1. Two-Way Analysis of Variance
Sec 12.3
Two-Way Analysis of Variance
Bluman, Chapter 12

2. 12-3 Two-Way Analysis of Variance
In doing a study that involves a two-way analysis of variance, the researcher is able to test the effects of two independent variables or factors on one dependent variable.
In addition, the interaction effect of the two variables can be tested.
Bluman, Chapter 12

3. A case study:
Suppose we want to test the effects of two different kinds of plant food and two different kinds of soil on the growth of a certain plant.
The factors are: the different kind of plant food and the different kind of soil.
The dependent variable is the plant growth.
Bluman, Chapter 12

4. Contributing factors:
Tempterature , water, sunlight etc. are kept constant.
There will be four groups
Bluman, Chapter 12

5. Soil type I II A1 A2 P L A N T F O D
Bluman, Chapter 12

6. 2X2 Design
There are two factors and two treatments, therefore this is a two by two design.
Bluman, Chapter 12

7. Two-Way Analysis of Variance
Variables or factors are changed between two levels (i.e., two different treatments).
The groups for a two-way ANOVA are sometimes called treatment groups.
A two-way ANOVA has several null hypotheses. There is one for each independent variable and one for the interaction.
Bluman, Chapter 12

8. Hypotheses;
See page 648
Bluman, Chapter 12

9. Two-Way ANOVA Summary Table
Source
Sum of Squares
d.f.
Mean
Squares F A B
A X B
Within (error)
SSA
SSB
SSAXB
SSW
a – 1
b – 1
(a – 1)(b – 1)
ab(n – 1)
MSA
MSB
MSAXB
MSW FA FB
FAXB
Total
Bluman, Chapter 12

10. Assumptions for Two-Way ANOVA
The populations from which the samples were obtained must be normally or approximately normally distributed.
The samples must be independent.
The variances of the populations from which the samples were selected must be equal.
The groups must be equal in sample size.
Bluman, Chapter 12

11. Chapter 12 Analysis of Variance
Section 12-3
Example 12-5
Page #648
Bluman, Chapter 12

12. Example 12-5: 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. Use a two-way analysis of variance at α = 0.05.
Bluman, Chapter 12

13. Example 12-5: Gasoline Consumption
Step 1: State the hypotheses.
The hypotheses for the interaction are these:
H0: There is no interaction effect between type of gasoline used and type of automobile a person drives on gasoline consumption.
H1: There is an interaction effect between type of gasoline used and type of automobile a person drives on gasoline consumption.
Bluman, Chapter 12

14. Example 12-5: Gasoline Consumption
Step 1: State the hypotheses.
The hypotheses for the gasoline types are
H0: There is no difference between the means of gasoline consumption for two types of gasoline.
H1: There is a difference between the means of gasoline consumption for two types of gasoline.
Bluman, Chapter 12

15. Example 12-5: Gasoline Consumption
Step 1: State the hypotheses.
The hypotheses for the types of automobile driven are
H0: There is no difference between the means of gasoline consumption for two-wheel-drive and four-wheel-drive automobiles.
H1: There is a difference between the means of gasoline consumption for two-wheel-drive and four-wheel-drive automobiles.
Bluman, Chapter 12

16. Example 12-5: Gasoline Consumption
Step 2: Find the critical value for each.
Since α = 0.05, d.f.N. = 1, and d.f.D. = 4 for each of the factors, the critical values are the same, obtained from Table H as
Step 3: Find the test values.
Since the computation is quite lengthy, we will use the summary table information obtained using statistics software such as Minitab.
Bluman, Chapter 12

17. Example 12-5: Gasoline Consumption
Two-Way ANOVA Summary Table
Source
Sum of Squares
d.f.
Mean
Squares F
Gasoline A
Automobile B
Interaction A X B
Within (error)
3.920
9.680
54.080
3.300 1 4
0.825
4.752
11.733
65.552
Total
70.890 7
Bluman, Chapter 12

18. Example 12-1: Lowering Blood Pressure
Step 4: Make the decision.
Since FB = and FAXB = are greater than the critical value 7.71, the null hypotheses concerning the type of automobile driven and the interaction effect should be rejected.
Step 5: Summarize the results.
Since the null hypothesis for the interaction effect was rejected, it can be concluded that the combination of type of gasoline and type of automobile does affect gasoline consumption.
Bluman, Chapter 12