Download Two Treatment Factors in a Randomized Complete Block Design

Two Treatment Factors in a Randomized Complete Block Design
Example: An anesthesiologist made a comparative study of the effects of two factors, acupuncture and
codeine, on postoperative dental pain in male subjects. The four treatment combinations were as
follows:
1. ________________________ treatment (a sugar pill and two inactive acupuncture points)
2. ________________________ treatment (a codeine pill and two inactive acupuncture points)
3. ________________________ only (a sugar pill and two active acupuncture points)
4. _____________________________________ (a codeine pill and two active acupuncture points)
Thirty-two subjects were grouped into 8 blocks (each of size 4) according to an initial evaluation of their
pain tolerance. The subjects in each block were randomly assigned to the four treatment combinations.
The pain relief scores were obtained for all subjects two hours after dental treatment (data were
collected on a double-blind basis). The data are given in the file Dental.jmp on the course website. A
portion of the data is shown below.
Questions:
1. Identify the response variable.
2. Why do you think pain tolerance was used as a blocking variable?
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Statistical model for two treatment factors in a RCBD
yijk = μ + αi + β j + αβij + ρk + εijk
i = 1, 2, …, a; j = 1, 2, …, b; k = 1, 2, …,n where

yijk = the response value in the kth block with the ith level of Factor A and the jth level of Factor B
are used

µ = the overall mean

αi = the effect of the ith level of Factor A

βj = the effect of the jth level of Factor B

αβij = the effect due to the interaction of the ith level of Factor A and the jth level of Factor B

ρk = the effect of the kth block

εijk = the random error associated with the observation in the kth block that received the ith level
of Factor A and jth level of Factor B
Model Assumptions
1. The error terms are independent and normally distributed.
2. The error terms have constant variance.
3. The treatment and block effects are assumed to be additive (i.e. there is no block*treatment
interaction).
Fitting the model in JMP
To fit this model in JMP, choose Analyze  Fit Model and enter the following:
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Click Run and JMP should return the following output:
Questions:
3. What does the test for interaction indicate?
4. How should we proceed with the analysis? Explain.
We can investigate the significant main effects in JMP by clicking on the red drop-down arrows next to
Pill and Acupuncture and selecting the following:
Investigating the main effect of pill:
Investigating the main effect of acupuncture:
Note: The Tukey HSD method is NOT available because both factors only have two levels, so pairwise
comparisons for each factor only involve one test and Tukey’s adjustment is not needed.
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Output for comparing the two levels of pill:
Output for comparing two levels of acupuncture:
Summarize the results of this study, using confidence intervals in your discussion.
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Checking model assumptions
First, we can examine the plot of the residuals versus the predicted values. This plot is automatically
given when we fit the model.
Additionally, we can look at the plot of the residuals against the treatment combinations. First, we need
to obtain the residuals. Click on the red drop-down arrow next to Response score and choose Save
Columns  Residuals. Next, choose Chart  Graph Builder. Then put the residuals in the Y zone and
then put one treatment factor in the X zone and the remaining treatment factor in the Group X zone.
You should then get a graph like the one given below.
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Lastly, select Analyze  Distribution and place Residual score in the Y,Columns box. Then click on the
red drop-down arrow and choose Normal Quantile Plot.
Questions:
5. Does the assumption of constant variance appear to be violated? Explain.
6. Does the assumption of normality appear to be violated? Explain.
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Investigating the predicted values and residuals
Using the output given below, verify the predicted value and residual for the two observations
highlighted.
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