1. Lesson 13 - 4 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 1.Compute mean value of the response variable within each cell 2.Compute row mean value of the response variable and the column mean value of the response variable with each level of each factor 3.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 4.For each level of factor A, plot the mean value of the response variable for each level of factor B 5.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. Mean Response A 1 A 2 A 3 Interaction Plot Mean Response A 1 A 2 A 3 Interaction Plot Mean Response A 1 A 2 A 3 Interaction Plot Mean Response A 1 A 2 A 3 Interaction Plot No Interaction Significant Interaction Some Interaction Interaction Plots Significant Interaction

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 – 3031 – 4041 – 5051 – 60 Med A2, 6, 35, 9, 14, 8, 86, 10, 9 B1, 3, 22, 8, 35, 2, 36, 1, 1 C1, 1, 46, 4, 87, 2, 54, 2, 8

12. 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

13. 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

14. Example Interaction Plot

15. Summary and Homework Summary –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 723 - 727: 1-8, 10, 11, 17