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Unsorted UnsortedTreatments Random Numbers 1 0.533 1 0.683 2 0.702 2 0.379 3 0.411 3 0.962 3 0.139
Sorted Sorted ExperimentalTreatments Random Units Numbers 3 0.139 1 2 0.379 2 3 0.411 3 1 0.533 4 1 0.683 5 2 0.702 6 3 0.962 7
Randomization
Bread Rise Experiment
1. Mix The Dough
2. Divide the dough into 12 small loaves of the same size.
3. Randomly assign 4 loaves to rise 35 minutes, 4 to rise 40 minutes, etc.
4. After allowing each loaf to rise the specified time, measure the height of the loaf.
Model for CRD Design
ijiijY
tirj
N
i
ijij
,,1 ,,1
tindependenmutually s' ),,0(~ 2
Cell Means Model
Alternate Model
iiijiitY
tirj
N
i
ijij
,,1 ,,1
tindependenmutually s' ),,0(~ 2
Effects Model
),(~ 2 iit NY
Notation
Sample means
ir
jij
ii Yr
Y1
1
t
ti
r
ii
t
i
r
jit rnyy
ny
ny
n
ii
111 1
where,111
Grand Mean
Least Squares Estimates Cell Means
021 1
t
i
r
jiij
i
i
yssE
t
i
r
jiij
i
yssE1 1
2Choose estimates to minimize
ii y
3 ,4321 rrr
Matrix Notation for Alternate Model
34
33
32
31
24
23
22
21
14
13
12
11
3
2
1
34
33
32
31
24
23
22
21
14
13
12
11
1001
1001
1001
1001
0101
0101
0101
0101
0011
0011
0011
0011
εβXy
y
y
y
y
y
y
y
y
y
y
y
y
εXβy
LS Estimatorsare solutionto
yXβXX ˆ
Problem is singular
XX
yXβXX ˆ
21XXX
1001
1001
1001
1001
0101
0101
0101
0101
0011
0011
0011
0011
SAS proc glm
112212
2111 XXXXXX
XXXXXX
Non-singular
00
0XXXX
111 is a generalized inverse for XX
yXXXβ )(ˆ Biased Estimates
0
ˆˆ
ˆˆ
ˆˆ
ˆ 32
31
3
β
3
2
1
β
Bread Rise Experiment
0
ˆˆ
ˆˆ
ˆˆ
ˆ 32
31
3
β
=^
:
Matrix Notation for Estimable Functions
βL ˆ is an unbiased estimator for
when the rows of L are linear combination of the rows of for example
Lβ
X
1001
1001
1001
1001
0101
0101
0101
0101
0011
0011
0011
0011
X
1010
0110L
3
2
1
β
is a linear
))(,(~ˆ 2 LXXLLMVNL
ssE must represent varation in experimental units not subsamples, repeated measures or duplicates
Teaching Example (illustration of problems)
Classes randomized to different teaching methods experimental unit=class
No replicate classes no way to compute ssE
Teaching method confounded with difference in classes
Use of student to student variability (i.e. subsamples) to calculate ssECould be totally misleading
● Independence of error terms εij
● Equality of variance across levels of treatment factor
● Normal distribution of εij
Check equal variance assumption 1. plot data vs treatment factor level 2. plot residuals vs predicted values or cell means
Check normality with normal plot of residuals
ods graphics on;proc glm data=bread plots=diagnostics; class time; model height=time/solution;run;ods graphics off;
λ = 1 -1.294869
Solutions►Analysis►Design of Experiments
Two-Level FactorialResponse Surface
MixtureMixed-Level Factorial
Optimal Design
Split-Plot Design
GeneralFactorial
Define Variables►Add> ►Add qualitative factorial variable
Customize…►Replicate Runs
Edit Responses… Design►Randomize Design …
Fit …
Model ► Fit Details…Model►Check Assumptions►Perform Residual AnalysisModel►Check Transformation►Box-Cox Plot
Teaching Experiment
Objective: Compare student satisfaction between 3 different teaching methods
Experimental unit: class
Two replicate classes for each teaching method.
Response: rating given by each student, summarized over class as multinomial vector of counts
),,,,,(~ 54321
5
4
3
2
1
iiiiii
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i
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i
pppppnMN
y
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y
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iiiii yi
yi
yi
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i pppppyyyyy
nP 44321
5432154321
)(
y
power, 1-β
practicalsignificance
power
Size of a practical difference
3. H0: μ3 = μ4
Does a mix of artificial fertilizers enhance yield?
Is there a difference in plowed and broadcast?
Does Timing of Application change Yield?
^ ^
Option 1 Option 2
Option 1
Option 2
Review Important Concepts
• Experimental Unit
• Randomization
• Replication
• Practical Difference
• Determining the number of replicates by calculating the power