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  1. Example • Freshman, Sophomore, Junior, Senior • Measure Happiness (1-100)

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

  3. 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

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

  5. 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

  6. Multiple t-tests • Good if you have just a couple of planned comparisons • Do a normal t-test, but use the other groups to help estimate your error term • Helps increase you df

  7. Remember

  8. Note

  9. Proof Candy Gender 5.00 1.00 4.00 1.00 7.00 1.00 6.00 1.00 4.00 1.00 5.00 1.00 1.00 2.00 2.00 2.00 3.00 2.00 4.00 2.00 3.00 2.00 2.00 2.00

  10. t = 2.667 / .641 = 4.16

  11. t = 2.667 / .641 = 4.16

  12. t = 2.667 / .641 = 4.16

  13. t = 2.667 / .641 = 4.16

  14. Also, when F has 1 df between

  15. Within Variability • Within variability of all the groups represents “error” • You can therefore get a better estimate of error by using all of the groups in your ANOVA when computing a t-value

  16. Note: This formula is for equal n

  17. Hyp 1: Juniors and Seniors will have different levels of happiness Hyp 2: Seniors and Freshman will have different levels of happiness

  18. Hyp 1: Juniors and Seniors will have different levels of happiness

  19. Hyp 1: Juniors and Seniors will have different levels of happiness

  20. Hyp 1: Juniors and Seniors will have different levels of happiness

  21. Hyp 1: Juniors and Seniors will have different levels of happiness t crit (20 df) = 2.086

  22. Hyp 1: Juniors and Seniors will have different levels of happiness t crit (20 df) = 2.086 Juniors and seniors do have significantly different levels of happiness

  23. Hyp 2: Seniors and Freshman will have different levels of happiness

  24. Hyp 2: Seniors and Freshman will have different levels of happiness

  25. Hyp 2: Seniors and Freshman will have different levels of happiness

  26. Hyp 2: Seniors and Freshman will have different levels of happiness t crit (20 df) = 2.086

  27. Hyp 2: Seniors and Freshman will have different levels of happiness t crit (20 df) = 2.086 Freshman and seniors do not have significantly different levels of happiness

  28. Hyp 1: Juniors and Sophomores will have different levels of happiness Hyp 2: Seniors and Sophomores will have different levels of happiness PRACTICE!

  29. 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

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

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

  32. Linear Contrasts a = weight given to a group

  33. Linear Contrasts 1 2 3 4 a1 = 0, a2 = 0, a3 = 1, a4 = -1 L = -23 a1 = 1, a2 = 0, a3 = 0, a4 = -1 L = -9 a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5

  34. 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

  35. SS Contrast

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

  37. SS Contrasts a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5 n = 6 L = 13.5 Sum a2 = .52+-.52+ -.52 + .52 = 1

  38. SS Contrasts a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5 n = 6 L = 13.5 Sum a2 = .52+-.52+ -.52 + .52 = 1

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

  40. SS Contrasts a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27 n = 6 L = 27 Sum a2 = 12+-12+ -12 + 12 = 4

  41. SS Contrasts a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27 n = 6 L = 27 Sum a2 = 12+-12+ -12 + 12 = 4

  42. F Test Note: MS contrast = SS contrast

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

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

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

  46. SPSS