2. SPSS Homework
12.1

3. Practice Data from exercise 11.1
1) Use linear contrasts to compare 5 days vs 20 and 35 days
2) Imagine you had no hypotheses and you were concerned about Type 1 error. Compare all conditions to each other using Bonferroni’s t.
5 days vs 20 days
5 days vs 35 days
20 days vs 35 days

4. Practice 1) Use linear contrasts to compare 5 days vs 20 and 35 days
F(1,15) = 4.54

5. Practice
2) Imagine you had no hypotheses and you were concerned about Type 1 error. Compute all possible t-tests using Bonferroni’s t.
5 vs 20
Bonferroni t critical
3 comp, df = 15
2.69
5 vs 35
20 vs 35

7. 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!

8. 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

9. But, what if. . .
You were asked to determine if psychology majors had significantly different class attendance than sociology and biology majors.
You would do an ANOVA with contrast codes

10. 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

11. 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

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

13. 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

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

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

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

17. 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

18. 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

19. 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
Each subject in a cell = Xij
i = level of A; j = level of B B B1 B2 B3 A1 A2 A

20. 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
Each subject in a cell = Xij
i = level of A; j = level of B B B1 B2 B3 A1 A2 A

21. 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
Each subject in a cell = Xijk
i = level of A; j = level of B; k = observation in that cell B B1 B2 B3 A1
X111,X112 X113
X121,X122 X123
X131,X132 X133 A2
X211,X212 X213
X221,X222 X223
X231,X232 X233 A

22. Notation n = 3; N = nab = 18 The means for each of the cells are: B1
X111,X112 X113
X121,X122 X123
X131,X132 X133 A2
X211,X212 X213
X221,X222 X223
X231,X232 X233 A

23. Notation n = 3; N = nab = 18 The means for each of the cells are: B1
X11
X12
X13 A2
X21
X22
X23 A

24. Notation n = 3; N = nab = 18 The means for row (level of A): B1 B2 B3
X11
X12
X13
X1. A2
X21
X22
X23
X2.

25. Notation n = 3; N = nab = 18 The means for column (level of B): B1 B2
X11
X12
X13
X1. A2
X21
X22
X23
X2.
X.1
X.2
X.3

26. Notation n = 3; N = nab = 18 The grand mean: B1 B2 B3 A1 X11 X12 X13

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

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

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

30. 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

31. 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

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

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
Main effect of gender

34. 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

35. 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

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

37. 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?

38. 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!

39. 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

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
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

41. 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

42. 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

43. 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

44. 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

45. 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)

46. 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

47. 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?

48. 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

49. 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

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

51. 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 = – ( )

52. Sum of Squares
SSWithin

53. 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

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
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

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

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

57. 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

58. 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

59. 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)

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

61. 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

62. 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

63. 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.

64. 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

65. 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

66. In SPSS

67. Interpreting the Results
Main Effects
Easy – just like a one-way ANOVA

68. 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

69. Interpreting the Results
Interaction
Does the effect of one IV on the DV depend on the level of another IV?

70. 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

71. Sociology Psychology Biology

74. Practice 2 x 2 Factorial Determine if 1) there is a main effect of A
2) there is a main effect of B
3) if there is an interaction between AB

75. Practice
A: NO
B: NO
AB: NO

76. Practice
A: YES
B: NO
AB: NO

77. Practice
A: NO
B: YES
AB: NO

78. Practice
A: YES
B: YES
AB: NO

79. Practice
A: YES
B: YES
AB: YES

80. Practice
A: YES
B: NO
AB: YES

81. Practice
A: NO
B: YES
AB: YES

82. Practice
A: NO
B: NO
AB: YES

84. 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

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