The Randomized Block Design
• Suppose a researcher is interested in how
several treatments affect a continuous
response variable (Y).
• The treatments may be the levels of a single
factor or they may be the combinations of
levels of several factors.
• Suppose we have available to us a total of
N = nt experimental units to which we are
going to apply the different treatments.
The Completely Randomized (CR) design
randomly divides the experimental units into t
groups of size n and randomly assigns a
treatment to each group.
The Randomized Block Design
• divides the group of experimental units into
n homogeneous groups of size t.
• These homogeneous groups are called
blocks.
• The treatments are then randomly assigned
to the experimental units in each block one treatment to a unit in each block.
Example 1:
• Suppose we are interested in how weight gain (Y)
in rats is affected by Source of protein (Beef,
Cereal, and Pork) and by Level of Protein (High
or Low).
• There are a total of t = 32 = 6 treatment
combinations of the two factors (Beef -High
Protein, Cereal-High Protein, Pork-High Protein,
Beef -Low Protein, Cereal-Low Protein, and
Pork-Low Protein) .
• Suppose we have available to us a total of N = 60
experimental rats to which we are going to apply the
different diets based on the t = 6 treatment
combinations.
• Prior to the experimentation the rats were divided
into n = 10 homogeneous groups of size 6.
• The grouping was based on factors that had
previously been ignored (Example - Initial weight
size, appetite size etc.)
• Within each of the 10 blocks a rat is randomly
assigned a treatment combination (diet).
• The weight gain after a fixed period is
measured for each of the test animals and is
tabulated on the next slide:
Randomized Block Design
Block
1
107
(1)
96
(2)
112
(3)
83
(4)
87
(5)
90
(6)
Block
6
128
(1)
89
(2)
104
(3)
85
(4)
84
(5)
89
(6)
2
102
(1)
72
(2)
100
(3)
82
(4)
70
(5)
94
(6)
7
56
(1)
70
(2)
72
(3)
64
(4)
62
(5)
63
(6)
3
102
(1)
76
(2)
102
(3)
85
(4)
95
(5)
86
(6)
8
97
(1)
91
(2)
92
(3)
80
(4)
72
(5)
82
(6)
4
93
(1)
70
(2)
93
(3)
63
(4)
71
(5)
63
(6)
9
80
(1)
63
(2)
87
(3)
82
(4)
81
(5)
63
(6)
5
111
(1)
79
(2)
101
(3)
72
(4)
75
(5)
81
(6)
10
103
(1)
102
(2)
112
(3)
83
(4)
93
(5)
81
(6)
Example 2:
• The following experiment is interested in
comparing the effect four different
chemicals (A, B, C and D) in producing
water resistance (y) in textiles.
• A strip of material, randomly selected from
each bolt, is cut into four pieces (samples)
the pieces are randomly assigned to receive
one of the four chemical treatments.
• This process is replicated three times
producing a Randomized Block (RB) design.
• Moisture resistance (y) were measured for
each of the samples. (Low readings indicate
low moisture penetration).
• The data is given in the diagram and table on
the next slide.
Diagram: Blocks (Bolt Samples)
9.9
10.1
11.4
12.1
C
A
B
D
13.4
12.9
12.2
12.3
D
B
A
C
12.7
12.9
11.4
11.9
B
D
C
A
Table
Chemical
A
B
C
D
Blocks (Bolt Samples)
1
2
3
10.1
12.2
11.9
11.4
12.9
12.7
9.9
12.3
11.4
12.1
13.4
12.9
The Model for a randomized Block Experiment
yij     i   j   ij
i = 1,2,…, t
j = 1,2,…, b
yij = the observation in the jth block receiving the
ith treatment
yij     i   j   ij
 = overall mean
i = the effect of the ith treatment
j = the effect of the jth Block
ij = random error
The Anova Table for a randomized Block Experiment
Source
S.S.
d.f.
M.S.
F
Treat
Block
Error
SST
SSB
SSE
t-1
n-1
(t-1)(b-1)
MST
MSB
MSE
MST /MSE
MSB /MSE
p-value
• A randomized block experiment is assumed to be
a two-factor experiment.
• The factors are blocks and treatments.
• The is one observation per cell. It is assumed that
there is no interaction between blocks and
treatments.
• The degrees of freedom for the interaction is used
to estimate error.
The Anova Table for Diet Experiment
Source
Block
Diet
ERROR
S.S
5992.4167
4572.8833
3147.2833
d.f.
9
5
45
M.S.
F
665.82407
9.52
914.57667 13.076659
69.93963
p-value
0.00000
0.00000
The Anova Table forTextile Experiment
SOURCE
Blocks
Chem
ERROR
SUM OF SQUARES
7.17167
5.20000
0.53500
D.F.
2
3
6
MEAN SQUARE
3.5858
1.7333
0.0892
F
40.21
19.44
TAIL PROB.
0.0003
0.0017
• If the treatments are defined in terms
of two or more factors, the treatment
Sum of Squares can be split
(partitioned) into:
– Main Effects
– Interactions
The Anova Table for Diet Experiment
terms for the main effects and interactions between
Level of Protein and Source of Protein
Source
Block
Diet
ERROR
Source
Block
Source
Level
SL
ERROR
S.S
5992.4167
4572.8833
3147.2833
S.S
5992.4167
882.23333
2680.0167
1010.6333
3147.2833
d.f.
9
5
45
d.f.
9
2
1
2
45
M.S.
F
665.82407
9.52
914.57667 13.076659
69.93963
M.S.
665.82407
441.11667
2680.0167
505.31667
69.93963
F
9.52
6.31
38.32
7.23
p-value
0.00000
0.00000
p-value
0.00000
0.00380
0.00000
0.00190
Repeated Measures Designs
In a Repeated Measures Design
We have experimental units that
• may be grouped according to one or several
factors (the grouping factors)
Then on each experimental unit we have
• not a single measurement but a group of
measurements (the repeated measures)
• The repeated measures may be taken at
combinations of levels of one or several
factors (The repeated measures factors)
Example
In the following study the experimenter was
interested in how the level of a certain enzyme
changed in cardiac patients after open heart
surgery.
The enzyme was measured
• immediately after surgery (Day 0),
• one day (Day 1),
• two days (Day 2) and
• one week (Day 7) after surgery
for n = 15 cardiac surgical patients.
The data is given in the table below.
Table: The enzyme levels -immediately after surgery (Day
0), one day (Day 1),two days (Day 2) and one week (Day 7)
after surgery
Subject
1
2
3
4
5
6
7
8
Day 0 Day 1 Day 2 Day 7
108
63
45
42
112
75
56
52
114
75
51
46
129
87
69
69
115
71
52
54
122
80
68
68
105
71
52
54
117
77
54
61
Subject
9
10
11
12
13
14
15
Day 0 Day 1 Day 2 Day 7
106
65
49
49
110
70
46
47
120
85
60
62
118
78
51
56
110
65
46
47
132
92
73
63
127
90
73
68
• The subjects are not grouped (single group).
• There is one repeated measures factor -Time
– with levels
–
–
–
–
Day 0,
Day 1,
Day 2,
Day 7
• This design is the same as a randomized
block design with
– Blocks = subjects
The Anova Table for Enzyme Experiment
Source
Subject
Day
ERROR
SS
4221.100
36282.267
390.233
df
MS
14
301.507
3 12094.089
42
9.291
F
32.45
1301.66
p-value
0.0000
0.0000
The Subject Source of variability is modelling the
variability between subjects
The ERROR Source of variability is modelling the
variability within subjects
Example :
(Repeated Measures Design - Grouping Factor)
• In the following study, similar to example 3,
the experimenter was interested in how the
level of a certain enzyme changed in cardiac
patients after open heart surgery.
• In addition the experimenter was interested in
how two drug treatments (A and B) would
also effect the level of the enzyme.
• The 24 patients were randomly divided into three
groups of n= 8 patients.
• The first group of patients were left untreated as a
control group while
• the second and third group were given drug
treatments A and B respectively.
• Again the enzyme was measured immediately after
surgery (Day 0), one day (Day 1), two days (Day 2)
and one week (Day 7) after surgery for each of the
cardiac surgical patients in the study.
Table: The enzyme levels - immediately after surgery (Day 0),
one day (Day 1),two days (Day 2) and one week (Day 7)
after surgery for three treatment groups (control, Drug A,
Drug B)
0
122
112
129
115
126
118
115
112
Control
Day
1
2
87
68
75
55
80
66
71
54
89
70
81
62
73
56
67
53
7
58
48
64
52
71
60
49
44
0
93
78
109
104
108
116
108
110
Group
Drug A
Day
1
2
56
36
51
33
73
58
75
57
71
57
76
58
64
54
80
63
7
37
34
49
60
65
58
47
62
0
86
100
122
101
112
106
90
110
Drug B
Day
1
2
46
30
67
50
97
80
58
45
78
67
74
54
59
43
76
64
7
31
50
72
43
66
54
38
58
• The subjects are grouped by treatment
– control,
– Drug A,
– Drug B
• There is one repeated measures factor -Time
– with levels
–
–
–
–
Day 0,
Day 1,
Day 2,
Day 7
The Anova Table
Source
Drug
Error1
Time
Time x Drug
Error2
SS
1745.396
df
2
MS
872.698
10287.844
47067.031
357.688
21
3
6
489.897
15689.010
59.615
668.031
63
10.604
F
1.78
p-value
0.1929
1479.58
5.62
0.0000
0.0001
There are two sources of Error in a repeated
measures design:
The between subject error – Error1 and
the within subject error – Error2
Tables of means
Drug
Control
A
B
Overall
Day 0
118.63
103.25
103.38
108.42
Day 1
77.88
68.25
69.38
71.83
Day 2
60.50
52.00
54.13
55.54
Day 7
55.75
51.50
51.50
52.92
Overall
78.19
68.75
69.59
72.18
120
Time Profiles of Enzyme Levels
100
Control
Enzyme Level
Drug A
Drug B
80
60
40
0
1
2
3
Day
4
5
6
7