STAT 305: Chapter 14 – Randomized Complete Block Designs & Their Analysis
Spring 2014
EXAMPLE 14.1 – Comparing Methods of Determining Blood Serum Level
Data File: Serum-Meth.JMP
The goal of this study was determine if four different methods for determining
blood serum levels significantly differ in terms of the readings they give.
Suppose we wish to have 6 readings for each method which we will use to make
our comparisons. One approach we could take would be to find 24 volunteers
and randomly allocate six subjects to each method and compare the readings
obtained using the four methods. (Note: this is called a completely randomized
design). There is one major problem with this approach, what is it?
Instead of taking this approach it would clearly be better to use each method on
the sample from the same subject. This removes subject to subject variation from
the results and will allow us to get a clearer picture of the actual differences in
the methods. Also if we truly only wish to have 6 readings for each method, this
approach will only require the use of 6 subjects versus the 24 subjects the
completely randomized approach discussed above requires, thus reducing the
“cost” of the experiment.
The experimental design where each patient’s serum level is determined using
each method is called a randomized complete block (RCB) design. Here the patients
serve as the blocks; the term randomized refers to the fact that the methods will be
applied to the samples from the patients in a random order, and complete refers to
the fact that each method is used on each subject. In some experiments where
blocking is used it is not possible to apply each treatment to each block resulting
in what is called an incomplete block design. These are less common and we will
not discuss them in this class.
The table below contains the raw data from the RCB experiment to compare the
serum determination methods.
Method
Subject
1
2
3
4
1
360 435 391 502
2
1035 1152 1002 1230
3
632 750 591 804
4
581 703 583 790
5
463 520 471 502
6
1131 1340 1144 1300
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STAT 305: Chapter 14 – Randomized Complete Block Designs & Their Analysis
Spring 2014
Visualizing the need for Blocking
Select Fit Y by X from the Analyze menu and place Serum Level in the Y,
Response box and Method in the X, Factor box. The resulting comparative plot
is shown below. Does there appear to be any differences in the serum levels
obtained from the four methods?
This plot completely ignores the fact that the same six blood samples were used
for each method. We can incorporate this information visually by selecting
Oneway Analysis > Matching Column... > then highlight Patient in the list. This
will have the following effect on the plot.
Now we can clearly see that ignoring the fact the blood samples were used for
each method is a big mistake!
On the next page we will show how to correctly analyze these data.
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STAT 305: Chapter 14 – Randomized Complete Block Designs & Their Analysis
Spring 2014
Correct Analysis of RCB Design Data in JMP
First select Fit Y by X from the Analyze menu and place Serum Level in the Y,
Response box, Method in the X, Factor box, and Patient in the Block box. The
results from JMP are shown below.
Notice the Y axis is “Serum Level – Block Centered”. These means that the
results we are seeing in the display are the differences in the serum level
readings adjusting for the fact that the readings for each method came from the
same 6 patients. Examining the data in this way we can clearly see that the
methods differ in the serum level reported when measuring blood samples from
the same patient.
The results of the ANOVA clearly show we have strong evidence that the four
methods do not give the same readings when measuring the same blood sample
(p < .0001). We do not test the blocking factor generally, so even though the
patient p-value is highly significant we really don’t care – we know already that
the patients differ, that is why we used them as blocks in the first place.
However, the fact that much of the variation in the response is due to patient-topatient differences and reaffirms the fact that blocking on patients was a good
idea!
The tables below give the block corrected mean for each method and the block
means used to make the adjustment.
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STAT 305: Chapter 14 – Randomized Complete Block Designs & Their Analysis
Spring 2014
As was the case with one-way ANOVA (completely randomized) we may still
wish to determine which methods give significantly different means when
measuring the same blood sample. Select Compare Means... > All Pairs,
Tukey’s HSD.
We can see that methods 4 & 2 differ significantly from methods 1 & 3 but not
each other. The same can be said for methods 1 & 3 when compared to methods
4 & 2. The confidence intervals quantify the size of the difference we can expect
on average when measuring the same blood samples. For example, we see that
method 4 will give between 90.28 and 255.05 higher serum levels than method 3
on average when measuring the same blood sample. Other comparisons can be
interpreted in similar fashion.
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STAT 305: Chapter 14 – Randomized Complete Block Designs & Their Analysis
Spring 2014
Example 14.2 – Glucose Concentrations During Four Stages of Labor
Data File: Glucose Labor.JMP
The purpose of a study conducted by Maheux et al. in their paper “Glucose
Homeostasis During Spontaneous Labor in Normal Human Pregnancy” published in
Journal of Clinical Endocrinology and Metabolism (1996) was to evaluate the effect of
labor on glucose production and utilization. Subjects in the study were six
normal pregnant women. Among the data collected were the following glucose
concentrations during four stages of labor: latent (A1) and active (A2) phases of
cervical dilatation, fetal expulsion (B), and placental expulsion (C). The raw data
collected are presented below:
Labor Phase
Subject A1
A2
B
C
1
3.60 4.40 5.30 6.20
2
3.53 3.70 4.10 3.80
3
4.02 4.80 5.40 5.27
4
4.90 5.33 6.30 6.20
5
4.06 4.65 6.10 6.90
6
3.97 5.20 4.90 4.60
These data entered into JMP are shown below.
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STAT 305: Chapter 14 – Randomized Complete Block Designs & Their Analysis
Spring 2014
Again we use Analyze > Fit Y by X with Y = Glucose Concentration and X =
Labor Phase and place Subject in the Block box. The dialog box set up correctly
is shown below.
The results from JMP are shown below.
Sample sizes are really too small to assess normality, so we will just have to
assume that assumption is valid for these data. We can check equality of the
variance of the glucose levels within each labor phase using the UnEqual
Variances option as in one-way ANOVA.
None of the p-values are less
than  = .05 so we will use
ANOVA to test for
differences in the mean
glucose levels across labor
phase.
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STAT 305: Chapter 14 – Randomized Complete Block Designs & Their Analysis
Spring 2014
The ANOVA table and multiple comparisons are shown below.
Using Tukey’s HSD to perform the multiple comparisons.
We can see that the mean glucose levels during fetal and placental expulsion differ
significantly from the mean levels during the latent dilatation phase. The mean level
during placental expulsion for example is estimated to between .612 and 2.35 units larger
than the mean glucose level at the latent dilatation phase.
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