Lecture 24: Analysis of Block Designs
Example
Reusing
Response: torque on knee
Conditions: placement of feet
Reuse the experimental
material so that each piece of
experimental material
experiences all the treatments
in a random order.
Feet back, feet neutral, feet
staggered
Experimental Material: men who
have had knee replacement.
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2
Reusing Material
Control of Outside Variables
Each person will sit in a chair
and rise to a standing position.
Each person will do this three
times, once with each foot
position.
The order of the three foot
positions will be randomized for
each person.
Older males.
All have had total knee
arthroplasty (replacement).
Height of chair constant.
All participants wear tennis
shoes and comfortable clothing.
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Randomization
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Randomization
Use three colored chips;
Each participant
experiences all three
conditions (placement of
feet) in a random order.
 Red – Feet Neutral
 White – Feet Back
 Blue – Feet Staggered
Each participant draws chips
without replacement to determine
his order of treatments.
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1
Lecture 24: Analysis of Block Designs
Replication
Sample Size Tables
There are 15 participants.
The treatments (foot
positions) are replicated 15
times.
Although constructed for
completely randomized
designs, the sample size tables
can give us insight into the
size of the detectable
difference in treatment means.
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Sample Size Tables
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Informal Analysis
3 groups.
Alpha = 0.05, Beta = 0.10
n = 15 per group.
Can detect between a 1.2 and 1.4
standard deviation difference in
treatment sample means.
Method Neutral
Back
Staggered
Mean
24.1 Nm 21.7 Nm 20.7 Nm
Std. Dev. 3.8582
2.9835
2.3719
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Estimated Effects
Neutral: 24.1 – 22.167 =
1.933 Nm
1.933
Back: 21.7 – 22.167 =
–0.467 Nm
–1.467 Nm
– 0.467
Staggered: 20.7 – 22.167 =
– 1.467
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2
Lecture 24: Analysis of Block Designs
Estimated Effects
Analysis of Variance
Staggered reduces the torque
the most, on average.
Neutral increases the torque
the most, on average.
Are these effects statistically
significant?
Source
Method
Subject
Error
C. Total
df
SS
MS
F Prob > F
2 91.60
45.8 31.84 < 0.0001
14 371.51
28 40.27 1.4383
44 503.38
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14
Test of Hypothesis
Test of Hypothesis
H0: all method effects are 0
HA: some method effects
are not 0
F = 31.84, P-value < 0.0001
Because the P-value is so
small, we should reject the
null hypothesis and
conclude that some of the
method effects are not zero.
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Multiple Comparisons
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Multiple Comparisons
Compute the LSD for comparing the
Method Means.
∗
1
Level
Neutral
Back
Staggered
1
LSD = 2.04841 1.4383
Mean
24.1
21.7
20.7
A
B
C
Levels not connected by the same
letter are significantly different.
= 2.04841(0.43792) = 0.897
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Lecture 24: Analysis of Block Designs
Conclusion
JMP
Analyze – Fit Model
Response – Torque
Construct Model Effects
Each method produces a
different mean torque which
is statistically different from
the other methods.
Method
Subject
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Response Torque
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
Comment
0.919994
0.874277
1.199305
22.16667
45
Analysis of Variance
Source
Model
Error
C. Total
DF
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28
44
The summary RSquare and
Model F-Ratio are not
helpful in summarizing the
effects of Methods on
Torque.
Sum of
Squares Mean Square
F Ratio
28.9442 20.1234
463.10667
1.4383 Prob > F
40.27333
503.38000
<.0001*
Effect Tests
Source
Method
Subject
Nparm
2
14
DF
2
14
Sum of
Squares
91.60000
371.50667
F Ratio Prob > F
<.0001*
31.8424
<.0001*
18.4493
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Comment
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Comment
The Subject term quantifies
the variation in experimental
material.
The Error term quantifies the
lack of consistency of Method
effects across Subjects.
Method RSquare
. ⁄
0.182
.
18.2% of the variation in
Torque is explained by the
different Methods.
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Lecture 24: Analysis of Block Designs
Analysis of Variance
Source
Method
Subject
Interaction
Error
C. Total
df
2
14
28
0
44
Replication
SS
91.60
371.51
40.27
0
503.38
Although there is replication
of the Methods (15 subjects do
all three Methods) there is not
replication of the Method by
Subject combinations.
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“Error”
“Error”
There is no way to estimate
error due to different trials
of Method for each Subject
because each Subject uses
each Method only once.
The Error term is actually the
interaction between Method
and Subject.
In a Block design this
interaction is assumed to be no
different from random error.
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Interpretation
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Interpretation
If some treatments are better
with some blocks and worse
for others, the “Error” term
will be large and so there may
be no statistically significant
consistent treatment effects.
If the F test for treatment
effects turns out to be not
statistically significant, this
could be due to no treatment
effects or to no consistent
treatment effects.
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