Factorial Analysis of Variance

1. Factorial Analysis of Variance One dependent variable, more than one independent variable (“factor”)

2. Two factors, more reality • Imagine you want to describe what influences GPA, body fat, a team’s winning %, the outcome of an electoral poll… • Surely, the person who says “it was because [just one thing]” is a fool? • Well, then people who conduct one way ANOVA’s are fools! • More variables simply get closer to the truth

3. Two factors, more reality • But there’s more to it than that (of course) • Consider this experiment: • Take 2 sets of golfers: 1 set (A1) is high anxious, 1 set (A2) is low anxious • Assign 1/3 of each set of golfers to a different performance scenario: Low pressure (B1), Moderate pressure (B2), High pressure (B3)

4. Two factors, more reality • So for assignment to groups we get:

5. Two factors, more reality • Suppose that the performance scores are…

6. MAIN EFFECTS • What do we find? • We can consider the overall effect of anxiety (Factor A) on performance • The null hypothesis here would be • This is analogous to doing a 1-way ANOVA on the row means of MA1 (8) and MA2 (4) NB: if you were to do a 1-way ANOVA, you’d ignore the effect of situation completely

7. MAIN EFFECTS • This overall effect of anxiety is called the main effect of anxiety

8. MAIN EFFECTS • What do we find? • We can also consider the overall effect of situation (Factor B) on performance • The null hypothesis here would be • This is analogous to doing a 1-way ANOVA on the row means of MB1 (4.5), MB2 (7) and MB3 (6.5) NB: here, you’d ignore the effect of anxiety completely

9. MAIN EFFECTS • This overall effect of situation is called the main effect of situation • In each of the main effects, note that each mean within the main effect has been computed by averaging across levels of the factor not considered in the main effect • This is how it is ignored, statistically. Its effects are, quite literally, averaged out WHENEVER YOU INTERPRET A MAIN EFFECT, YOU SHOULD PAY ATTENTION TO THE FACT THAT IT AVERAGES ACROSS LEVELS OF THE OTHER FACTOR – ESPECIALLY WHEN YOU GET…

10. 8-6 = 2 11-2 = 9 5-4 = 1 INTERACTIONS • Note the difference between each pair of means in our original table of data

11. INTERACTIONS • The magnitude of the difference changes depending on the situation 8-6 = 2 11-2 = 9 5-4 = 1

12. INTERACTIONS • The magnitude of the difference changes depending on the situation • In other words, the effect of anxiety on performance depends on the situation in which the participants are asked to perform • In other words, the situation moderates the effect of anxiety on performance • In other words, the anxiety-performance relationship differs depending on the situation

13. INTERACTIONS • Also, the magnitude of the difference changes depending on anxiety level • In other words, the effect of situation on performance depends on the anxiety level of the participants • In other words, anxiety level moderates the effect of situation on performance • In other words, the situation-performance relationship differs depending on anxiety level

14. INTERACTIONS • Whatever way you slice it, it’s the same thing, and it’s easier to see in a graph: Ordinalinteraction = lines do not cross

15. INTERACTIONS • The essential point is, when the lines are non-parallel (significantly so), you have an interaction, and the effect of one factor on the dependent variable depends on the level of other factor being considered Non-parallelism implies interaction

16. INTERACTIONS • So, is this an interaction?

17. INTERACTIONS • How about this?

18. INTERACTIONS • This is a disordinal interaction

19. Why bother with interactions? • With figure B, it seems we have a main effect of anxiety level • That implies that the effect of anxiety on performance can be generalized across different pressure conditions. • With figures A and C, generalization across situations would be a serious mistake • A main effect would fail to acknowledge that the effect of anxiety changes across situations • In which figure, A or C, would the main effect of anxiety be more likely?

20. Why bother with interactions? • Note: With disordinal interactions, post-hoc tests can be confusing, as they may result in no apparent significant differences. Here the interaction is not necessarily caused by a large difference between pairs of means, but may be a function of the change in direction of differenceacross pairs of means as well

21. Statistically speaking… • With Factor A, Factor B, and the interaction A x B, the sums of squares are as follows: Recall: But: So:

22. Statistically speaking… • These are converted to variance estimates by dividing by the d. of f.: So… And:

23. Statistically speaking… • And remember, these are estimates:

24. Statistically speaking… • Or, in another form: Total sum of squares Within-groups SS Between-groups SS SS associated with A SS associated with B SS associated with interaction

25. Statistically speaking… • So the F-tests are: For Factor A:

26. Statistically speaking… • So the F-tests are: For Factor B:

27. Statistically speaking… • So the F-tests are: …and for the interaction:

28. Assumptions • Similar to 1-way ANOVA and independent samples t-test • Assumptions now stated for cells, not groups. It is assumed that… • Observations are independent (uncorrelated) from one cell to the next • 2 is the same for all cells (homogeneity of variance) • Cell populations are normally distributed • Last 2 are mostly a concern for small sample sizes

29. Assumptions • Note – we have been discussing equal cell sizes throughout. • This is important, as it guarantees independence of statistical effects in the analysis • If cell sizes are unequal, certain adjustments must be made… • These involve reporting different types of sums of squares (Types 1 to 3 as reported in SPSS, Stevens, 1986, 1996) But that is probably a step too far for this class…just make sure your cell sizes are equal!

30. Factorial ANOVA in SPSS • Data considerations • Dependent variable: interval/ratio, normally distributed within each cell of the analysis) • Independent variables: must be discrete categories • If the independent variable is continuous, the number of categories can be created artificially by doing a median split or quartile split, BUT you’d be better off with regression • Must be independent (see slide 28)

31. There is now more than one grouping factor, so “group” is not a good name for either Factorial ANOVA in SPSS Dependent variable • Data considerations Each row is one subject

32. Factorial ANOVA in SPSS 1 dependent variable, so choose univariate • Performing the analysis The general linear model is the same family of statistical techniques as we used in regression – we could dummy code these variables and get exactly the same answer using regression techniques

33. Factorial ANOVA in SPSS • Performing the analysis Here are the variables as listed in the data file They just need to be slid over to the right places

34. Further options This is just a descriptor – it has no function in the analysis Here is the dependent variable Here are the factors Factorial ANOVA in SPSS • Performing the analysis

35. Factorial ANOVA in SPSS • Performing the analysis “Plots” lets you request a graph of the 2 factors

36. 1. Slide the factors across to the correct boxes Factorial ANOVA in SPSS 4. Click “continue” to do so! • Performing the analysis 2. Click on “Add” to request the plot 3. Selected plot appears here

37. Factorial ANOVA in SPSS • Performing the analysis “Post Hoc” lets you request follow-ups, but only to the main effects – and that is pointless here, because…?

38. Factorial ANOVA in SPSS 4. Continue • Performing the analysis If you wanted to do a post hoc on the main effects, you’d simply: 1. select the variables 2. Slide them over 3. Select the post hoc test

39. Factorial ANOVA in SPSS • Performing the analysis “Options” lets you request descriptives, effect sizes, power statistics, and homogeneity tests. It is the most important box to choose

40. Factorial ANOVA in SPSS • Performing the analysis Here you see them: Descriptives Effect sizes Power calculation Homogeneity tests

41. Factorial ANOVA in SPSS • Performing the analysis Finally, click “ok” once you’ve specified what you want

42. Factorial ANOVA in SPSS • The output Here’s the table showing the allocation of subjects to levels of each factor Here’s the table showing the descriptive stats – note the balanced design!

43. Factorial ANOVA in SPSS • The output Here’s the homogeneity test – we’re OK – it is not significant, which means the assumption is met

44. Observed Power Significance values Main Effects 2 - like R2’s for each effect Interaction Factorial ANOVA in SPSS • The output Here’s the ANOVA summary table

45. Factorial ANOVA in SPSS • The output Here, we have significant main effects AND a significant interaction

46. Factorial ANOVA in SPSS • The output It is normal in such circumstances to report the means for the main effects but to note that these are superceded by the interaction, which implies that the main effects are not genuine

47. Factorial ANOVA in SPSS • The output LASTLY, here is the plot. You can clearly (?) see the main effects are not genuine, and that the only way to really describe what is going on here is through the interaction

48. Factorial ANOVA in SPSS • More on follow-up tests • What to do if the homogeneity assumption is not met • What to do if you have multiple measures on each subject (repeated measures!)

49. More on follow-up tests • SPSS does not do pair-wise comparisons for interactions, BUT many periodicals expect some kind of follow-up • So here’s how: q is looked up in table;  is sig. level, k is n cells, N is total people in expt, n is # subjects p/cell, MSW is from SPSS output

50. More on follow-up tests Let’s say we want to look at the interaction MSW from SPSS output = 2.25