_____________________________
Randomized Block
Design
_____________________________
1. Experimental Units (Subjects) Are
Assigned Randomly to Treatments
2. Uses Blocking Variable Besides
Independent (Treatment) Variable
„
Permits Better Assessment of Treatment
3. Analyzed by Randomized Block F Test
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ANOVA - 1
_____________
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Randomized Block
Design Example
Factor (Diskette Brand)
Level 2
Level 3
Factor
Level 1
Levels
(Treatments)
IBM
Stores
Experimental
Units
_____________________________
_____________________________
Stores
Stores
Blocking
Variable
(Store)
NEC FUJI
Dependent
Variable
$ 6
$ 4
$ 2
Store 1
$ 11
$ 7
$ 3
Store 2
(Price)
$ 15
$ 11
$ 7
Store 3
$ 24
$ 22
$ 20
Store 4
ANOVA - 2
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_____________
Randomized Block F Test
_____________________________
_____________________________
1. Tests the Equality of 2 or More (p)
Population Means When Blocking
Variable Used
_____________________________
2. More Efficient Analysis Than One-Way
ANOVA
_____________________________
„
Error Variation Is Reduced
3. Used to Analyze Randomized Block
Designs
ANOVA - 3
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_____________________________
_____________________________
_____________________________
_____________
Randomized Block F Test
Assumptions
Populations are Normally Distributed
_____________________________
2. Homogeneity of Variance
„
_____________________________
Populations have Equal Variances
3. Independence of Errors
„
_____________________________
_____________________________
1. Normality
„
_____________________________
_____________________________
Independent Random Samples are Drawn
_____________________________
4. No Interaction Between Blocks &
Treatments
_____________________________
ANOVA - 4
_____________
_____________________________
Randomized Block F Test
Hypotheses
H0: µ1 = µ2 =... = µp
„
„
All Population Means
are Equal
No Treatment Effect
At Least 1 Population
Mean is Different
„
Treatment Effect
µ1 ≠ µ2 ≠ ... ≠ µp Is
Wrong
„
_____________________________
f(X)
_____________________________
µ 1 = µ2 = µ3
Ha: Not All µj Are Equal
„
_____________________________
X
_____________________________
_____________________________
f(X)
µ1 = µ2 µ3
ANOVA - 5
X
_____________________________
_____________________________
_____________
Randomized Block F Test
Basic Idea
1. SS(Total) & SST Are Same As
Completely Randomized Design
2. Error Variation (SSE) Is Different
„
„
Blocking Effect (SSB) Comes Out of Error
Variation (SSE) Reducing Error, SSE
In Completely Randomized Design, Error
Variation Includes Blocking Effect
3. By Reducing Error, F May Increase
ANOVA - 6
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_____________________________
_____________________________
_____________________________
_____________________________
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_____________
_____________________________
Randomized Block F Test
Total Variation Partitioning _____________________________
_____________________________
Total
Total Variation
Variation
_____________________________
SS(Total)
_____________________________
Variation
VariationDue
Dueto
to
Treatment
Treatment
SST
_____________________________
Variation
VariationDue
Dueto
to
Blocking
Blocking
Variation
VariationDue
Dueto
to
Random
RandomSampling
Sampling
SSB
SSE
_____________________________
_____________________________
ANOVA - 7
_____________
Randomized Block F Test
Summary Table
Source of
Variation
Degrees
of
Freedom
Sum of
Squares
Mean
Square
(Variance)
F
Among
Treatments
p-1
SST
MST
MST
MSE
b-1
SSE
n - p - b +1
SSB
MSB
MSB
MSE
SSE
MSE
Among
Blocks
Error
Total
n- 1
SS(Total)
_____________________________
_____________________________
_____________________________
_____________________________
Same as
Completely
Randomized
Design
ANOVA - 8
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_____________________________
_____________________________
_____________________________
_____________
_____________________________
Formula
_____________________________
_____________________________
Sum of squares between Treatments(SST):
p
SST = j =1 b ⋅ ( x j − x )
∑
2
Sum of squares for Blocks (SSB):
p
SSB =
ANOVA - 9
p ⋅ (x
i =1
∑
i
− x)
2
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_____________________________
_____________________________
_____________
_____________________________
Formula
_____________________________
_____________________________
Sum of squares Total (SS(Total)):
b
_____________________________
p
SS (Total ) = ∑∑ ( x ij − x ) 2
_____________________________
i =1 j =1
_____________________________
Sum of squares of sampling error:
SSE = SS(Total) - SST - SSB
_____________________________
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ANOVA - 10
_____________
_____________________________
Randomized Block F Test
Critical Value
_____________________________
_____________________________
Degrees of Freedom Are
(p -1) & (n
(n - b - p + 1)
_____________________________
Reject H0
α
Do Not
Reject H0
0
Fα,( p−1, n - b - p + 1)
_____________________________
F
Always One-Tail!
© 1984-1994 T/Maker Co.
ANOVA - 11
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_____________________________
_____________________________
_____________
Randomized Block F Test
Thinking Challenge
You’re a market research
analyst. Using the computer,
is there a difference in mean
diskette price at 4 stores (.05)?
Store
1
2
3
4
ANOVA - 12
IBM NEC
6
4
11
7
15
11
24
22
FUJI
2
3
7
20
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_____________________________
_____________________________
FUJI
NEC
IBM
_____________________________
_____________________________
_____________________________
_____________________________
_____________________________
_____________
... toner was low. Only a
portion of output printed.
Source of
Variation
Degrees
of
Freedom
Sum of
Squares
Mean
Square
(Variance)
F
Among
Blocks
3
_____________________________
_____________________________
_____________________________
72
Among
Treatments
_____________________________
_____________________________
186
_____________________________
Error
Total
_____________________________
638
_____________________________
ANOVA - 13
_____________
Randomized Block F Test
Solution*
Source of
Variation
Degrees
of
Freedom
Sum of
Squares
Mean
Square
(Variance)
F
3-1=2
Among
Treatments
72
36
27
Among
Blocks
4-1=3
558
186
139.5
Error
12-3-4+1
=6
8
1.33
Total
12- 1 = 11
638
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ANOVA - 14
_____________
Randomized Block F Test
Solution*
H0: µ1 = µ2 = µ3
Ha: Not All Equal
α = .05
ν1 = 2 ν2 = 6
Critical Value(s):
α = .05
0
ANOVA - 15
5.14
F
Test Statistic:
F=
MST 36
=
= 27
MSE 1.33
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_____________________________
_____________________________
_____________________________
_____________________________
Decision:
Reject at α = .05
Conclusion:
There Is Evidence Mean
Prices Are Different
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_____________________________
_____________________________
_____________
Factorial Design
1. Experimental Units (Subjects) Are
Assigned Randomly to Treatments
„
Subjects are Assumed Homogeneous
2. Two or More Factors or Independent
Variables
„
Each Has 2 or More Treatments (Levels)
3. Analyzed by Two-Way ANOVA
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_____________________________
_____________________________
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_____________________________
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ANOVA - 16
_____________
Factorial Design
Example
Factor 2 (Training Method)
Factor Level 1 Level 2 Level 3
Levels
(High)
Factor 1
11 hr.(Motivation) Level 2 27 hr./
(Low)
29 hr./
Level 1 19 hr.
- 22 hr.17 hr.- 31 hr.25 hr./ 31 hr./
30 hr./ 49 hr./
20 hr.
Treatment
ANOVA - 17
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_____________________________
_____________
Advantages
of Factorial Designs
1. Saves Time & Effort
„
e.g., Could Use Separate Completely
Randomized Designs for Each Variable
2. Controls Confounding Effects by Putting
Other Variables into Model
3. Can Explore Interaction Between
Variables
ANOVA - 18
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_____________________________
_____________________________
_____________________________
_____________________________
_____________
_____________________________
Two-Way ANOVA
_____________________________
_____________________________
1. Tests the Equality of 2 or More
Population Means When Several
Independent Variables Are Used
_____________________________
2. Same Results as Separate One-Way
ANOVA on Each Variable
„
No Interaction Can Be Tested
3. Used to Analyze Factorial Designs
_____________________________
_____________________________
_____________________________
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ANOVA - 19
_____________
Two-Way ANOVA
Assumptions
Populations are Normally Distributed
_____________________________
2. Homogeneity of Variance
„
_____________________________
Populations have Equal Variances
3. Independence of Errors
„
_____________________________
_____________________________
1. Normality
„
_____________________________
Independent Random Samples are Drawn
_____________________________
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ANOVA - 20
_____________
_____________________________
Two-Way ANOVA
Data Table
Factor
A
1
X111
1
X112
X211
2
X212
:
:
Xa11
a
Xa12
ANOVA - 21
Factor B
2
...
X121 ...
X122 ...
X221 ...
X222 ...
:
:
Xa21 ...
Xa22 ...
_____________________________
_____________________________
b
X1b1
X1b2
X2b1
X2b2
:
Xab1
Xab2
Observation k
Xijk
Level i Level j
Factor Factor
A
B
_____________________________
_____________________________
_____________________________
_____________________________
_____________________________
_____________
_____________________________
Two-Way ANOVA
Null Hypotheses
_____________________________
1. No Difference in Means Due to Factor A
„
_____________________________
H0: µ1.. = µ2.. =... = µa..
2. No Difference in Means Due to Factor B
„
H0: µ.1. = µ.2. =... = µ.b.
_____________________________
_____________________________
3. No Interaction of Factors A & B
„
_____________________________
H0: ABij = 0
_____________________________
_____________________________
ANOVA - 22
_____________
Two-Way ANOVA
Total Variation Partitioning
_____________________________
_____________________________
_____________________________
Total
Total Variation
Variation
_____________________________
SS(Total)
Variation
VariationDue
Dueto
to
Treatment
TreatmentAA
Variation
VariationDue
Dueto
to
Treatment
TreatmentBB
_____________________________
SSB
_____________________________
Variation
Variation Due
Due to
to
Interaction
Interaction
Variation
VariationDue
Dueto
to
Random
RandomSampling
Sampling
_____________________________
SS(AB)
SSE
_____________________________
SSA
ANOVA - 23
_____________
Two-Way ANOVA
Summary Table
Source of Degrees of Sum of
Variation
Freedom Squares
Mean
Square
F
_____________________________
_____________________________
_____________________________
A
(Row)
a-1
SS(A)
MS(A)
MS(A)
MSE
_____________________________
B
(Column)
b-1
SS(B)
MS(B)
MS(B)
MSE
_____________________________
SS(AB)
MS(AB)
MS(AB)
MSE
_____________________________
AB
(a-1)(b-1)
(Interaction)
Error
n - ab
SSE
Total
n-1
SS(Total)
ANOVA - 24
MSE
Same as
Other
Designs
_____________________________
_____________________________
_____________
_____________________________
Interaction
_____________________________
1. Occurs When Effects of One Factor Vary
According to Levels of Other Factor
2. When Significant, Interpretation of Main
Effects (A & B) Is Complicated
3. Can Be Detected
In Data Table, Pattern of Cell Means in One
Row Differs From Another Row
In Graph of Cell Means, Lines Cross
„
„
ANOVA - 25
_____________________________
_____________________________
_____________________________
_____________________________
_____________________________
_____________________________
_____________
_____________________________
Graphs of Interaction
_____________________________
Effects of Motivation (High or Low) & Training
Method (A, B, C) on Mean Learning Time
Interaction
Average
Response
No Interaction
High
Average
Response
High
Low
A
B
C
Low
A
B
C
_____________________________
_____________________________
_____________________________
_____________________________
_____________________________
_____________________________
ANOVA - 26
_____________
Conclusion
1. Described Analysis of Variance (ANOVA)
_____________________________
_____________________________
_____________________________
2. Explained the Rationale of ANOVA
_____________________________
3. Compared Experimental Designs
_____________________________
4. Tested the Equality of 2 or More Means
„
„
„
ANOVA - 27
Completely Randomized Design
Randomized Block Design
Factorial Design
_____________________________
_____________________________
_____________________________
_____________