Randomized Block Design & Factorial Design-1
Randomized Block
Design
Randomized Block
Design Example
Factor (Diskette Brand)
Level 2
Level 3
1. Experimental Units (Subjects) Are
Assigned Randomly to Treatments
Factor
Level 1
Levels
(Treatments)
IBM
2. Uses Blocking Variable Besides
Independent (Treatment) Variable
Experimental
Units
Permits Better Assessment of Treatment
3. Analyzed by Randomized Block F Test
ANOVA - 1
Stores
Stores
Blocking
Variable
(Store)
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
Randomized Block F Test
Assumptions
Randomized Block F Test
1. Tests the Equality of 2 or More (p)
Population Means When Blocking
Variable Used
1. Normality
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
Randomized Block F Test
Basic Idea
1. SS(Total) & SST Are Same As
Completely Randomized Design
f(X)
µ1 = µ2 = µ3
Ha: Not All µj Are Equal
Independent Random Samples are Drawn
4. No Interaction Between Blocks &
Treatments
ANOVA - 3
Populations have Equal Variances
3. Independence of Errors
Error Variation Is Reduced
3. Used to Analyze Randomized Block
Designs
Populations are Normally Distributed
2. Homogeneity of Variance
2. More Efficient Analysis Than One-Way
ANOVA
ANOVA - 5
Stores
NEC FUJI
X
2. Error Variation (SSE) Is Different
f(X)
µ1 = µ 2 µ 3
X
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
Randomized Block Design & Factorial Design-2
Randomized Block F Test
Total Variation Partitioning
Randomized Block F Test
Summary Table
Source of
Variation
Total
Total Variation
Variation
SS(Total)
Variation
Variation Due
Dueto
to
Treatment
Treatment
SST
Variation
Variation Due
Dueto
to
Blocking
Blocking
Variation
Variation Due
Dueto
to
Random
Random Sampling
Sampling
SSB
SSE
ANOVA - 7
Degrees
of
Freedom
Sum of
Squares
Mean
Square
(Variance)
F
Among
Treatments
p-1
SST
MST
MST
MSE
Among
Blocks
b-1
SSB
MSB
SSE
n - p - b +1
MSB
MSE
SSE
MSE
Error
Total
n- 1
SS(Total)
Same as
Completely
Randomized
Design
ANOVA - 8
Formula
Formula
Sum of squares between Treatments(SST):
Sum of squares Total (SS(Total)):
p
SST = ∑ b ⋅ ( x j − x ) 2
b
p
SS (Total ) = ∑∑ ( x ij − x ) 2
j =1
i =1 j =1
Sum of squares for Blocks (SSB):
p
Sum of squares of sampling error:
SSE = SS(Total) - SST - SSB
SSB = ∑ p ⋅ ( x i − x ) 2
i =1
ANOVA - 9
ANOVA - 10
Randomized Block F Test
Thinking Challenge
Randomized Block F Test
Critical Value
Degrees of Freedom Are
(p -1) & (n
(n - b - p + 1)
You’re a market research
analyst. Using the computer,
is there a difference in mean
diskette price at 4 stores (.05)?
Reject H0
Do Not
Reject H0
0
α
Fa ( p−1,an - b - p + 1f
Always OneOne-Tail!
F
Store
1
2
3
4
© 19841984-1994 T/Maker Co.
ANOVA - 11
ANOVA - 12
IBM NEC
6
4
11
7
15
11
24
22
FUJI
2
3
7
20
FUJI
NEC
IBM
Randomized Block Design & Factorial Design-3
... toner was low. Only a
portion of output printed.
Source of
Variation
Degrees
of
Freedom
Sum of
Squares
Among
Treatments
Mean
Square
(Variance)
F
72
Among
Blocks
3
186
Error
Total
638
ANOVA - 13
Randomized Block F Test
Solution*
Source of
Variation
Sum of
Squares
Mean
Square
(Variance)
F
Among
3-1=2
Treatments
72
36
27
Among
Blocks
4-1=3
558
186
139.5
Error
1212-3-4+1
=6
8
1.33
Total
1212- 1 = 11
638
ANOVA - 14
Randomized Block F Test
Solution*
H0: µ1 = µ2 = µ3
Ha: Not All Equal
α = .05
ν1 = 2 ν2 = 6
Critical Value(s):
Test Statistic:
F=
α = .05
0
5.14
F
MST 36
=
= 27
MSE 1.33
Decision:
Reject at α = .05
Conclusion:
There Is Evidence Mean
Prices Are Different
ANOVA - 15
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
ANOVA - 16
Factorial Design
Example
Factor 2 (Training Method)
Factor Level 1 Level 2 Level 3
Levels
☺
Factor 1
11 hr.☺
(Motivation) Level 2 27 hr.
(Low)
29 hr.
Level 1 19 hr.
(High)
ANOVA - 17
Degrees
of
Freedom
☺ 22 hr.☺
17 hr.☺ 31 hr.☺
25 hr. 31 hr.
30 hr. 49 hr.
20 hr.
Treatment
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
Randomized Block Design & Factorial Design-4
Two-Way ANOVA
Assumptions
Two-Way ANOVA
1. Normality
1. Tests the Equality of 2 or More
Population Means When Several
Independent Variables Are Used
2. Homogeneity of Variance
2. Same Results as Separate One-Way
ANOVA on Each Variable
Populations are Normally Distributed
No Interaction Can Be Tested
Populations have Equal Variances
3. Independence of Errors
Independent Random Samples are Drawn
3. Used to Analyze Factorial Designs
ANOVA - 19
ANOVA - 20
Two-Way ANOVA
Data Table
Factor
A
1
X111
1
X112
X211
2
X212
:
:
Xa11
a
Xa12
Factor B
2
...
X121 ...
X122 ...
X221 ...
X222 ...
:
:
Xa21 ...
Xa22 ...
Two-Way ANOVA
Null Hypotheses
1. No Difference in Means Due to Factor A
b
X1b1
X1b2
X2b1
X2b2
:
Xab1
Xab2
Observation k
Xijk
Level i Level j
Factor Factor
B
A
ANOVA - 21
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
Source of Degrees of Sum of
Variation
Freedom Squares
Total
Total Variation
Variation
SS(Total)
Variation
Variation Due
Dueto
to
Treatment
Treatment AA
Variation
Variation Due
Dueto
to
Treatment
Treatment BB
SSB
SSA
ANOVA - 23
Two-Way ANOVA
Summary Table
Variation
Variation Due
Dueto
to
Interaction
Interaction
Variation
Variation Due
Dueto
to
Random
Random Sampling
Sampling
SS(AB)
SSE
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
MSE
AB
(a(a-1)(b1)(b-1)
(Interaction)
Error
n - ab
SSE
Total
n-1
SS(Total)
ANOVA - 24
Same as
Other
Designs
Randomized Block Design & Factorial Design-5
Interaction
Graphs of Interaction
1. Occurs When Effects of One Factor Vary
According to Levels of Other Factor
Effects of Motivation (High or Low) & Training
Method (A, B, C) on Mean Learning Time
2. When Significant, Interpretation of Main
Effects (A & B) Is Complicated
Interaction
Average
Response
No Interaction
High
Average
Response
High
3. Can Be Detected
ANOVA - 25
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
A
ANOVA - 26
Conclusion
Low
In Data Table, Pattern of Cell Means in One
Row Differs From Another Row
In Graph of Cell Means, Lines Cross
Completely Randomized Design
Randomized Block Design
Factorial Design
B
C
Low
A
B
C