1. Lesson ANOVA - D
Two-Way ANOVA

2. Objectives Analyze a two-way ANOVA design Draw interaction plots
Perform the Tukey test

3. Vocabulary
Factorial design – a design of experiment that will test n levels of k factors in a n x k factorial design
Cells – in a factorial design, a cell contains different levels of the factors
Crossed – a condition when all levels of one factor are combined with all levels of another factor
Main effect – the effect of changing levels of a single factor
Interaction effect – when changes in level of one factor result in different changes in the response variable for different levels of the second factor
Interaction Plots – graphical depictions of the interaction between factors in a factorial design

4. One-way ANOVA A one-way ANOVA is appropriate when
We wish to analyze k population means
The k populations are described by one factor that has k different levels
The k different populations do not have any particular relationship to each other … they are just different from each other
We analyze whether at least one of the population means is significantly different from the others

5. Two-way ANOVA
Sometimes, however, the populations are described by two different factors
One factor could be which of the medications is given
One factor could be the age group of the patient
We still have a set of different populations, but there is a definite structure to these populations
The related populations given the same medication
The related populations of the same age group
This is a situation for two-way ANOVA
Two-way ANOVA is an applicable method to analyze two factors and their interactions

6. Requirements Two-way Analysis of Variance (ANOVA):
The populations from which the samples are drawn must be normally distributed
The samples are independent
The populations have the same variance

7. Interaction Effect
Always test the hypothesis regarding the interaction effect. If the null hypothesis of no interaction is rejected, we do not interpret the results of the hypotheses involving the main effects because the interaction clouds those results. An interaction plot is a chart that graphically represents the interactions between the factors

8. Constructing Interaction Plots
Compute mean value of the response variable within each cell
Compute row mean value of the response variable and the column mean value of the response variable with each level of each factor
On a Cartesian plane, label the horizontal axis for each level of factor A. Vertical axis will represent the mean value of the response variable
For each level of factor A, plot the mean value of the response variable for each level of factor B
Connect the points with straight lines (you should have as many lines as you have levels of factor B)
The more the difference there is in the slope of the two lines, the stronger the evidence of interaction

9. Interaction Plots Some No Interaction Interaction Significant
Mean Response
Mean Response
A1 A2 A3
A1 A2 A3
Interaction Plot
Interaction Plot
Significant
Interaction
Significant
Interaction
Mean Response
Mean Response
A1 A2 A3
A1 A2 A3

10. Interaction Plots Summary
Parallel lines – factors A and B have no interaction
Somewhat parallel lines – factors A and B have some interaction
Significantly non-parallel lines – factors A and B have a significant interaction

11. Example
3 patients in each age group are given each of the 3 different medications
The measured effects are
Enter into Excel and run Two-way ANOVA
Age group
21 – 30
31 – 40
41 – 50
51 – 60
Med A
2, 6, 3
5, 9, 1
4, 8, 8
6, 10, 9 B
1, 3, 2
2, 8, 3
5, 2, 3
6, 1, 1 C
1, 1, 4
6, 4, 8
7, 2, 5
4, 2, 8

12. Main Effects
The effect of one of the three medications may be consistently different from the others
This is the main effect of the medication factor
Age group
21 – 30
31 – 40
41 – 50
51 – 60
Med A B C

13. Main Effects
The effect on one of the four age groups may be consistently different from the others
This is the main effect of the age group factor
Age group
21 – 30
31 – 40
41 – 50
51 – 60
Med A B C

14. Interaction Effect
Two different medications can have two different effects on two different age groups
This is the interaction effect
Age group
21 – 30
31 – 40
41 – 50
51 – 60
Med A B C

15. Two-way ANOVA Outcomes
There are three possible effects in a two-way ANOVA
The interaction between factors A and B
The main effect of factor A
The main effect of factor B
Two-way ANOVA is more complex than one- way ANOVA because of this possible interaction

16. Using Excel The data was entered into Excel as follows
The columns are the age groups
Groups of 3 rows together are the medications
Every combination of medication / age group has the same number of subjects

17. Example Summary From Excel:
“Sample” refers to the rows – the medications
“Columns” refers to the columns – the age groups
We conclude that there is no interaction effect
We conclude that there is a no age group effect
We conclude that there is medication effect

18. Example Interaction Plot
The interaction plot shows somewhat, but not significant, interaction because the lines are largely parallel but do have some intersections

19. Summary and Homework Summary Homework
A two-way analysis of variance analyzes whether two factors affect the means
The main effect of Factor A
The main effect of Factor B
The interaction of Factor A with Factor B
The main effects can be interpreted only when there is no significant interaction between the factors
Homework
pg : 1-8, 10, 11, 17