Balanced Incomplete Block
Design
Ford Falcon Prices Quoted by 28 Dealers to
8 Interviewers (2 Interviewers/Dealer)
Source: A.F. Jung (1961). "Interviewer Differences Among Automile Purchasers," JRSSC (Applied Statistics), Vol 10, #2, pp. 93-97
Balanced Incomplete Block Design (BIBD)
• Situation where the number of treatments exceeds
number of units per block (or logistics do not allow
for assignment of all treatments to all blocks)
• # of Treatments  g
• # of Blocks  b
• Replicates per Treatment  r < b
• Block Size  k < g
• Total Number of Units  N = kb = rg
• All pairs of Treatments appear together in
l = r(k-1)/(g-1) Blocks for some integer l
BIBD (II)
• Reasoning for Integer l:
 Each Treatment is assigned to r blocks
 Each of those r blocks has k-1 remaining positions
 Those r(k-1) positions must be evenly shared among the
remaining g-1 treatments
• Tables of Designs for Various g,k,b,r in Experimental
Design Textbooks (e.g. Cochran and Cox (1957) for a
huge selection)
• Analyses are based on Intra- and Inter-Block
Information
Interviewer Example
• Comparison of Interviewers soliciting prices from Car
Dealerships for Ford Falcons
• Response: Y = Price-2000
• Treatments: Interviewers (g = 8)
• Blocks: Dealerships (b = 28)
• 2 Interviewers per Dealership (k = 2)
• 7 Dealers per Interviewer (r = 7)
• Total Sample Size N = 2(28) = 7(8) = 56
• Number of Dealerships with same pair of
interviewers: l = 7(2-1)/(8-1) = 1
Interviewer Example
Dealer\Interviewer
A
1
100
2
235
3
50
4
133
5
50
6
25
7
140
8
*
9
*
10
*
11
*
12
*
13
*
14
*
15
*
16
*
17
*
18
*
19
*
20
*
21
*
22
*
23
*
24
*
25
*
26
*
27
*
28
*
Interviewer Mean 104.714
Block Total
1331
B
125
*
*
*
*
*
*
41
180
65
50
100
170
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
104.429
1497
C
*
95
*
*
*
*
*
50
*
*
*
*
*
75
25
132
145
100
*
*
*
*
*
*
*
*
*
*
88.857
1346
D
*
*
30
*
*
*
*
*
195
*
*
*
*
95
*
*
*
*
99
100
50
35
*
*
*
*
*
*
86.286
1344
E
*
*
*
*
30
*
*
*
*
75
*
*
*
*
*
50
*
*
235
*
*
*
150
135
70
*
*
*
106.429
1542
F
*
*
*
80
*
*
*
*
*
*
100
*
*
*
55
*
*
*
*
100
*
*
163
*
*
50
75
*
89.000
1246
G
*
*
*
*
*
88
*
*
*
*
*
96
*
*
*
*
*
152
*
*
50
*
*
150
*
100
*
100
105.143
1285
H
*
*
*
*
*
*
150
*
*
*
*
*
150
*
*
*
96
*
*
*
*
50
*
*
138
*
65
89
105.429
1473
Dealer Mean
112.5
165.0
40.0
106.5
40.0
56.5
145.0
45.5
187.5
70.0
75.0
98.0
160.0
85.0
40.0
91.0
120.5
126.0
167.0
100.0
50.0
42.5
156.5
142.5
104.0
75.0
70.0
94.5
98.786
Intra-Block Analysis
• Method 1: Comparing Models Based on Residual
Sum of Squares (After Fitting Least Squares)
 Full Model Contains Treatment and Block Effects
 Reduced Model Contains Only Block Effects
 H0: No Treatment Effects after Controlling for Block Effects
Full Model: yij     i   j   ij i  1,..., g j  1,...,b (Not e: Not all pairs i, j )

 ^ ^ ^ 
SSEF    yij      i   j  


i 1 j 1 
Reduced Model: yij     j   ij
g
b

 ^ ^ 
SSER    yij      j  


i 1 j 1 
g
b
T est St at ist ic: Fobs
2
dfF  N  (1  ( g  1)  (b  1))  rg  g  b  1
i  1,..., g
j  1,...,b
2
dfR  N  (1  (b  1))  rg  b
 SSER  SSEF 
 SSER  SSEF 
 df  df 


g 1
R
F






 SSEF 


SSEF
 df 
 g (r  1)  (b  1) 


 F 
H0
~
Fg 1, g ( r 1) (b 1)
Least Squares Estimation (I) – Fixed Blocks
Model: nij yij  nij    i   j   ij 
1 if T rti in Blk j
i  1,..., g ; j  1,...,b nij  
0 otherwise
Q   n    nij  yij     i   j 
g
g
b
i 1 j 1
2
ij ij
b
2
i 1 j 1
g
b
set
Q
 2 nij  yij     i   j   0 

i 1 j 1
^
g
b
 n
i 1 j 1
ij
g
^
k
^
^
yij  N   r   i  k   j
i 1
j 1
^
 y   N     y  
b
set
Q
 2 nij  yij     i   j   0 
 i
j 1
a
set
Q
 2 nij  yij     i   j   0 
 j
i 1
b
^
n
j 1
ij
g
n
i 1
ij
^
g
^
^
yij  y j  k    nij  i  k  j
^
^ 
1
1 g
 kyi  kr   kr  i  k  nij  y j     nij  i 
k i 1
j 1
k

k
^
i  1,..., g
j 1
^
^
1
1 g
  j  y j     nij  i
k
k i 1
^
b
yij  yi  r   r  i   nij  j
^
^
^
i 1
j  1,...,b
Least Squares Estimation (II)
^
^ 
1
1 g
kyi  kr   kr  i  k  nij  y j     ni ' j  i ' 
k i '1
j 1
k

^
b
^
g
^
^ 

Consider the Last Term:  nij  y j  k    ni ' j  i ' 
j 1
i '1


b
b
1)
n
ij
j 1
y j  Bi
b
2)
^
^
^
n k   k  n
g
b
^
 k  r  kr 
i
ij
j 1
3)
 Sum of Block Totals that Trt i appears in
b
^
g
b
^
^
 nij ni ' j  i '   n  i   nij ni ' j  i '
j 1 i '1
j 1
g
Notes: (a)

i 1
b
g
^
^
2
ij
j 1 i '1
i ' i
g
^
i
^
 0    i   i '
i '1
i ' i
b
^
g
^
b
1 if Trts i, i ' in Blk j
nij ni ' j  
0 otherwise
g
(b):  nij ni ' j  l
i '1
i ' i
^
g
^
^
  nij ni ' j  i '   nij  i    i '  nij ni ' j  r  i  l   i '   r  l   i
j 1 i '1
j 1
i '1
i ' i
j 1
i '1
i ' i
Least Squares Estimation (III)
kyi  kr   kr  i  Bi  kr   r  l  i
^
^
^
^
 kyi  Bi   i kr  r  l    i r k  1  l  
^
^
 i l g  1  l    i lg
^
^
kyi  Bi kQi
i 

lg
lg
1
Qi  yi  Bi
k
^
Analysis of Variance (Fixed or Random Blocks)
^
^
^ Full 

Full Model: SSEF   nij  yij     i   j 
i 1 j 1


g
b
2
^ Full

j
^
1 g
 y  j  y    nij  i
k i 1
2
^
^ Reduced 
^ Reduced


Reduced Model: SSER   nij  yij     j
j
 y  j  y 
i 1 j 1


Difference: SSER  SSEF  SST rt sAdjusted for Blocks 
g
b
g
 SST(Adjusted) 
k  Qi2
i 1
lg
Source
df
Blks (Unadj)
b-1
Qi  yi 
1
Bi
k
1
Bi  Sum of Block MeanscontainingT rti
k
SS
MS
b
 ^ Reduced 

k
 j

j 1 

2
SSB/(b-1)
g
Trts (Adj)
g-1
Error
gr-(b-1)-(g-1)-1
Total
gr-1
k  Qi2 lg
SST(Adj)/(g-1)
i 1
^
^
^ Full 

nij  yij     i   j 

i 1 j 1


g
b
 n y
g
b
i 1
j 1
ij
ij
 y 

2
2
SSE/(g(r-1)-(b-1))
ANOVA F-Test for Treatment Effects
H 0 : 1  ...   g  0 H A : Not all  i  0
TS : Fobs
MST (Adj)

MSE
H0
~
Fg 1, g ( r 1)(b 1)
Note: This test can be obtained directly from the
Sequential (Type I) Sum of Squares When Block is entered
first, followed by Treatment
Interviewer Example
mu
Interviewer
A
B
C
D
E
F
G
H
ANOVA
Source
Blocks (Unadj)
Trts(Adj)
Error
Total
y(i*)
733
731
622
604
745
623
736
738
B(i)
1331
1497
1346
1344
1542
1246
1285
1473
df
27
7
21
55
Q(i)
67.5
-17.5
-51
-68
-26
0
93.5
1.5
SS
106244.43
5377.00
30030.00
141651.43
alpha(i)
16.875
-4.375
-12.750
-17.000
-6.500
0.000
23.375
0.375
Sum
MS
3934.98
768.14
1430.00
98.786
SST(Adj) SST(Unadj)
1139.063 246.036
76.563
222.893
650.250
690.036
1156.000 1093.750
169.000
408.893
0.000
670.321
2185.563 282.893
0.563
308.893
5377.000 3923.714
F
P-Value
0.5372
0.7967
Dealer
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Beta(j)Red
13.714
66.214
-58.786
7.714
-58.786
-42.286
46.214
-53.286
88.714
-28.786
-23.786
-0.786
61.214
-13.786
-58.786
-7.786
21.714
27.214
68.214
1.214
-48.786
-56.286
57.714
43.714
5.214
-23.786
-28.786
-4.286
Beta(j)Full
7.464
64.152
-58.723
-0.723
-63.973
-62.411
37.589
-44.723
99.402
-23.348
-21.598
-10.286
63.214
1.089
-52.411
1.839
27.902
21.902
79.964
9.714
-51.973
-47.973
60.964
35.277
8.277
-35.473
-28.973
-16.161
Sum
SSB(Unadj)
376.163
8768.663
6911.520
119.020
6911.520
3576.163
4271.520
5678.735
15740.449
1657.235
1131.520
1.235
7494.378
380.092
6911.520
121.235
943.020
1481.235
9306.378
2.949
4760.092
6336.163
6661.878
3821.878
54.378
1131.520
1657.235
36.735
106244.429
SSE(Full)
1069.531
6091.320
96.258
652.508
5.695
1596.125
351.125
150.945
381.570
73.508
1040.820
504.031
306.281
294.031
148.781
3894.031
1929.758
126.008
7875.125
144.500
815.070
2.820
21.125
110.633
1868.133
354.445
53.820
72.000
30030.000
Car Pricing Example
The GLM Procedure
Dependent Variable: price
Source
Model
Error
Corrected Total
DF
34
21
55
Sum of
Squares
111621.4286
30030.0000
141651.4286
Source
DF
Type I SS
Mean Square
F Value
Pr > F
dlr_blk
intrvw_trt
27
7
106244.4286
5377.0000
3934.9788
768.1429
2.75
0.54
0.0101
0.7967
Source
DF
Type III SS
Mean Square
F Value
Pr > F
dlr_blk
intrvw_trt
27
7
107697.7143
5377.0000
3988.8042
768.1429
2.79
0.54
0.0093
0.7967
Mean Square
3282.9832
1430.0000
F Value
2.30
Pr > F
0.0241
Recall: Treatments: g = 8 Interviewers, r = 7 dealers/interviewer
Blocks: b = 28 Dealers, k = 2 interviewers/dealer
l = 1 common dealer per pair of interviewers
Comparing Pairs of Trt Means & Contrasts
• Variance of estimated treatment means depends on
whether blocks are treated as Fixed or Random
• Variance of difference between two means DOES NOT!
• Algebra to derive these is tedious, but workable. Results
are given here:
1
k
1
i   i 
y 
yi  
Bi
rg
lg
lg
^
^
^
2
^ ^ 
 ^ ^  2k
V   i   j   V  i   j  
lg




g
2kMSE
  g (r  1)  (b  1) C   
lg
2
 Bonferroni' s Bij  t / 2C ,
^
 ^ ^  2kMSE
 V  i   j  
lg


^
g
^
For generalContrast:    wi  i
i 1
2
^  k a 2
V   
wi

  lg i 1
Car Pricing Example
g  8 r  7 k  2 l  1 MSE  1430
^
^
^
i   i 
1
k
1
1
2
1
y  
yi  
Bi 
y  yi  Bi
rg
lg
lg
56
8
8
2
4 2  2
 ^ ^  2k
V  i   j  



lg
8
2


^
 ^ ^  2kMSE 1430
V  i   j  

 715
lg
2


 Bonferroni' s Bij  t / 2C ,
2kMSE
 t.05 / 56 , 21 715  3.58(26.7)  95.73
lg
  g (r  1)  (b  1)  8(7  1)  (28  1)  48  27  21
C
g ( g  1) 8(7)

 28
2
2
Car Pricing Example – Adjusted Means
The GLM Procedure
Least Squares Means
intrvw_
trt
1
2
3
4
5
6
7
8
price LSMEAN
115.660714
94.410714
86.035714
81.785714
92.285714
98.785714
122.160714
99.160714
Note: The largest difference (122.2 - 81.8 = 40.4) is not
even close to the Bonferroni Minimum significant
Difference = 95.7
Recovery of Inter-block Information
• Can be useful when Blocks are Random
• Not always worth the effort
• Step 1: Obtain Estimated Contrast and Variance
based on Intra-block analysis
• Step 2: Obtain Inter-block estimate of contrast and
its variance
• Step 3: Combine the intra- and inter-block estimates,
with weights inversely proportional to their variances
Inter-block Estimate
g


y j   nij yij  k    nij i  k  j   nij  ij 
i 1
i 1
i 1


g
g
1 if Trt i occurs in Blk j
nij  
0 otherwise
 j ~ N  0, k 2 2  k 2 
g
 k    nij i   j
i 1
This is a multiple regression with g predictors which leads to estimates:
~
~
  y 
~
g
B  rk 
i  i
r l
~
~
   wi  i
i 1
E  MSE    2
g
Bi   nij y j
i 1
2 2
2
 ~  k    k
V   
r l
 
g
w
i 1
2
i
Ng 2
E  MS  Blks|Trts     2  
 
 b 1 
 b 1 
     MS  Blks|Trts   MSE  

N

g


^ 2
Combined Estimate


 1 ^
1 ~

 ^ 
~
 
V   
V   
  
  
 


 1
1 
 ^ 

~
V    V    
  
  
where :
^
g
^
   wi  i
i 1
~
g
~
   wi  i
i 1
^ ~
V   V   
1
V 
 ^   ~


 
 
V


V
 
 
 1

1
 
 
 ^ 

~
V    V    
  
  

2
^  k a 2
V   
wi

  lg i 1

V 

~



k 2 2  k 2
r l
^
i 
g
~
w
i 1
 b 1 

   MS Blks | T rts  MSE
Ng
^ 2
k
1
yi  
Bi
lg
lg
i 
2
i
^ 2
Bi  rk y 
r l
  MSE
Interviewer Example
ANOVA
Source
Trts(Unadj)
Blocks(Adj)
Error
Total
^
g
df
7
27
21
55
^
   wi  i
i 1
^
V 

SS
MS
3923.714286 560.5306
107697.71 3988.804
30030.00
1430
141651.43
2
a
  k
wi2


 l g i 1
Interviewer
A
B
C
D
E
F
G
H
alpha-hat
16.875
-4.375
-12.750
-17.000
-6.500
0.000
23.375
0.375
mu+alpha-hat
115.661
94.411
86.036
81.786
92.286
98.786
122.161
99.161
alpha-tilda
-8.667
19.000
-6.167
-6.500
26.500
-22.833
-16.333
15.000
alpha-bar
11.767
0.300
-11.433
-14.900
0.100
-4.567
15.433
3.300
mu+alpha-bar
110.552
99.086
87.352
83.886
98.886
94.219
114.219
102.086
 ^  1430(2)
V   
wi2  357.5 wi2

1(8) i 1
 
i 1
a
^
a
2 2
2 g
 ~  k    k
   wi  i V    
wi2

r l
 
i 1
i 1
 28  1 
2
2
3988.8

1430



  2(1430) g
g
^
~
56  8 
 

2
V   
wi  1436.2 wi2

7 1
 
i 1
i 1
~
g
~
1 ~
 1 ^



 357.5
^
~
1436.2 
 
 0.80   0.20 
1 
 1
 357.5  1436.2 
g
g

2 
2
357.5
w
1436.2
w


i 
i 

g
i 1
i 1



2
V  

286.25
w

i
g
g

i 1
2 
2
357.5
w

1436.2
w


i  
i 

i 1
i 1

 

 