_____________________________

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

_____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

ANOVA - 1

_____________ _____________________________

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

_____________________________ _____________________________

_____________________________ _____________________________ _____________________________ _____________

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

_____________________________ _____________________________ _____________________________ _____________________________ _____________

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

_____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________

_____________________________ 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

_____________________________ _____________________________ _____________________________ _____________________________ _____________ _____________________________

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

_____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________

_____________________________

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

_____________________________ _____________________________

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

_____________________________ _____________________________ _____________________________ _____________

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

_____________________________ _____________________________ _____________________________

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

_____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________

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

_____________________________ _____________________________ _____________________________ _____________________________ _____________________________

Decision: Reject at α = .05 Conclusion: There Is Evidence Mean Prices Are Different

_____________________________ _____________________________ _____________________________ _____________

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

(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

_____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________

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

_____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________________________ _____________

_____________________________

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

_____________________________ _____________________________ _____________________________ _____________________________

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

_____________________________ _____________________________ _____________________________

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

_____________________________ _____________________________ _____________________________ _____________