2. Chapter 12 A Priori and Post Hoc Comparisons Multiple t-tests
Linear Contrasts
Orthogonal Contrasts
Trend Analysis
Bonferroni t
Fisher Least Significance Difference
Studentized Range Statistic
Dunnett’s Test

4. Contrasts Some contrasts are independent Some are not
Freshman vs. Sophomore
(1, -1, 0, 0)
Junior vs. Senior
(0, 0, 1, -1)
Some are not
Freshman vs. Sophomore, Junior, Senior
(3, -1, -1, -1)
Freshman vs. Sophomore & Junior
(2, -1, -1, 0)

5. Orthogonal Contrasts
If you have a complete set of orthogonal contrasts
The sum of SScontrast = SSbetween

6. Orthogonal Contrasts 1) ∑ aj = 0 2) ∑ aj bj = 0
Already talked about
2) ∑ aj bj = 0
Ensures contrasts of independent of one another
3) Number of comparisons = K -1
Ensures enough comparisons are used

7. Orthogonal Contrasts ∑ aj bj = 0 Fresh, Sophomore, Junior, Senior
(3, -1, -1, -1) and (2, -1, -1, 0)
(3*2)+(-1*-1)+(-1*-1) = 8

8. Orthogonal Contrasts ∑ aj bj = 0 Fresh, Sophomore, Junior, Senior
(-1, 1, 0, 0) & (0, 0, -1, 1)
(-1*0)+(1*0)+(-1*0)+(1*0) = 0
*Note: this is not a complete set of contrasts (rule 3)

9. Orthogonal Contrasts Lets go to five groups
What would the complete set contrasts be that would satisfy the earlier rules?

10. Orthogonal Contrasts
General rule
There is more than one right answer

11. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen

12. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen
2 limbs are created
The elements on different limbs can not be combined with each other
Elements on the same limbs can be combined with each other (making new limbs)

13. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen

14. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen

15. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen

16. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen
3, 3, -2, -2, -2

17. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen
3, 3, -2, -2, -2
1, -1, 0, 0, 0

18. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen
3, 3, -2, -2, -2
1, -1, 0, 0, 0
0, 0, 1, 1, -2

19. Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad
Fresh & Soph vs. Jun, Sen, & Grad
Fresh vs. Soph Jun & Sen vs. Grad
Jun vs. Sen
3, 3, -2, -2, -2
1, -1, 0, 0, 0
0, 0, 1, 1, -2
0, 0, 1, -1, 0

20. Orthogonal Contrasts 1) ∑ aj = 0 2) ∑ aj bj = 0
3) Number of comparisons = K -1
3, 3, -2, -2, -2
1, -1, 0, 0, 0
0, 0, 1, 1, -2
0, 0, 1, -1, 0

21. Orthogonal Contrasts 1) ∑ aj = 0 2) ∑ aj bj = 0
3) Number of comparisons = K -1
3, 3, -2, -2, -2 = 0
1, -1, 0, 0, 0 = 0
0, 0, 1, 1, -2 = 0
0, 0, 1, -1, 0 = 0

22. Orthogonal Contrasts A) 3, 3, -2, -2, -2 B) 1, -1, 0, 0, 0
D) 0, 0, 1, -1, 0
A, B = 0; A, C = 0; A, D = 0
B, C = 0; B, D = 0
C, D = 0

23. Orthogonal Contrasts
If you have a complete set of orthogonal contrasts
The sum of SScontrast = SSbetween

24. Compute a complete set of orthogonal contrasts for the following data.
Test each of the contrasts you create for significance

25. Orthogonal Contrasts Fresh, Soph, Jun, Sen Fresh & Soph vs. Jun & Sen
Fresh vs. Soph Jun vs Sen
1, 1, -1, -1
1, -1, 0, 0
0, 0, 1, -1

26. 1, 1, -1, -1
L = 1
SScontrast = 1.5; F = .014
1, -1, 0, 0
L = 4
SScontrast = 48; F = .48
0, 0, 1, -1
L = -23
SScontrast = 1587; F = 15.72*
F crit (1, 20) = 4.35

27. SScontrast = 1.5
SScontrast = 48
SScontrast = 1587 =
F crit (1, 20) = 4.35

28. Orthogonal Contrasts Why use them? People like that they sum together
People like that they are independent
History
I would rather have contrasts based on reason then simply because they are orthogonal!

30. Chapter 12 A Priori and Post Hoc Comparisons Multiple t-tests
Linear Contrasts
Orthogonal Contrasts
Trend Analysis
Bonferroni t
Fisher Least Significance Difference
Studentized Range Statistic
Dunnett’s Test

31. Trend Analysis
The logic of trend analysis is exactly the same logic we just talked about with contrasts!

32. Example
You collect axon firing rate scores from rats in one of four conditions.
Condition 1 = 10 mm of Zeta inhibitor
Condition 2 = 20 mm of Zeta inhibitor
Condition 3 = 30 mm of Zeta inhibitor
Condition 4 = 40 mm of Zeta inhibitor
Condition 5 = 50 mm of Zeta inhibitor
You think Zeta inhibitor reduces the number of times an axon fires – are you right?

33. What does this tell you ?

35. Trend Analysis
Contrast Codes!

36. Trend Analysis

37. a1 = -2, a2 = -1, a3 = 0, a4 = 1, a5 = 2
L = 7.2
F crit (1, 20) = 4.35

39. Note

40. Example
You place subjects into one of five different conditions of anxiety.
1) Low anxiety
2) Low-Moderate anxiety
3) Moderate anxiety
4) High-Moderate anxiety
5) High anxiety
You think subjects will perform best on a test at level 3 (and will do worse at both lower and higher levels of anxiety)

41. What does this tell you ?

43.
Contrast Codes!

44. Trend Analysis
Create contrast codes that will examine a quadratic trend.
-2, 1, 2, 1, -2

45. a1 = -2, a2 = 1, a3 = 2, a4 = 1, a5 = -2
L = 10
F crit (1, 20) = 4.35

47. Trend Analysis
How do you know which numbers to use?
Page 742

48. Linear
(NO BENDS)

49. Quadratic
(ONE BEND)

50. Cubic
(TWO BENDS)

52. Practice
You believe a balance between school and one’s social life is the key to happiness. Therefore you hypothesize that people who focus too much on school (i.e., people who get good grades) and people who focus too much on their social life (i.e., people who get bad grades) will be more depressed.
You collect data from 25 subjects
5 subjects = F
5 subjects = D
5 subjects = C
5 subjects = B
5 subjects = A
You measured their depression

53. Practice
Below are your findings – interpret!

54. Trend Analysis
Create contrast codes that will examine a quadratic trend.
-2, 1, 2, 1, -2

55. a1 = -2, a2 = 1, a3 = 2, a4 = 1, a5 = -2
L = -12.8
F crit (1, 20) = 4.35

59. Remember
Freshman, Sophomore, Junior, Senior
Measure Happiness (1-100)

61. ANOVA Traditional F test just tells you not all the means are equal
Does not tell you which means are different from other means

62. Why not Do t-tests for all pairs Fresh vs. Sophomore Fresh vs. Junior
Fresh vs. Senior
Sophomore vs. Junior
Sophomore vs. Senior
Junior vs. Senior

63. Problem What if there were more than four groups?
Probability of a Type 1 error increases.
Maximum value = comparisons (.05)
6 (.05) = .30

64. Chapter 12 A Priori and Post Hoc Comparisons Multiple t-tests
Linear Contrasts
Orthogonal Contrasts
Trend Analysis
Bonferroni t
Fisher Least Significance Difference
Studentized Range Statistic
Dunnett’s Test

65. Bonferoni t
Controls for Type I error by using a more conservative alpha

66. Do t-tests for all pairs
Fresh vs. Sophomore
Fresh vs. Junior
Fresh vs. Senior
Sophomore vs. Junior
Sophomore vs. Senior
Junior vs. Senior

67. Maximum probability of a Type I error
6 (.05) = .30
But what if we use
Alpha = .05/C
= .05 / 6
6 (.00855) = .05

68. t-table Compute the t-value the exact same way
Problem: normal t table does not have these p values
Test for significance using the Bonferroni t table (page 751)

69. Practice

71. Practice Fresh vs. Sophomore t = .69 Fresh vs. Junior t = 2.41
Fresh vs. Senior
t = -1.55
Sophomore vs. Junior
t = 1.72
Sophomore vs. Senior
t = -2.24
Junior vs. Senior
t = -3.97*
Critical t = 6 comp/ df = 20 = 2.93

72. Bonferoni t Problem Silly What should you use as the value in C?
Increases the chances of the Type II error!

77. 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

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

79. 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