Unsorted Treatments Random Numbers 1 0.533 1 0.683 2 0.702 2 0.379 3 0.411 3 0.962 3 0.139 Sorted...
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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
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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.
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Model for CRD Design
ijiijY
tirj
N
i
ijij
,,1 ,,1
tindependenmutually s' ),,0(~ 2
Cell Means Model
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Alternate Model
iiijiitY
tirj
N
i
ijij
,,1 ,,1
tindependenmutually s' ),,0(~ 2
Effects Model
),(~ 2 iit NY
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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
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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
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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
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21XXX
1001
1001
1001
1001
0101
0101
0101
0101
0011
0011
0011
0011
SAS proc glm
112212
2111 XXXXXX
XXXXXX
Non-singular
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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
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Bread Rise Experiment
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0
ˆˆ
ˆˆ
ˆˆ
ˆ 32
31
3
β
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=^
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:
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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
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is a linear
))(,(~ˆ 2 LXXLLMVNL
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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
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● Independence of error terms εij
● Equality of variance across levels of treatment factor
● Normal distribution of εij
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Check equal variance assumption 1. plot data vs treatment factor level 2. plot residuals vs predicted values or cell means
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Check normality with normal plot of residuals
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ods graphics on;proc glm data=bread plots=diagnostics; class time; model height=time/solution;run;ods graphics off;
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λ = 1 -1.294869
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Solutions►Analysis►Design of Experiments
Two-Level FactorialResponse Surface
MixtureMixed-Level Factorial
Optimal Design
Split-Plot Design
GeneralFactorial
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Define Variables►Add> ►Add qualitative factorial variable
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Customize…►Replicate Runs
Edit Responses… Design►Randomize Design …
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Fit …
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Model ► Fit Details…Model►Check Assumptions►Perform Residual AnalysisModel►Check Transformation►Box-Cox Plot
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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
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power, 1-β
practicalsignificance
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power
Size of a practical difference
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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?
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^ ^
Option 1 Option 2
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Option 1
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Option 2
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Review Important Concepts
• Experimental Unit
• Randomization
• Replication
• Practical Difference
• Determining the number of replicates by calculating the power