Lecture 31: Split Plot/Repeated Measures
Split Plot
Split Plot/Repeated Measures
In a split plot design there
are experimental units of
two different sizes.
The term Split Plot usually
refers to an Agricultural
experiment while the term
Repeated Measures is used
by Social Scientists.
Whole plot
Sub plot
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Example
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Treatments
Response: Yield, bushels of
soybeans per acre.
Conditions:
Factorial crossing is used to
make treatments from the 6
combinations of variety of
soy bean and type of
insecticide.
Variety (3 levels)
Insecticide (2 levels)
Experimental Material: fields
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Block Design?
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Treatments in one field
We might consider a block
design where the fields are the
blocks.
Blocks are made by subdividing
each field into 6 plots.
Treatments are assigned, at
random, to plots in each field.
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V1,I2
V2,I1
V2,I2
V3,I1
V1,I1
V3,I2
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Lecture 31: Split Plot/Repeated Measures
Inconvenience
Real Problem
Farmer will have to load and
empty the planter 6 times for
each field.
Farmer will have to load and
empty the sprayer 6 times for
each field.
If there is any wind, the
insecticide applied to one
plot may drift onto an
adjacent plot, biasing the
experiment.
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Run Two Experiments
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Example Field Layouts
In the spring plant the 3
varieties of soy bean using a
randomized complete block
design with fields as blocks.
Blocks made by subdividing
fields.
V1
V3
V2
V2
V1
V3
V3
V2
V1
V3
V1
V2
Have 8 fields total.
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Run Two Experiments
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Example Field Layouts
After the soy beans are up and
growing, apply insecticide 1 to
four of the fields, chosen at
random from the 8, and
Insecticide 2 to the other
fields.
Insecticide 1
V1 V3 V2
Insecticide 2
V2 V1 V3
V3 V2 V1
Insecticide 2
V3 V1 V2
Insecticide 1
Have 8 fields total.
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Lecture 31: Split Plot/Repeated Measures
Factorial Crossing
Two Experiments in One
Notice that each level of
Insecticide appears with each
level of Variety.
The Insecticide*Variety
treatment combinations are
replicated on several plots.
Completely Randomized
Design
Insecticides are applied
using whole fields as
experimental units.
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Two Experiments in One
Randomized Complete Block
Design.
Varieties are planted using
fields as blocks. All three
varieties are assigned to plots
within each block at random.
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Two Experiments in One
Because there are two
experiments using two
different sizes (whole fields
and plots in fields) of
experimental units, the
ANOVA needs to reflect this.
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Completely Randomized
Sources:
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Error Variation
Error variation comes from
differences in fields treated
(getting the same
Insecticide) the same.
Insecticide, 2 levels, 1 df
Error, 4 fields within each
Insecticide, 3 + 3 = 6 df
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Lecture 31: Split Plot/Repeated Measures
Error Variation
Randomized Complete Block
Variety, 3 levels, 2 df
Variety*Insecticide, 2 df
Error, 12 df
Error comes from the
inconsistency of Variety
effect across different fields.
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Partial ANOVA
Source
Insecticide
Fields[Insecticide]
Variety
Insecticide*Variety
Error
C. Total
df
1
6
2
2
12
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Repeated Measures
F Ratio
MSInsect/MSFields
MSVariety/MSError
MSInsect*Var/MSError
In a repeated measures
design experimental units
are measured several times.
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Example
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Exam Aids
Response: Score on a statistics
exam.
Conditions:
What exam aids are used?
Nothing
Formula sheet
Calculator
Both a formula sheet and a
calculator.
Exam Aids (4 levels)
Type of Test (2 levels)
Experimental Material:
students
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Lecture 31: Split Plot/Repeated Measures
Type of Exam
Two Experiments in One
Multiple choice (100 pt)
Problem solving (100 pt)
Each exam measures
competency with the
statistics material.
Completely randomized
design.
Assign exam aids at random to
the 100 students so that 25
students experience each of
the levels.
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Two Experiments in One
Randomized complete block
design.
Have each student take both
tests. The order is determined
at random for each students.
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Repeated Measures
Measuring each students
competency on statistics
twice, once with a multiple
choice exam and once with
a problem solving exam.
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Analysis of Variance
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Analysis of Variance
If we were only looking at the
effects of exam aids in using a
completely randomized
design, there would be two
sources of variation; exam aids
and random error.
Source
Exam Aids
Error
C. Total
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df
3
96
99
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Lecture 31: Split Plot/Repeated Measures
Random Error
Analysis of Variance
The variation in students treated
the same (using the same exam
aid) goes into the sum of squares
for error.
This variation could result from
different study habits, different
test taking abilities, different
content knowledge, etc.
Source
Exam Aids
Students[Exam Aids]
C. Total
df
3
96
99
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Second Factor
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Analysis of Variance
The second factor, type of
test, is examined by reusing
students in a randomized
complete block design.
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Error
Source
Exam Aids
Students[Exam Aids]
Type of Test
Aids*Test Interaction
Error
C. Total
df
3
96
1
3
96
199
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Comment
 The exam aid levels are really
combinations of factors;
The sums of squares for error
in this part of the design is due
to the interaction between type
of test and the students (df =
1*96), the inconsistency of
test effect across different
students.
 Formulas – Yes, No
 Calculator – Yes, No
 The exam aids sums of squares can be
broken down into sums of squares for
Formulas, Calculator and a
Formulas*Calculator interaction.
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