Randomized Complete Block Designs (RCB)
EXAMPLE – Comparing Methods of Determining Blood Serum Level
Data File:Serum-Meth.JMP in the Biometry JMP folder
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 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 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 / 41 / 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
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 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.
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 is 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).
The tables below give the block corrected mean for each method and the block means used to make the adjustment.
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 13 but not each other. The same can be said for methods 13 when compared to methods 42. 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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