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

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

4. 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?

5. What does this tell you ?

7. Trend Analysis
Contrast Codes!

8. Trend Analysis

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

11. Note

12. 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)

13. What does this tell you ?

15.
Contrast Codes!

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

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

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

20. Linear
(NO BENDS)

21. Quadratic
(ONE BEND)

22. Cubic
(TWO BENDS)

24. 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

25. Practice
Below are your findings – interpret!

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

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

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

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

34. 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

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

36. 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

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

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

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

40. 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)

41. Practice

43. 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

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

46. 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

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

48. 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

52. 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

53. Fisher Least Significance Difference
Simple
1) Do a normal omnibus ANOVA
2) If there it is significant you know that there is a difference somewhere!
3) Do individual t-test to determine where significance is located

54. Fisher Least Significance Difference
Problem
You may have an ANOVA that is not significant and still have results that occur in a manner that you predict!
If you used this method you would not have “permission” to look for these effects.

55. Remember

56. Remember

57. Studentized Range Statistic
Which groups would you likely select to determine if they are different?

58. Studentized Range Statistic
Which groups would you likely select to determine if they are different?
This statistics controls for Type I error if (after looking at the data) you select the two means that are most different.

59. Studentized Range Statistic
Easy!
1) Do a normal t-test

60. Studentized Range Statistic
Easy!
2) Convert the t to a q

61. Studentized Range Statistic
3) Critical value of q (note: this is a two-tailed test)
Figure out df (same as t)
Example = 20
Figure out r
r = the number of groups

62. Studentized Range Statistic
3) Critical value of q (note: this is a two-tailed test)
Figure out df (same as t)
Example = 20
Figure out r
r = the number of groups
Example = 4

63. Studentized Range Statistic
3) Critical value of q
Page 744
Example
q critical = +/- 3.96

64. Studentized Range Statistic
4) Compare q obs and q critical same way as t values
q = -5.61
q critical = +/– 3.96

65. Practice
You collect axon firing rate scores from rates 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 are simply interested in determining if any two groups are different from each other – use the Studentized Range Statistic

66. Studentized Range Statistic
Easy!
1) Do a normal t-test

67. Studentized Range Statistic
Easy!
2) Convert the t to a q

68. Studentized Range Statistic
3) Critical value of qnote: this is a two-tailed test)
Figure out df (same as t)
Example = 20
Figure out r
r = the number of groups
Example = 5

69. Studentized Range Statistic
3) Critical value of q
Page 744
Example
q critical = +/- 4.23

70. Studentized Range Statistic
4) Compare q obs and q critical same way as t values
q = -4.34
q critical = +/– 4.23

71. Dunnett’s Test
Used when there are several experimental groups and one control group (or one reference group)
Example:
Effect of psychotherapy on happiness
Group 1) Psychoanalytic
Group 2) Humanistic
Group 3) Behaviorism
Group 4) Control (no therapy)

73. Psyana vs. Control
Human vs. Control
Behavior vs. Control

74. Psyana vs. Control = 47.8 – 51.4 = -3.6
Human vs. Control = 50.8 – = -0.6
Behavior vs. Control = 59 – 51.4 = 7.6

75. Psyana vs. Control = 47.8 – 51.4 = -3.6
Human vs. Control = 50.8 – = -0.6
Behavior vs. Control = 59 – 51.4 = 7.6
How different do these means need to be in order to reach significance?

79. Dunnett’s t is on page 753
df = Within groups df / k = number of groups

80. Dunnett’s t is on page 753
df = 16 / k = 4

81. Dunnett’s t is on page 753
df = 16 / k = 4

82. Psyana vs. Control = 47.8 – 51.4 = -3.6
Human vs. Control = 50.8 – = -0.6
Behavior vs. Control = 59 – 51.4 = 7.6*
How different do these means need to be in order to reach significance?

83. Practice
As a graduate student you wonder what undergraduate students (freshman, sophomore, etc.) have different levels of happiness then you.

85. Dunnett’s t is on page 753
df = 25 / k = 5

86. Fresh vs. Grad = -17.5*
Soph vs. Grad = -21.5*
Jun vs. Grad = -31.5*
Senior vs. Grad = -8.5