Randomized Complete Block Design
Up to this point in the semester we’ve been investigating completely randomized designs (CRD). The
CRD is a type of experiment where the experimental units (subjects) are randomly assigned to one of
the treatment groups. The randomization of treatments or experimental conditions is key because it
reduces the likelihood that the results will be affected by __________________ confounding variables
and other sources of bias that are often present in observational studies.
However, there are times when we ______________ that there are other factors related to the
response. In these situations, we can do better than to simply use a CRD, hoping to balance out the
effects of these confounding factors.
Example: Blood pressure and dental visits
Many people have high anxiety about visiting the dentist. Researchers want to know if this affects blood
pressure in such a way that, on average, blood pressure while waiting to see the dentist is higher than it
is an hour after the visit. Suppose that to investigate this, researchers plan to randomly sample 20
visitors to the dentist. They will then randomly select 10 of these subjects and take their blood pressure
while in the waiting room, while the other 10 will have their blood pressure measured after the
conclusion of the visit to the dentist. Note that this is a completely randomized design with 10
replications.
Question:
1. What is the major flaw of this study design?
A better approach would be to use a __________________________ design. In general with this type of
set-up, individuals would be matched based on the confounding variable we want to control, and each
member of the pair would be randomly assigned to one of the two different treatment groups. In this
example, the confounding variable we want to control is _____________________________________.
So, we could take baseline measurements for all 20 subjects, and then place the two with the highest
blood pressure in a pair, the next highest two, etc. Then, one person in the pair would be randomly
assigned to have their blood pressure measured while waiting to visit the dentist and the other have
their blood pressure measured after the appointment.
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This seems sort of silly doesn’t it? If we planned on 10 replications, it would make much more sense to
take before and after measurements on the same 10 people. This is a special case of a matched-pair
design, when the same subject is measured twice under different conditions. This is what the
experimenters actually did in the study. Ten subjects agreed to have their systolic blood pressure taken
while they were in the dentist’s waiting room and again an hour after the conclusion of their visit to the
dentist. The data are shown below.
Subject
BP before
BP after
1
132
118
2
135
137
3
149
140
4
133
139
5
119
107
6
121
116
7
128
122
8
132
124
9
119
115
10
110
103
Source: Exercise 13.45 of Mind on Statistics by Utts and Heckard (4th Edition)
In your introductory course, you most likely learned how to analyze data collected in this way using a
matched pair _____-test. We will briefly review this method.
Matched pair t-test
First, we need to calculate the differences between the before and after measurements for each
subject.
Subject
BP before
BP after
Difference
1
132
118
2
135
137
3
149
140
4
133
139
5
119
107
6
121
116
7
128
122
8
132
124
9
119
115
10
110
103
Questions:
2. Find the average of these differences. This is the observed average difference, x d .
3. Is this average positive or negative? What does the sign of the average difference indicate in
the context of the problem?
4. What does the magnitude (or size) of the average difference indicate in the context of the
problem?
5. What should the average difference be if blood pressure is not affected by dental visits? This is
called the expected average difference under the null hypothesis, µd.
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To test whether the blood pressure is affected by dental visits, we will compare our observed result to
the expected average difference using the following test statistic:
where sd represents the standard deviation of the differences and n the number of subjects. The
denominator of the test statistic is the standard error of the differences. We can use JMP to compute
the average, the standard deviation, and the standard error of the differences. Open the file
DentalVisits_t_test.jmp found on the course website. Note that the variable of interest is the
DIFFERENCE in blood pressure (before – after).
Choose Analyze Distribution and place Difference = before-after in the Y, Columns box.
Click OK. Fill in the missing values in the JMP output below.
Next, compute the test statistic.
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To compute the test statistic and p-value using JMP, click on the red drop-down arrow next to
Difference=before – after and choose Test Mean. Enter the expected value of the difference under the
null hypothesis in the box next to Specify Hypothesized Mean as shown below.
Click OK and you should get the following JMP output.
Questions:
6. What is the p-value for testing whether the mean blood pressure while waiting to see the
dentist is higher than it is an hour after the visit?
7. Is there evidence to support the hypothesis that the mean blood pressure while waiting to see
the dentist is higher than one hour after the visit?
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Example: Advertising strategies
“Suppose that a national chain of cafes wants to compare three methods for increasing sales at cafes
near college campuses. Each method involves placing a weekly ad in the campus newspaper. Method 1,
a coupon for a free cup of coffee is included with the ad. With Method 2, a coupon for a free cookie
with any coffee drink purchase is included in the ad. The third method is to place the ad without any
coupons. They place the ads for one full semester and measure how much sales change from the same
semester the previous year.”
Source: Mind on Statistics by Utts and Heckard (4th Edition) pg. 202
Questions:
8. How would you set this up as a completely randomized design?
9. Can you think of any other factors besides advertising method that might affect sales?
To account for these other known variables, we once again want to do better than a completely
randomized design. However, a matched-pair design is not possible for this experiment because we can
to compare _____ methods! So, we will instead use what is called a randomized ___________________
________________design.
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Randomized Complete Bloc Design (RCBD)
In an RCBD, the subjects (i.e. experimental units) are first divided into __________________________
groups called _____________ according to the confounding variable you want to control. The term
randomized refers to the fact that treatments will be randomly assigned to the experimental units
_______________ each block, and complete refers to the fact that each block consists of one complete
replication of the set of treatments. Therefore, each treatment will show up __________ within each
block. In some experiments where blocking is used, it is not possible to apply each treatment once in
each block, resulting in what is called an ________________ block design. These are less common and
we will not discuss them in this class.
When grouping experimental units into blocks, keep the following objectives in mind:
1. Within blocks, make the experimental units as _______________________ as possible with
respect to the response variable.
2. Make the different blocks as _____________________ as possible with respect to the response
variable.
Back to the advertising example
Suppose the café company decides to use a total of 12 campuses in their study, and they block the
campuses on the basis of size. This means that the three smallest campuses are placed in Block 1, the
next three smallest in Block 2, etc. Next, the treatments are randomly assigned to the experimental
units within each block, using a separate randomization for each block. In the end, each treatment
shows up once in each block. For example, we could have the following arrangement of treatments:
Block 1
(smallest campuses)
Block 2
1
Experimental Unit
2
3
Method 1
Method 3
Method 2
Method 1
Method 2
Method 3
Method 3
Method 1
Method 2
Method 2
Method 3
Method 1
Block 3
Block 4
(largest campuses)
Question:
10. What is the advantage of using the randomized complete block design over the completely
randomized design?
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In general, blocking allows us to reconcile two somewhat opposing objectives of experimental design.
1. One objective is to reduce ____________ variability. Thus, one strategy for experimentation
might be to work with homogenous experimental units. For example, the café company may
decide to use only small campuses in their study to reduce variability.
2. Another objective is to make inferences relevant to the population of interest, and this leads to
a seemingly contradictory strategy: to deliberately use a ______________ range of experimental
units so that inferences will apply to a wide range of situations. For example, if the café
company only used small campuses in their study, would it be fair for them to generalize the
results to larger campuses? Why or why not?
Blocking allows us to include a ________ variety of experimental units while at the same time
______________ the unexplained variation as much as possible.
Example: Alcohol consumption and reaction time
Researchers were interested in investigating the effect of the amount of alcohol consumed on reaction
time. Age is thought to be another variable which could affect the reaction time, so observations were
blocked according to age. A randomized complete block design is used, and reaction time was
measured in seconds. The data are given below and can be found in the file AlcoholReaction.jmp on
the course website.
Amount of Alcohol
None
1 oz.
2 oz.
20 – 39
0.42
0.47
0.65
Age
40 – 59
0.51
0.62
0.66
(Block)
≥60
0.57
0.73
0.79
Using JMP to analyze a RCBD
We’re still trying to determine if there is a treatment effect. The only thing we’ve done differently is
remove some variation that would otherwise have been thrown into error. In JMP select Analyze  Fit
Model and enter the following into the dialogue box.
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Click Run and JMP should return the following output:
Questions:
11. What does the p-value of 0.0112 indicate?
12. What does the p-value of 0.0152 indicate?
As with the case of the one-way ANOVA (for a CRD), we may still wish to determine ______________
treatment groups are significantly different form the others. We can again use multiple comparison
procedures, such as Tukey’s approach.
Question:
13. What are the conclusions from the study?
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