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Chapter 2. Randomized Complete Block Design
2.1 Randomized Complete Block Design
2.1.1 Examples
When examining the effect of a factor, it is often helpful to remove the effect of excess
variation through the use of blocking. A blocking variable is one that may affect the
variation of the response, but is unrelated to the primary hypothesis of interest. The
desired result is to have homogeneous experimental units within each block so that when
the blocking effect is removed (through modeling), all individuals can be considered
homogeneous before receiving the treatments. The term randomized complete block
design usually refers to a design where there is a single fixed factor of interest and a
single random blocking effect. The number of experimental units in each block is such
that within each block each of the treatments can be randomly assigned the same number
of times. Some examples of a randomized complete block designs follow.
Examples of Randomized Complete Block Design
Snakebite Venom
To compare the immune response of mice to the venom of four poisonous snakes, venom
is taken from adult male coral, copperhead, sidewinder, and pit viper snakes. One mouse
from each of seven litters are randomly assigned to the four snakes. Each receives minute
amounts of the venom of the corresponding snake by injection. The increase in antibody
activity as measured from a blood sample is the response. The four treatments that
constitute the fixed factor are the four snakes of interest. The seven litters represent all
litters and thus make up a random effect. Litter is a blocking effect since it is not of
primary interest to determine the variation in antibody activity between litters, but instead
to remove the added variation that comes with differing litters.
Recyclable Scrap Metal
Before beginning a full-scale operation to promote recycling of metals in a large county,
a recycling agency conducts a study to compare recycling opportunities in the six largest
cities of that county. The primary question of interest is which of the cities should be the
major focus of the agency. To answer this question, the agency wishes to compare the
amount of scrap metal wasted by individuals in each of the cities. Ten days of the year
are randomly selected for scrap metal examination. On each of the ten days, one
randomly chosen garbage truck load (of equal size) from each city is scoured for
recyclable scrap metal. The material is then weighed for each load. Only the six cities are
of interest. These make up the fixed factor. The day of collection is a random effect since
it is a sample of all possible days. Further, day of collection is a blocking effect since the
variation among days is not of primary interest.
Rice fertilizer
A rice farmer has a choice among four fertilizers. To compare the fertilizers he randomly
selects four rows of his field which have been planted with the same seed. The plants on a
particular row can be expected to have identical environmental conditions, i.e., sunlight,
water, etc. Each row is divided into four segments. The four fertilizers are randomly
assigned to the four segments of each row.
Table 2.1: Randomized complete block design set up for rice fertilizer example.
Segment
Row 2 F4: 16.27 F3: 15.43 F1: 13.54 F2: 14.
Row 3 F4: 13.60 F1:11.53 F3: 13.06 F2: 12.
Row 4 F3: 16.27 F2: 13.43 F4: 16.84 F1: 14.
etc. See the video RCB for the setup and discussion of the analyses.
2.1.4 Hypotheses Testing
In this example, we are interested in testing if there is an effect due to Fertilizer. Thus the
null hypothesis is: o 1 2 3 4
H : α α α α (^). The alternative hypothesis is that at least 1 pair
of the α'ss are not equal. We could test:
2
o
H : 0
B
σ . However, this factor is not important
other than reducing the overall variability.
The F ratio and significance value for testing o 1 2 3 4
H : α α α α (^) are 17.8 and .000.
Thus, we will reject the null hypothesis and say that there are differences in the
Fertilizers. The pairwise comparisons indicate that all of the populations unequal. The
best estimates of
2
σ and^
2
B
σ (^) are .2907 and 1.403; where as the true values were .25 and 1.
2.1.5 Simulation
Using the RCB.sps file, you can increase the number of blocks and see what effect that
has on your estimates of
2
σ and^
2
B
σ (^). Then increase the number of treatments and see
what effect that has. Keeping the number of blocks and treatments at the original level,
change the variance of
2
σ and
2
B
σ (^) to 1 and 3 and see what effect occurs. Try different
combinations.
2.1.6 Matrix Notation
In this example, the matrix notation is:
16x1 16x1 1x1 16x4 4x1 16x4 4x1 16x
length =J μ + X α + Z b + ε
where:
length J | X | | Z |
2 2
b
2 2 2
b b
2 2 2 2
b b b
2 2 2 2 2
b b b b
2 2 2
b b
0 0 0 +
ε
ε
ε
ε
ε
σ σ
σ σ σ
σ σ σ σ
σ σ σ σ σ
σ σ σ
2 2 2
b b
2 2 2 2
b b b
0 0 0 0 +
0 0 0 0 +
0 0 0
ε
ε
σ σ σ
σ σ σ σ
2 2 2 2 2
b b b b
2 2
b
0 0 +
0 0 0 0 0 0 0 0 +
0 0
ε
ε
σ σ σ σ σ
σ σ
2 2 2
b b
0 0 0 0 0 0 +
0 0 0 0 0 0 0 0
ε
σ σ σ
2 2 2 2
b b b
2 2 2 2 2
b b b b
0 0 0 0 0 0 0 0 +
0
ε
ε
σ σ σ σ
σ σ σ σ σ
2 2
b
0 0 0 0 0 0 0 0 0 0 0 +
0 0 0 0
ε
σ σ
2 2 2
b b
0 0 0 0 0 0 0 0 +
0 0 0 0 0 0 0
ε
σ σ σ
2 2 2 2
b b b
0 0 0 0 0 +
0 0 0 0 0 0 0 0
ε
σ σ σ σ
2 2 2 2 2
b b b b
0 0 0 0 + ε
σ σ σ σ σ
2.2 Randomized Complete Block Design with Subsampling
2.2.1 Examples
Subsampling in the randomized complete block design occurs when there is more than
one individual in each treatment/block combination.
Examples of Randomized Complete Block Design with Subsampling
Internet Advertising
An internet advertising company wishes to compare worldwide internet usage time for
four age groups: < 20 years, 20-40 years, 40-60 years, and > 60 years. There are many
factors which may also influence internet usage time, but in this case the only other easily
selected information about the individuals surveyed is the country of use. The company
selects five countries that they expect will represent most other countries well. A question
about internet usage time is sent to twenty individuals within each age category of each
country. The average daily internet usage time of each individual is the response. The
factor of interest is age. Since the only age levels of interest are the four age groups
considered, this factor is fixed. Additional variation is removed by considering country of
use. This is the blocking effect. Because the five countries represent all countries, it is a
random blocking effect. Individuals within each age/country combination are assumed to
be homogeneous. The result is a randomized complete block design with subsampling
since there are 20 individuals in each age/country combination.
Rice fertilizer
Consider the rice fertilizer example of Section 2.1.1. Suppose that instead of sampling a
single plant from each fertilizer/row segment, three plants are sampled from each
segment. The three samples in each segment are subsamples.
2.2.2. Model
The model used for a randomized complete block design with subsamples is
ijk i j ij ijk
Y B , i^ ^1 ,^ , a ; j^ ^1 ,^ , b ; k^ ^1 ,^ , n
where
is the true overall mean, ^ i is the true fixed effect of the i th treatment of the
fixed factor, ~^ (^0 , )
2
j B
B N (^) is the true random effect of the j th block,
2
~ (0, ) ij
N
(^) is
the random effect of each treatment/block combination, and
~ ( 0 , )
2 N ijk is the true
error for the k th individual of the j th block receiving the i th treatment. Again, we assume
that a
(^) is constrained to be zero.
2.2.3 Example Data Set
If we sample three plants from each segment in the rice fertilizer example, the simulated
results are as follows (measurements are lengths of fruiting period in days).
Table 2.3: Three lengths of fruiting period for each of the fertilizer/row combinations for
the rice fertilizer example
and see what effect that has. Keeping the number of blocks and treatments at the original
level, change the variance of
2
σ ,^
2
B
σ (^) and
2
η
σ (^) to 1 and 3 and 2 and see what effect occurs.
Try different combinations.
2.2.6 Matrix Notation
To illustrate this we will use 2 Fertilizers, 2 Rows and 2 Plants per Fertilizer, Row
combination. Here the model is:
8x1 8x1 1x1 8x2 2x1 8x2 2x1 8x4 4x1 8x
length =J μ + X α + Z b + Z d +ε
Now:
J | X | Z | Z
and the Variance of (Zb) is:
V Zb
2
b
σ
and the Variance of ( (^) Zd
) is:
V Zd
2
η
σ
Now
V Y ( ) V Zb ( ) V Zd ( ) V ( ) ε
and since this is a symmetric matrix we will give the lower
triangle part of that matrix. Given the V(Y) below, we see that
2
's 's
for j = j's
0 for j j's
b
ij i j
σ
COV Y Y
***** add some more:
In other words, Y’s in the same block are correlated and Y’s in the same Fertilizer/Row
combination are correlated.
