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