1. Factorial ANOVA
2 or More IVs

2. Questions (1) What are main effects in ANOVA?
What are interactions in ANOVA? How do you know you have an interaction?
What does it mean for a design to be completely crossed? Balanced? Orthogonal?
Describe each term in a linear model like this one:  

3. Questions (2)
Correctly interpret ANOVA summary tables. Identify mistakes in such tables. What’s the matter with this one?
Source SS Df MS F A
512
2 (J-1)
128 B
108
1 (K-1) 54
AxB 96
2 (J-1)(K-1) 48 24
Error 12
5 N-JK 2

4. Questions (3)
Find correct critical values of F from a table or software for a given design.
How does post hoc testing for factorial ANOVA differ from post hoc testing in one-way ANOVA?
Describe a concrete example of a two-factor experiment. Why is it interesting and/or important to consider both factors in one experiment?

5. 2-way ANOVA
So far, 1-Way ANOVA, but can have 2 or more IVs. IVs aka Factors.
Example: Study aids for exam
IV 1: workbook or not
IV 2: 1 cup of coffee or not
Workbook (Factor A)
Caffeine (Factor B) No
Yes
Caffeine only
Both
Neither (Control)
Workbook only

6. Main Effects (R & C means)
N=30 per cell
Workbook (Factor A)
Row Means
Caffeine
(Factor B) No
Yes
Caff
=80
SD=5
Both
=85
82.5
Control
=75
Book
77.5
Col Means 80

7. Main Effects and Interactions
Main effects seen by row and column means; Slopes and breaks.
Interactions seen by lack of parallel lines.
Interactions are a main reason to use multiple IVs

8. Single Main Effect for B
(Coffee only)

9. Single Main Effect for A
(Workbook only)

10. Two Main Effects; Both A & B
Both workbook and coffee

11. Interaction (1)
Interactions take many forms; all show lack of parallel lines.
Coffee has no effect without the workbook.

12. Interaction (2)
People with workbook do better without coffee; people without workbook do better with coffee.

13. Interaction (3)
Coffee always helps, but it helps more if you use workbook.

14. Labeling Factorial Designs
Levels – each IV is referred to by its number of levels, e.g., 2X2, 3X2, 4X3 designs. Two by two factorial ANOVA.
Cell – treatment combination.
Completely Crossed designs –each level of each factor appears at all levels of other factors (vs. nested designs or confounded designs).
Balanced – each cell has same n.
Orthogonal design – random sampling and assignment to balanced cells in completely crossed design.

15. Review What are main effects in ANOVA?
What are interactions in ANOVA? How do you know you have an interaction?
What does it mean for a design to be completely crossed? Balanced? Orthogonal?

16. Population Effects for 2-way
Population main effect associated with the treatment Aj (first factor):
Population main effect associated with treatment Bk (second factor):
The interaction is defined as , so the linear model is:
Each person’s score is a deviation from a cell mean (error). The cell means vary for 3 reasons.
The interaction is a residual:

17. Expected Mean Squares
(Factor A has J levels; factor B has K levels; there are n people per cell.)
E(MS error) =
Note how all terms estimate error.
E(MS A) =
E(MS B) =
E(MS Interaction) =

18. F Tests Effect F df A J-1, N-JK B K-1, AxB Interaction
For orthogonal designs, F tests for the main effects and the interaction are simple. For each, find the F ratio by dividing the MS for the effect of interest by MS error.
Effect F df A
J-1,
N-JK B
K-1,
AxB
Interaction
(J-1)(K-1), N-JK

19. Example Factorial Design (1)
Effects of fatigue and alcohol consumption on driving performance.
Fatigue
Rested (8 hrs sleep then awake 4 hrs)
Fatigued (24 hrs no sleep)
Alcohol consumption
None (control)
2 beers
Blood alcohol .08 %
DV - performance errors on closed driving course rated by driving instructor.

20. Cells of the Design M=13 M=7 M=2 M=10 M=18 M=10 Alcohol (Factor A)
Orthogonal design; n=2
Fatigue (Factor B)
None
(J=1)
2 beers
(J=2)
.08 %
(J=3)
Tired
(K=1)
Cell 1
2, 4
M=3
Cell 2
16, 18
M=17
Cell 3
18, 20
M=19
Rested
(K=2)
Cell 4
0, 2
M=1
Cell 5
Cell 6
M=13
M=7
M=2
M=10
M=18
M=10

21. Factorial Example Results
Main Effects?
Interactions?
Both main effects and the interaction appear significant. Let’s look.

22. Data Person DV Cell A alc B rest 1 2 4 3 16 18 5 6 20 7 8 9 10 11 128 9 10 11 12
Mean

23. Total Sum of Squares Person DV Mean D D*D 1 2 10 -8 64 4 -6 36 3 16 618 8 5 20
100 7
-10 9 11 12
Total
728

24. SS Within Cells Person DV Cell Mean D*D 1 2 3 4 16 17 18 5 19 6 20 7 88 9 10 11 12
Total

25. SS A – Effects of Alcohol
Person M
Level (A)
Mean (A)
D*D 1 10 2 64 3 4 5 18 6 7 8 9 11 12
Total
512

26. SS B – Effects of Fatigue
Person M
Level (B)
Mean (B)
D*D 1 10 13 9 2 3 4 5 6 7 8 11 12
Total
108

27. SS Cells – Total Between
Person M
Cell
Mean (Cell)
D*D 1 10 3 49 2 17 4 5 19 81 6 7 8 9 11 12
Total
716

28. Summary Table Source SS df MS F A 512 2 (J-1) 256 128 B 108 1(K-1) 54
Check: Total=Within+Between
728 = 
Source SS
Total
728
Between
716 A
512 B
108
Within 12
Interaction = Between – (A+B).
Interaction = 716-( ) = 96.
Source SS df MS F A
512
2 (J-1)
256
128 B
108
1(K-1) 54
AxB 96
2 (J-1)(K-1) 48 24
Error 12
6 (N-JK) 2

29. Interactions
Choose the factor with more levels for X, the horizontal axis. Plot the means. Join the means by lines representing the other factor. The size of the interaction SS is proportional to the lack of parallel lines.
If interactions exist, the main effects must be qualified for the interactions. Here, effect of alcohol depends on the amount of rest of the participant.

30. Software skills Run the drink & drive problem in R
Data and code are in Canvas

31. Review
Correctly interpret ANOVA summary tables. Identify mistakes in such tables. What’s the matter with this one?
Source SS Df MS F A
512
2 (J-1)
128 B
108
1 (K-1) 54
AxB 96
2 (J-1)(K-1) 48 24
Error 12
5 N-JK 2

32. Proportions of Variance
We can compute R-squared for magnitude of effect, but it’s biased, so the convention is to use omega-squared.

33. Planned Comparisons Cell Mean C1 C2 1 A1B1 3 -1/3 1/2 2 A2B1 17 -1/419
4 A1B2 1
1/3
5 A2B2
6 A3B2 -6
-12
B1 v B2
No alc v others

34. Planned Comparisons (2)
The first comparison (B1 v B2) has a value of –6. For any comparison,
Note this is the same as SS for B in the ANOVA.
Note squared is 54, which is our value of F from the ANOVA.
(critical t has dfe)

35. Planned Comparisons (3)
We can substitute planned comparisons for tests of main effects; they are equivalent (if you choose the relevant means). We can also do the same for interactions. In general, there are a total of (Cells-1) independent comparisons we can make (6-1 or 5 in our example). Our second test compared no alcohol to all other conditions.
This looks to be the largest comparison with these data.

36. Post Hoc Tests
For post hoc tests about levels of a factor, we pool cells. The only real difference for Tukey HSD and Newman –Keuls is accounting for this difference. For interactions, we are back to comparing cells. Don’t test unless F for the effect is significant.
For comparing cells in the presence of an interaction:

37. Post Hoc (2)
In our example A has 3 levels, B has 2. Both were significant. No post hoc for B (2 levels). For A, the column means were 2, 10, and 18. Are they different?
A: Yes, all are different because the differences are larger than But because of the interaction, the interpretation of differences in A or B are tricky.

38. Post Hoc (3)
For the rested folks, is the difference between no alcohol and 2 beers significant for driving errors? The means are 1 and 3.
A: They are not significantly different because 2 is less than Note: data are fictitious. Do not drink and drive.

39. Review Describe each term in a linear model like this one:
How does post hoc testing for factorial ANOVA differ from post hoc testing in one-way ANOVA?
Describe a concrete example of a two-factor experiment. Why is it interesting and/or important to consider both factors in one experiment?

40. Higher Order Factorials
If you can do ANOVA with 2 factors, you can do it with as many as you like.
For 3 factors, you have one 3-way interaction and three 2-way interactions.
Computations are simple but tedious.
For orthogonal, between-subject designs, all F tests have same denominator.
We generally don’t do designs with more than 3 factors. Complex & expensive.

41. Review R code
Download and run R code for driving (post hoc tests), run chocolate example

42. Hays (5th ed) p. 520 A study involved children of three age groups:
A1=5yrs, A2= 6 years, A3 = 7 years.
Second Factor was formal preschool experience
B1=none, B2=1 year, B3 = more than 1 year
Five children were sampled from each combination and given a score on ‘social maturity’ (high scores are more mature)

43. B1 B2 B3 A1 2 4 6 7 8 11 10 5 9 3 A2 14 15 12 A3 13 18 21

44. Enter the data into R Print the data to be sure they are correct
Be sure the data are arranged in the way R needs to analyze the problem

45. Run a 2-Way ANOVA on the data
What terms are significant
What is the interpretation at the level of main effects and interactions from the printout?

46. Graph the means
What is the interpretation of the means according to the graph?

47. What is the magnitude of effect
What is R-square for A, B, and A*B?
What is estimated omega-squared for each?

48. Unbalanced Designs
When you can assign to treatment, assign equal numbers in each cell
If unequal, unbalanced, and problems with tests
Participant mortality, design flaw, population characteristics (e.g., some types of cows more common in the herd)
Can delete or impute (but still problem)

49. All SS Types same if balanced
Type I SS – simply sequential in terms of model entry (two-factor study: A, B, AxB)
Test A; test B|A ; test AxB|A, B
Type II SS- partly sequential, considers terms at the same ‘level’ e.g.,
A, B test of A considers B but not AxB (A|B; B|A; AxB|A, B)
Type III SS – last in – regression SS
(A|B, AxB; B|A, AxB; AxB|A, B)
Dispute about best approach. SAS uses Type III, R uses Type II. Arguments are difficult to understand (hypotheses tested and model selection).