Randomized Block Design
Block Design: A block is a group of experimental units or subjects that are known before the
experiment to be similar in some way that is expected to affect the response to the treatments. In a
block design, the random assignment of units to treatments is carried out separately within each block.
Example It is common in nutrition studies to compare diets by feeding them to newly weaned male
rats and measuring the weight gained by the rats over a 28-day period. The initial weight of the rats is
an extraneous factor that is especially important as an influence on weight gain. Rather than randomly
assign 10 rats to each diet, we first divide the animals into 10 groups of 3 rats each based on weight.
Three lightest rats form the first group, the next lightest three form the second group, and so on, with
the tenth group consisting of the three heaviest rats. These groups of rats of similar weight are called
blocks. Now we randomly assign one rat from each block to each diet. This design is used to screen out
the effects of the extraneous factor, initial weight.
Nonrandom Assignment
of Units to Homogeneous Blocks
Block 1
Random
Allocation
Diet A
Diet B
…….
Block 2
Random
Allocation
Diet C
Diet A
Diet B
Block 10
Random
Allocation
Diet C
Diet A
Diet B
Diet C
Example An accounting firm, prior to introducing in the firm widespread training in statistical
sampling for auditing, tested three training methods: (1) study at home with programmed training
materials, (2) training sessions at local offices conducted by local staff, (3) training session in Chicago
conducted by national staff.Thirty auditors were grouped into 10 blocks of three, according to time
elasped since college graduation, and the auditors in each block were randomly assigned to the three
training methods. At the end of the training, each auditor was asked to analyze a complex case
involving statistical applications: a proficiency measure based on this analysis was obtained for each
auditor. The results were (block 1 consists of auditors graduated most recently, block 10 consists of
those graduated most distantly):
Block
1
2
3
4
5
6
7
8
9
10
Training Method
2
81
78
76
77
71
75
72
74
73
69
1
73
76
75
74
76
73
68
64
65
62
3
92
89
87
90
88
86
88
82
81
78
Normal Probability Plot
(response is score)
99
95
90
Percent
80
70
60
50
40
30
20
10
5
1
-5.0
-2.5
0.0
Residual
2.5
5.0
Versus Fits
(response is score)
5.0
Residual
2.5
0.0
-2.5
-5.0
60
65
70
75
80
Fitted Value
85
90
Source
Method
Block
Error
Total
DF
2
9
18
29
Seq SS
1295.00
433.37
112.33
1840.70
Adj SS
1295.00
433.37
112.33
Adj MS
647.50
48.15
6.24
F
103.75
7.72
P
0.000
0.000
Tukey 95.0% Simultaneous Confidence Intervals
Response Variable score
All Pairwise Comparisons among Levels of method
method = 1 subtracted from:
method
2
3
Lower
1.148
12.648
method = 2
method
3
Center
4.000
15.500
Upper
6.852
18.352
--------+---------+---------+-------(-----*-----)
(-----*-----)
--------+---------+---------+-------5.0
10.0
15.0
subtracted from:
Lower
8.648
Center
11.50
Upper
14.35
--------+---------+---------+-------(-----*-----)
--------+---------+---------+-------5.0
10.0
15.0
Tukey Simultaneous Tests
Response Variable score
All Pairwise Comparisons among Levels of method
method = 1 subtracted from:
method
2
3
Difference
of Means
4.000
15.500
method = 2
method
3
SE of
Difference
1.117
1.117
T-Value
3.580
13.874
Adjusted
P-Value
0.0058
0.0000
T-Value
10.29
Adjusted
P-Value
0.0000
subtracted from:
Difference
of Means
11.50
SE of
Difference
1.117