2. What if. . .
You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person misses class)
You would simply do a two-sample t-test
two-tailed
Easy!

3. But, what if. . .
You were asked to determine if psychology, sociology, and biology majors have significantly different class attendance
You would do a one-way ANOVA

4. But, what if. . .
You were asked to determine the effects of both college major (psychology, sociology, and biology) and gender (male and female) on class attendance
You now have 2 IVs and 1 DV
You could do two separate analyses
Problem: “Throw away” information that could explain some of the “error”
Problem: Will not be able to determine if there is an interaction

5. Factorial Analysis of Variance
Factor = IV
Factorial design is when every level of every factor is paired with every level of every other factor
Psychology
Sociology
Biology
Male X
Female

6. Factorial Analysis of Variance
Currently
Different people in each cell
Equal n in each cell
Psychology
Sociology
Biology
Male X
Female

7. Factorial Analysis of Variance
2 X 3 Factorial
“2” is because one IV has 2 levels (male and female)
“3” because one IV has 3 levels (psychology, biology, sociology)
Psychology
Sociology
Biology
Male X
Female

8. Factorial Analysis of Variance
2 X 3
4 X 5
2 X 2 X 7

9. Notation One factor is A and the other is B Psychology Sociology
Biology
Male
Female

10. Notation
One factor is A and the other is B B B1 B2 B3 A1 A2 A

11. Notation One factor is A and the other is B
Any combination of A and B is called a cell B B1 B2 B3 A1 A2 A

12. Notation One factor is A and the other is B
Any combination of A and B is called a cell
The number of subjects in each cell = n B B1 B2 B3 A1 A2 A

13. Sociology
Psychology
Biology
Female
2.00
1.00
3.00
.00
Males
4.00
n = 3
N = 18

14. Sociology
Psychology
Biology
Female
2.00
1.00
3.00
.00
Males
4.00
n = 3
N = 18

15. Sociology
Psychology
Biology
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
Males
4.00
Mean2j
3.67
0.33

16. Sociology
Psychology
Biology
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33

17. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33

18. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06

19. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
Main effect of gender

20. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
Main effect of major

21. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
Interaction between gender and major

22. Sum of Squares SS Total Computed the same way as before
The total deviation in the observed scores
Computed the same way as before

23. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SStotal = (2-2.06)2+ (3-2.06) (1-2.06)2 = 30.94
*What makes this value get larger?

24. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SStotal = (2-2.06)2+ (3-2.06) (1-2.06)2 = 30.94
*What makes this value get larger?
*The variability of the scores!

25. Sum of Squares
SS A
Represents the SS deviations of the treatment means around the grand mean
Its multiplied by nb to give an estimate of the population variance (Central limit theorem)
Same formula as SSbetween in the one-way

26. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSA = (3*3) (( )2+ ( )2)=1.36
*Note: it is multiplied by nb because that is the number of scores each mean is based on

27. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSA = (3*3) (( )2+ ( )2)=1.36
*What makes these means differ?
*Error and the effect of A

28. Sum of Squares
SS B
Represents the SS deviations of the treatment means around the grand mean
Its multiplied by na to give an estimate of the population variance (Central limit theorem)
Same formula as SSbetween in the one-way

29. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSB = (3*2) (( )2+ ( )2+ ( )2)= 14.16
*Note: it is multiplied by na because that is the number of scores each mean is based on

30. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSB = (3*2) (( )2+ ( )2+ ( )2)= 14.16
*What makes these means differ?
*Error and the effect of B

31. Sum of Squares
SS Cells
Represents the SS deviations of the cell means around the grand mean
Its multiplied by n to give an estimate of the population variance (Central limit theorem)

32. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSCells = (3) (( )2+ ( ) ( )2)= 24.35

33. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSCells = (3) (( )2+ ( ) ( )2)= 24.35
What makes the cell means differ?

34. Sum of Squares SS Cells What makes the cell means differ? 1) error
2) the effect of A (gender)
3) the effect of B (major)
4) an interaction between A and B

35. Sum of Squares Have a measure of how much cells differ
SScells
Have a measure of how much this difference is due to A
SSA
Have a measure of how much this difference is due to B
SSB
What is left in SScells must be due to error and the interaction between A and B

36. Sum of Squares
SSAB = SScells - SSA – SSB
8.83 =

37. Sum of Squares SSWithin SSWithin = SSTotal – (SSA + SSB + SSAB)
The total deviation in the scores not caused by
1) the main effect of A
2) the main effect of B
3) the interaction of A and B
SSWithin = SSTotal – (SSA + SSB + SSAB)
6.59 = – ( )

38. Sum of Squares
SSWithin

39. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) ((1-.33)2 + (0-.33)2 + ( )2 = 6.667

40. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) ((1-.33)2 + (0-.33)2 + ( )2 = 6.667
*What makes these values differ from the cell means?
*Error

41. Compute df
Source df SS A
1.36 B
14.16 AB
8.83
Within
6.59
Total
30.94

42. Source df SS A 1.36 B 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N - 1

43. Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N – 1
dfA = a – 1
dfB = b - 1

44. Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N – 1
dfA = a – 1
dfB = b – 1
dfAB = dfa * dfb

45. Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 12 6.59 Total 17 30.94
dftotal = N – 1
dfA = a – 1
dfB = b – 1
dfAB = dfa * dfb
dfwithin= ab(n – 1)

46. Compute MS Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 12 6.59
Total 17
30.94

47. Compute MS Source df SS MS A 1 1.36 B 2 14.16 7.08 AB 8.83 4.42 Within12
6.59
.55
Total 17
30.94

48. Compute F Source df SS MS A 1 1.36 B 2 14.16 7.08 AB 8.83 4.42 Within12
6.59
.55
Total 17
30.94

49. Test each F value for significance
Source df SS MS F A 1
1.36
2.47 B 2
14.16
7.08
12.87 AB
8.83
4.42
8.03
Within 12
6.59
.55
Total 17
30.94
F critical values (may be different for each F test)
Use df for that factor and the df within.

50. Test each F value for significance
Source df SS MS F A 1
1.36
2.47 B 2
14.16
7.08
12.87 AB
8.83
4.42
8.03
Within 12
6.59
.55
Total 17
30.94
F critical A (1, 12) = 4.75
F critical B (2, 12) = 3.89
F critical AB (2, 12) = 3.89

51. Test each F value for significance
Source df SS MS F A 1
1.36
2.47 B 2
14.16
7.08
12.87* AB
8.83
4.42
8.03*
Within 12
6.59
.55
Total 17
30.94
F critical A (1, 12) = 4.75
F critical B (2, 12) = 3.89
F critical AB (2, 12) = 3.89

52. Interpreting the Results
Source df SS MS F A 1
1.36
2.47 B 2
14.16
7.08
12.87* AB
8.83
4.42
8.03*
Within 12
6.59
.55
Total 17
30.94
F critical A (1, 12) = 4.75
F critical B (2, 12) = 3.89
F critical AB (2, 12) = 3.89

53. Interpreting the Results
Source df SS MS F A 1
1.36
2.47 B 2
14.16
7.08
12.87* AB
8.83
4.42
8.03*
Within 12
6.59
.55
Total 17
30.94
F critical A (1, 12) = 4.75
F critical B (2, 12) = 3.89
F critical AB (2, 12) = 3.89

54. Sociology
Psychology
Biology
Mean
Female
2.00
1.00
3.00
.00
Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06

55. Interpreting the Results
Source df SS MS F A 1
1.36
2.47 B 2
14.16
7.08
12.87* AB
8.83
4.42
8.03*
Within 12
6.59
.55
Total 17
30.94
F critical A (1, 12) = 4.75
F critical B (2, 12) = 3.89
F critical AB (2, 12) = 3.89

56. Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67
1.67
1.78
Males
4.00
Mean2j
Mean.j
3.67
3.17
0.33
2.33
2.06
Want to plot out the cell means

57. Sociology Psychology Biology

59. Practice
These are sample data from Diener et. al (1999). Participants were asked their marital status and how often they engaged in religious behavior. They also indicated how happy they were on a scale of 1 to 10.
Examine the data

60. Frequency of religious behavior
Never
Occasionally
Often
Married 6 3 7 2 8 4 5 9
Unmarried 1

61. Interpreting the Results
Source df SS MS F
Married
18.00
Relig
31.00 AB
3.00
Within
Total
88.00

63. Interpreting the Results
Source df SS MS F
Married 1
18.00
6.00*
Relig 2
31.00
15.50
5.17* AB
3.00
1.50
.50
Within 12
36.00
Total 17
88.00
F critical A (1, 12) = 4.75
F critical B (2, 12) = 3.89
F critical AB (2, 12) = 3.89