ANOVA - BI FACTORIAL ANOVA (2- WAY ANOVA)

BI FACTORIAL ANOVA (2- WAY ANOVA)

1. BI FACTORIAL ANOVA (2- WAY ANOVA)

2. Factorial ANOVA One-Way ANOVA = ANOVA with one IV with 1+ levels and one DV Factorial ANOVA = ANOVA with 2+ IV’s and one DV Factorial ANOVA Notation: 2 x 3 x 4 ANOVA The number of numbers = the number of IV’s The numbers themselves = the number of levels in each IV

3. Factorial ANOVA 2 x 3 x 4 ANOVA = an ANOVA with 3 IV’s, one of which has 2 levels, one of which has 3 levels, and the last of which has 4 levels Why use a factorial ANOVA? Why not just use multiple one-way ANOVA’s? Increased power – with the same sample size and effect size, a factorial ANOVA is more likely to result in the rejection of H o aka with equal effect size and probability of rejecting H o if it is true ( α ), you can use fewer subjects (and time and money)

4. Factorial ANOVA Why use a factorial ANOVA? Why not just use multiple one-way ANOVA’s? With 3 IV’s, you’d need to run 3 one-way ANOVA’s, which would inflate your α -level However, this could be corrected with a Bonferroni Correction 3. The best reason is that a factorial ANOVA can detect interactions , something that multiple one-way ANOVA’s cannot do

5. Factorial ANOVA Interaction: when the effects of one independent variable differ according to levels of another independent variable Ex. We are testing two IV’s, Gender (male and female) and Age (young, medium, and old) and their effect on performance If males performance differed as a function of age, i.e. males performed better or worse with age, but females performance was the same across ages, we would say that Age and Gender interact , or that we have an Age x Gender interaction

6. Factorial ANOVA Interaction: Presented graphically: Note how male’s performance changes as a function of age while females does not Note also that the lines cross one another, this is the hallmark of an interaction, and why interactions are sometimes called cross-over or disordinal interactions

7. Factorial ANOVA Interactions: However, it is not necessary that the lines cross, only that the slopes differ from one another I.e. one line can be flat, and the other sloping upward, but not cross – this is still an interaction See Fig. 17.2 on page 410 in the text for more examples

8. Factorial ANOVA As opposed to interactions, we have what are called main effects : the effect of an IV independent of any other IV’s This is what we were looking at with one-way ANOVA’s – if we have a significant main effect of our IV, then we can say that the mean of at least one of the groups/levels of that IV is different than at least one of the other groups/levels

9. Factorial ANOVA Finally, we also have simple effects : the effect of one group/level of our IV at one group/level of another IV Using our example earlier of the effects of Gender (Men/Women) and Age (Young/Medium/Old) on Performance, to say that young women outperformed other groups would be to talk about a simple effect

10. Factorial ANOVA We then calculate the Grand Mean ( ) This remains ( Σ X)/N, or all of our observations added together, divided by the number of observations We can also calculate SS total , which is also calculated the same as in a one-way ANOVA

11. Factorial ANOVA Next we want to calculate our SS terms for our IV’s, something new to factorial ANOVA SS IV = nx Σ ( - ) 2 n = number of subjects per group/level of our IV x = number of groups/levels in the other IV

12. Factorial ANOVA SS IV = nx Σ ( - ) 2 Subtract the grand mean from each of our levels means For SS gender , this would involve subtracting the mean for males from the grand mean, and the mean for females from the grand mean Note: The number of values should equal the number of levels of your IV Square all of these values Add all of these values up Multiply this number by the number of subjects in each cell x the number of levels of the other IV Repeat for any IV’s Using the previous example, we would have both SS gender and SS age

13. Factorial ANOVA Next we want to calculate SS cells , which has a formula similar to SS IV SS cells = Subtract the grand mean from each of our cell means Note: The number of values should equal the number of cells Square all of these values Add all of these values up Multiply this number by the number of subjects in each cell

14. Factorial ANOVA Now that we have SS total , the SS’s for our IV’s, and SS cells , we can find SS error and the SS for our interaction term, SS int SS int = SS cells – SS IV1 – SS IV2 – etc… Going back to our previous example, SS int = SS cells – SS gender – SS age SS error = SS total – SS cells

15. Two Way ANOVA

16. Data required When 2 independent variables (Nominal/categorical) have an effect on one dependent variable (ordinal or ratio measurement scale) Compares relative influences on Dependent Variable Examine interactions between independent variables Just as we had Sums of Squares and Mean Squares in One-way ANOVA , we have the same in Two-way ANOVA .

17. Two way ANOVA Include tests of three null hypotheses: Means of observations grouped by one factor are same; Means of observations grouped by the other factor are the same; and There is no interaction between the two factors. The interaction test tells whether the effects of one factor depend on the other factor

18. Example - we have test score of boys & girls in age group of 10 yr,11yr & 12 yr. If we want to study the effect of gender & age on score. Two independent factors- Gender, Age Dependent factor - Test score

19. Ho -Gender will have no significant effect on student score Ha - Ho - Age will have no significant effect on student score Ha - Ho – Gender & a ge interaction will have no significant effect on student score Ha -

20. Two-way ANOVA Table Source of Variation Degrees of Freedom Sum of Squares Mean Square F -ratio P-value Factor A r - 1 SS A MS A F A = MS A / MS E Tail area Factor B c - 1 SS B MS B F B = MS B / MS E Tail area Interaction ( r – 1) ( c – 1) SS AB MS AB F AB = MS AB / MS E Tail area Error (within) rc ( n – 1) SS E MS E Total rcn - 1 SS T

21. Example with SPSS Example: Do people with private health insurance visit their Physicians more frequently than people with no insurance or other types of insurance ? N=86 Type of insurance - 1.No insurance 2.Private insurance 3. TRICARE No. of visits to their Physicians(dependent variable) Gender -M 1 -F

22. References Methods in Biostatistics by BK Mahajan Statistical Methods by SP Gupta Basic & Clinical Biostatistics by Dawson and Beth Munro’s statistical methods for health care research