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. Linear Contrasts
You think that Freshman and Seniors will have different levels of happiness than Sophomores and Juniors

5. Linear Contrasts
Allows for the comparison of one group or set of groups with another group or set of groups

6. Linear Contrasts
a = weight given to a group

7. Linear Contrasts a1 = 0, a2 = 0, a3 = 1, a4 = -1 L = -23

8. SS Contrast You can use the linear contrast to compute a SS contrast
SS contrast is like SS between
SS contrast has 1 df
SS contrast is like MS between

9. SS Contrast

10. SS Contrasts
a1 = .5, a2 = -.5, a3 = -.5, a4 = .5
L = 80.5 – 67 = 13.5

11. SS Contrasts a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5
Sum a2 = = 1

12. SS Contrasts a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5
Sum a2 = = 1

13. SS Contrasts
a1 = 1, a2 = -1, a3 = -1, a4 = 1
L = 161 – 134 = 27

14. SS Contrasts a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27 n = 6
Sum a2 = = 4

15. SS Contrasts a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27 n = 6
Sum a2 = = 4

16. F Test
Note: MS contrast = SS contrast

17. F Test
Fresh & Senior vs. Sophomore & Junior

18. F Test
Fresh & Senior vs. Sophomore & Junior

19. F Test
Fresh & Senior vs. Sophomore & Junior
F crit (1, 20) = 4.35

20. SPSS

21. Make contrasts to determine
If seniors are happier than everyone else?
2) If juniors and sophomores have different levels of happiness?

22. If seniors are happier than everyone else?
a1 = -1, a2 = -1, a3 = -1, a4 = 3
L = 45
F crit (1, 20) = 4.35

24. 2) If juniors and sophomores have different levels of happiness?
a1 = 0, a2 = -1, a3 = 1, a4 = 0
L = -10
F crit (1, 20) = 4.35

27. Practice
To investigate the maternal behavior of lab rats, we move the rat pup a fixed distance from the mother and record the time required for the mother to retrieve the pup. We run the study with 5, 20, and 35 day old pups.
Figure out if 5 days is different than 35 days.
SPSS #6 Homework (Do the ANOVA analysis in SPSS – use output to answer question above)
5 days 15 10 25 20 18
20 days 30 23
35 days 40 35 50 43 45

31. 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

33. 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)

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

35. 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

36. 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

37. 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)

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

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

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

41. 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)

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

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

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

45. 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

46. 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

47. 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

48. 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

49. 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

50. 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

51. 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

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

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

54. 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

55. 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

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

57. 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!