1. CHAPTER 3 Analysis of Variance (ANOVA) PART 3 = TWO-WAY ANOVA WITH REPLICATION (FACTORIAL EXPERIMENT)

2. Two-Way ANOVA with Replication (Factorial Experiment)
Factorial experiments and their corresponding ANOVA computations are valuable designs when simultaneous conclusions about two or more factors are required.
Replication means an independent repeat of each factor combination (interaction)
Purpose: Examines (1) the effect of Factor A on the dependent variable, y; (2) the effect of Factor B on the dependent variable, y; along with (3) the effects of the interactions between different levels of the two factors on the dependent variable , y.

3. Factorial Experiment, cont.
Effects model for factorial experiment:

4. Two-Factor Factorial Experiment
REMEMBER THIS
Three Sets of Hypothesis:
i. Factor A Effect:
H0: 1 =  2 = ... =  a =0
H1: at least one i  0
ii. Factor B Effect:
H0: 1 = 2 = ... = b =0
H1: at least one j ≠ 0
iii. Interaction Effect:
H0: ( )ij = 0 for all i,j
H1: at least one ( )ij  0
H0: µ1 = µ2 = ... = µa *
H1: µi  µk for at least one pair (i,k)
H0: µ1 = µ2 = ... = µb *
H1: µi  µk for at least one pair (i,k)
H0: µAB1 = µAB2 = ... = µABb *
H1: µABi  µABk for at least one pair (i,k)

5. Two-Factor Factorial Experiment
CAN USE THIS INSTEAD
Three Sets of Hypothesis:
i. Factor A Effect:
H0: 1 =  2 = ... =  a =0
H1: at least one i  0
ii. Factor B Effect:
H0: 1 = 2 = ... = b =0
H1: at least one j ≠ 0
iii. Interaction Effect:
H0: ( )ij = 0 for all i,j
H1: at least one ( )ij  0
H0: There is no difference in means of factor A
H1: There is a difference in means of factor A
H0: There is no difference in means of factor B
H1: There is a difference in means of factor B
H0: There is no interaction effect between
factor A and B for/on ………
H1: There is an interaction effect between

6. Two-Factor Factorial Experiment, cont.
Format for data: Data appear in a grid, each cell having two or more entries. The number of values in each cell is constant across the grid and represents r, the number of replications within each cell.
Calculations:
Sum of squares total (SST) = sum of squared differences between each individual data value (regardless of group membership) minus the grand mean, , across all data... total variation in the data (not variance).

7. Two-Factor Factorial Experiment, cont.
Calculations, cont.:
Sum of squares Factor A (SSA) = sum of squared differences between each group mean for Factor A and the grand mean, balanced by sample size... between-factor-groups variation (not variance).
Sum of squares Factor B (SSB) = sum of squared differences between each group mean for Factor B and the grand mean, balanced by sample size... between-factor-groups variation (not variance).

8. Two-Factor Factorial Experiment, cont.
Calculations, cont.:
Sum of squares Error (SSE) = sum of squared differences between individual values and their cell mean... within-groups variation (not variance).
Sum of squares Interaction:
SSAB = SST – SSA – SSB – SSE

9. Two-Factor Factorial Experiment, cont.
Calculations, cont.:
Mean Square Factor A (MSA) = SSA/(a – 1),
where a = the number of levels of Factor A .
Mean Square Factor B (MSB) = SSB/(b – 1),
where b = the number of levels of Factor B .
Mean Square Interaction (MSAB) = SSAB/(a – 1)(b – 1).
Mean Square Error (MSE) = SSE/ab(r – 1),
where ab(r – 1) = the degrees of freedom on error .

10. Two-Factor Factorial Experiment, cont.
Calculations - F-Ratios:
F-Ratio, Factor A = MSA/MSE ,
where numerator degrees of freedom are a – 1 and denominator degrees of freedom are ab(r – 1). This F-ratio is the test statistic for the hypothesis that the Factor A group means are equal. To reject the null hypothesis means that at least one Factor A group had a different effect on the dependent variable than the rest.
F-Ratio, Factor B = MSB/MSE,
where numerator degrees of freedom are b – 1 and denominator degrees of freedom are ab(r – 1). This F-ratio is the test statistic for the hypothesis that the Factor B group means are equal. To reject the null hypothesis means that at least one Factor B group had a different effect on the dependent variable than the rest.

11. Two-Factor Factorial Experiment, cont.
Calculations - F-Ratios:
F-Ratio, Interaction = MSAB/MSE,
where numerator degrees of freedom are (a – 1)( b – 1) and denominator degrees of freedom are ab(r – 1). This F-ratio is the test statistic for the hypothesis that Factors A and B operate independently. To reject the null hypothesis means that there is some relationship where levels of Factor A operate differently with different levels of Factor B.
If F-Ratio>F or p-value< , reject H0 at the  level.

12. Two-Factor Factorial Experiment, cont. Two-Factor ANOVA Table
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Square F
p-Value
Factor A
SSA
a-1
Factor B
SSB
b-1
Interaction
SSAB
(a-1)(b-1)
Error
SSE
ab(r-1)
Total
SST
abr-1

13. Two-Factor Factorial Experiment,
Example 4.4: State of Ohio Wage Survey
A survey was conducted of hourly wages for a
sample of workers in two industries at three locations
in Ohio. Part of the purpose of the survey was to
determine if differences exist in both industry type and location. The sample data are shown on the next slide.

14. Two-Factor Factorial Experiment- An Example
Industry
Cincinnati
Cleveland
Columbus I
$12.10
$11.80
$12.90
11.80
11.20
12.70
12.10
12.00
12.20 II
12.40
12.60
13.00
12.50
Factors:
Factor A: Industry Type (2 levels)
Factor B: Location (3 levels)
Replications:
Each experimental condition is repeated 3 times

15. Two-Factor Factorial Experiment- An Example
1. Hypothesis for this analysis:
i. Interaction Effect:
H0: There is no interaction effect between industry type and location to hourly wages.
H1: There is no interaction effect between industry type and location to hourly wages.
ii. Factor A Effect:
H0: There is no difference between the means of hourly wages for two types of industry
H1: There is a difference between the means of hourly wages for two types of industry
iii. Factor B Effect:
H0: There is no difference between the means of hourly wages for the three location
H1: There is a difference between the means of hourly wages for the three location

16. Two-Factor Factorial Experiment- An Example Result: Two-Way ANOVA Table
2. Test Statistics : (F test)
SSA – MSA v. SSE = SST – SSA – SSB – SSAB
SSB – MSB vi. MSE
iii. SSAB – MSAB vii. F test (A, B & AB)
iv. SST viii. ANOVA TABLE
Source of
Variation
Sum of
Squares
Degrees of
Freedom
Mean
Square F
p-Value
Factor A
0.50 1
4.19
0.06
Factor B
1.12 2
0.56
4.69
0.03
Interaction
0.37
0.19
1.55
0.25
Error
1.43 12
0.12
Total
3.42 17

17. Two-Factor Factorial Experiment-An Example
3. F (alfa) value (critical value)
Industries: Fa = 4.75
Locations: Fa = 3.89
Interaction: Fa = 3.89
4. Rejection region : (Draw picture)
Industries: F = 4.19 < Fa = Fail to Reject H0
Locations: F = 4.69 > Fa = Reject H0
Interaction: F = 1.55 < Fa = Fail to Reject H0
5. Conclusions Using the Critical Value Approach
The means of hourly wages do not differ by industry type.
The means of hourly wages do differ by location.
There is no interaction between industry type and location to hourly wages.

18. Two-Factor Factorial Experiment-An Example
5. Conclusions Using the p-Value Approach
Industries: p-value = .06 > a = .05
The means of hourly wages do not differ by industry type.
Locations: p-value = .03 < a = .05
The means of hourly wages do differ by location.
Interaction: p-value = .25 > a = .05
There is no interaction between industry type and location to hourly wages.

19. Two-Factor Factorial Experiment-An Example
Gasoline Consumption
A researcher wishes to see whether the type of gasoline used and the type of automobile driven have any effect on gasoline consumption. Two types of gasoline, regular and high-octane, will be used, and two types of automobiles, two-wheel- and four-wheel drive, will be used in each group. There will be two automobiles in each group, for a total of eight automobiles used. Using a two-way analysis of variance, perform the analyses. The data (in miles per gallon) are shown here, and the summary table is given in the following table.

20. Factor B = Independent Variable
Gasoline Type
Type of Automobile
Two-wheel-drive
Four-wheel-drive
Regular
26.7
28.6
25.2
29.3
High-Octane
32.3
26.1
32.8
24.2
Factor A = Independent Variable
Response Variable =
Dependent Variable

21. STEP 1 i. Interaction Effect:
H0: There is no interaction effect between type of gasoline used and type of
automobile a person drive on gasoline consumption.
H1: There is an interaction effect between type of gasoline used and type of
ii. Factor A Effect:
H0: There is no difference between the means of gasoline consumption for
the two types of gasoline
H1: There is a difference between the means of gasoline consumption for the
two types of gasoline
iii. Factor B Effect:
two-wheel- and four-wheel-drive automobiles.
H1: There is a difference between the means of gasoline consumption for two-
wheel- and four-wheel-drive automobiles.

22. STEP 2 B
Two-wheel-drive (mean)
Four-wheel-drive (mean)
Overall Mean A
Overall Mean A
Regular (mean)
25.95
28.95
27.45
High-Octane mean)
32.55
25.15
28.85
28.15
Overall Mean B
29.25
27.05

23. Disordinal Interaction
There is a SIGNIFICANCE interaction between automobile type and gasoline type for the gasoline consumption.

24. No Interaction (parallel)
Ordinal Interaction
There is an interaction but not significant. The main effect can be interpreted independently
No Interaction (parallel)
There is no significant interaction . The main effect can be interpreted independently

25. STEP 2
SSAB = SST – SSA – SSB – SSE

26. STEP 2 a=2, b=2, n=2 Source of Variation Sum of Squares Degrees of
Freedom
Mean
Square F
Factor A
(Gasoline)
3.920
a – 1 = 1
4.752
Factor B
(Automobile)
9.680
b – 1 = 1
11.733
Interaction
(AB)
54.080
(a–1)(b–1)
= 1
65.552
Error (within)
3.300
ab(n-1) = 4
0.825
Total
70.980 7

27. STEP 3, 4 & 5 Interaction (AxB) F(0.05, 1, 4) =7.71
F-ratio (65.552) > F-critical value(7.71)
There is an interaction effect between type of gasoline used and type of automobile a person drive on gasoline consumption.
Factor A
F-ratio (4.752) < F-critical value(7.71)
Fail to reject Ho. There is no difference between the means of gasoline consumption for the two types of gasoline.
Factor B
F-ratio (11.733) < F-critical value(7.71)
Reject Ho. There is a difference between the means of gasoline consumption for two-wheel- and four-wheel-drive automobiles.

28. EXERCISE TEXT BOOK PAGE 153 ( Example 9.6) PAGE 157 ( 1 & 2)
2 step
5 IMPORTANT STEP:
HYPOTHESIS TESTING
TEST STATISTIC – F TEST
F (alfa) – VALUE (CRITICAL VALUE)
REJECTION REGION
CONCLUSION
SSA – MSA
SSB – MSB
SSAB – MSAB
SST
SSE = SST – SSA – SSB - SSAB
MSE
F TEST SSA= MSA/MSE
F TEST SSB = MSB/MSE
F TEST SSAB = MSAB/MSE
BUILD ANOVA TABLE

29. THE END…………