3
2
1
1
2
3
2
3
1
3
1
2
Trt1
1
0
-1
0
-1
1
-1
1
0
1
-1
0
0
1
-1
-1
0
1
1
0
-1
0
-1
1
-1
1
0
Trt2
0
1
-1
1
-1
0
-1
0
1
0
-1
1
1
0
-1
-1
1
0
0
1
-1
1
-1
0
-1
0
1
Res1
0
1
0
0
0
-1
0
-1
1
0
1
-1
0
0
1
0
-1
0
0
1
0
0
0
-1
0
-1
1
Res2
0
0
1
0
1
-1
0
-1
0
0
0
-1
0
1
0
0
-1
1
0
0
1
0
1
-1
0
-1
0
Cow
10
10
10
11
11
11
12
12
12
13
13
13
14
14
14
15
15
15
16
16
16
17
17
17
18
18
18
Period
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
Trt
1
3
2
2
1
3
3
2
1
1
2
3
2
3
1
3
1
2
1
3
2
2
1
3
3
2
1
Trt1
1
-1
0
0
1
-1
-1
0
1
1
0
-1
0
-1
1
-1
1
0
1
-1
0
0
1
-1
-1
0
1
Trt2
0
-1
1
1
0
-1
-1
1
0
0
1
-1
1
-1
0
-1
0
1
0
-1
1
1
0
-1
-1
1
0
Res1
0
1
-1
0
0
1
0
-1
0
0
1
0
0
0
-1
0
-1
1
0
1
-1
0
0
1
0
-1
0
Res2
0
0
-1
0
1
0
0
-1
1
0
0
1
0
1
-1
0
-1
0
0
0
-1
0
1
0
0
-1
1
Note: Trt1,Trt2 and Res1,Res2 are coded so thatTrt and Res Effects sum to Zero
(“Trt3 = -Trt1-Trt2” and “Res3 = -Res1-Res2”, with no Res effects in Period 1)