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Chapter 4
Randomised Block Experiments
4
...
In the completely randomised design, the experimental units are assumed to
be homogeneous in their response to treatment
...
In this case, it is more efficient to use a blocked
design in which the experimental units are subdivided into groups called blocks
...
Treatments can then be randomly
allocated separately within each block
...
All comparisons are made within a block and not between blocks
...
2
Randomised Complete Block Experiment
In a randomised complete block experiment, the number of experimental units per
block is equal to the number of treatments, so that each treatment occurs once in each
block
...
The
measurements for the fifteen men are as follows:
Diet
A
B
C
33
1
0
...
67
0
...
86
0
...
21
Block
3
0
...
81
0
...
40
1
...
75
5
1
...
41
0
...
Each block contains k
experimental units to which k treatments are randomly allocated
...
b
y13
...
...
y 21
y 22
y 23
...
y kb
1
Model
y ij i j ij
,
Mean
Variance
i = 1, 2,
...
The model assumes that the treatment and the block effects are additive
...
We say that there is no
interaction between the treatment and block effects
...
f
b-1
SS
SS(blocks)
MS=SS/df
MS(blocks)
Between Treatments
Within Groups
Error
Total(about mean)
k-1
(b-1)(k1)
bk–1
(=N-1)
SS(treatments)
SS(Error)
MS(treatments)
MS(Error)
Test for differences due to treatments
34
MS Ratio
MS(Blocks)
MS(Error)
Ho : 1 2
...
F
MS(Treatments)
MS(Error) > F( k 1),(b 1)( k 1),
Reject if
Otherwise there is no evidence to reject
...
k
H1: not all i ' s are equal
...
36
The following table will be produced
...
Corrected Model
2
...
457
189
...
000
Intercept
10
...
685
4424
...
000
DIET
1
...
660
273
...
000
AGE
1
...
355
146
...
000
Error
1
...
415E-03
Total
13
...
759
14
a
...
993 (Adjusted R Squared =
...
k
H1: not all i ' s are equal
...
350, p= 0
...
001, reject at the 0
...
There is very strong evidence
that there is a difference, on average, in the total lipid level due to the three diets
...
k
H1: not all i ' s are equal
...
890, p = 0
...
001, reject at the 0
...
There is very strong evidence
that there is a difference, on average, between the ages
...
4
...
Example
An experiment is conducted to investigate the effects of different types of background
music on the productivity of bank tellers
...
The productivity of the workers was
measured
...
Model for a a Latin square
y ij i j k ijk
, i = 1, 2,
...
ANOVA table
Source
Between Rows
d
...
k
H1: not all i ' s are equal
...
Tests of Between-Subjects Effects
Dependent Variable: Producity of bank tellers
Type III Sum
Source
of Squares a
df
Mean Square
Corrected Model
1115
...
973
Intercept
9840
...
640
MUSIC
494
...
540
WEEK
261
...
340
DAY
360
...
040
Error
627
...
307
Total
11584
...
360
24
a
...
640 (Adjusted R Squared =
...
777
188
...
362
1
...
721
Sig
...
166
...
112
...
210
Practical 4: Randomised block design
A
...
Doughnuts were cooked on six different days using four
different cooking oils each day
...
Day
1
2
3
4
5
6
1
...
3
...
5
...
7
...
40
A
164
177
168
156
172
195
Cooking oil
B
C
172
177
197
184
167
187
161
169
180
179
190
197
D
178
196
177
181
184
191
Enter the data into one named column
...
Use Analyze/General Linear Model/Univariate and the option Model to open
the GLM-General Factorial Model dialog box
...
Select Main Effects in the Build term(s) box
...
Click OK
...
Variance-stabilising Transformation
Data set 1
Insects are caught in traps at three different sites and using four different types of trap
...
Type of trap
A
B
C
D
a)
b)
c)
d)
e)
f)
Site 2
47
62
87
43
Site 3
105
170
165
86
Enter the data into the first column and create two further variables to contain the
levels of the factors, Trap type and Site
...
1
...
Click on the response variable and then on to move it to the Dependent
List box
...
Click on the variables, trap and site and then on to move them to the
Independent List box
...
Click on Options
...
Check that Mean and Standard deviation are selected in the Cell
Statistics box
...
Click Continue
...
Click OK
...
Now make a log transformation on the count data in column 1 and store the
transformed data in another column using the following procedure
...
Click on Transform
Compute
2
...
3
...
4
...
5
...
Repeat steps b) and c)
...
Data set 2
41
Site 1
11
13
15
7
In an experiment on the effectiveness of insecticides, six different concentrations of weed
killer were used on four different sites and the resulting number of weeds counted
...
Investigate the relationship between the mean, the range and the standard
deviation by site and by treatment
...
Now use the General Linear Model/Univariate procedure to analyse the
transformed data
...
ii)
Comment on the ANOVA results using the untransformed data
...
iv)
Compare the results using the untransformed and transformed data
...