Multiple Comparisons Overall Risk of Type I Error

Multiple Comparisons

Overall Risk of Type I Error in Using Repeated t Tests at = 0. 05

ANOVA: Graphical

Example: ANOVA terms 1 2 3 4 Treatment 1 y 11 = 48 y 12 = 39 y 13 = 42 y 14 = 43 Treatment 2 y 21 = 40 y 22 = 48 y 23 = 44 Treatment 3 y 31 = 39 y 32 = 30 y 33 = 32 y 34 = 35 Overall n 1 = 4 y 1 = 43 s 1 = 3. 74 n 2 = 3 y 2 = 44 s 2 = 4 n 3 = 4 y 3 = 34 s 3 = 3. 92 11 40

ANOVA Table Source Between Within Total df 2 8 10 SS 228 120 348 MS 114 15

ANOVA Table: Formulas Source Between df SS (Sum of Squares) MS (Mean Square) I– 1 SS/df Within n • – I Total n • – 1

F distribution http: //www. vosesoftware. com/Model. Risk. Help/index. htm#Distributions/Continuous_distributions/F_distribution. htm

F Table

Scientific Conclusion for F test This study (does not) provide(s) evidence [(P = )] at the significance level that there is a difference in ____ among the ____ groups.

Example: ANOVA A random sample of 15 healthy young men are split randomly into 3 groups of 5. They receive 0, 20, and 40 mg of the drug Paxil for one week. Then their serotonin levels are measured to determine whether Paxil affects serotonin levels.

Example: ANOVA (cont). Dose ni y i si (ni-1)si 2 ni(y i - y )2 0 mg 48. 62 49. 85 64. 22 62. 81 62. 51 5 57. 60 7. 678 235. 78 492. 56 20 mg 58. 60 72. 52 66. 72 80. 12 68. 44 5 69. 28 7. 895 249. 32 15. 36 40 mg 68. 59 78. 28 82. 77 76. 53 72. 33 5 75. 70 5. 460 119. 24 333. 96 overall 15 67. 53 604. 34 841. 88

Example: ANOVA (cont) Source Between Within Total df 2 12 14 SS 841. 88 604. 34 1446. 23 MS 420. 94 50. 36

Example: ANOVA (cont) Does Paxil affect serotonin levels in healthy young men? Let 1 be the mean serotonin level for men receiving 0 mg of Paxil. Let 2 be the mean serotonin level for men receiving 20 mg of Paxil. Let 3 be the mean serotonin level for men receiving 40 mg of Paxil.

Example: ANOVA (cont) H 0: 1 = 2 = 3; mean serotonin levels are the same at all 3 dosage levels [or, mean serotonin levels are unaffected by Paxil dose] HA: The mean serotonin levels of the three groups are not all equal. [or, serotonin levels are affected by Paxil does]

Example: ANOVA (cont) Source Between Within Total df 2 12 14 SS 841. 88 604. 34 1446. 23 MS 420. 94 50. 36

Example: ANOVA (cont) Source Between Within Total df 2 12 14 SS MS F-Ratio 841. 88 420. 94 8. 36 604. 34 50. 36 1446. 23 P-Value 0. 0053 This study provides evidence (P = 0. 0053) at the 0. 05 significance level that there is a difference in serotonin levels among the groups of men taking 0, 20, and 40 mg of Paxil. This study provides evidence (P = 0. 0053) at the 0. 05 significance level that Paxil intake affects serotonin levels in young men.

Verification of Conditions

Example 11. 6. 1: Randomized Block Procedure Researchers are interested in the effect that acid has on growth rate of alfalfa plants. To control sunlight, the randomized block procedure is used.

Example 11. 6. 9: F test

Example 11. 7. 3: Two. Way ANOVA

Example 11. 7. 4: Two-Way ANOVA

Bonferroni t Table

Example: ANOVA A random sample of 15 healthy young men are split randomly into 3 groups of 5. They receive 0, 20, and 40 mg of the drug Paxil for one week. Then their serotonin levels are measured to determine whether Paxil affects serotonin levels.

Example: Bonferroni Adjustment Dose ni y i SSi Source Between Within Total 0 mg 5 57. 60 235. 78 df 2 12 14 20 mg 5 69. 28 249. 32 40 mg 5 75. 70 119. 24 SS MS F-Ratio 841. 88 420. 94 8. 36 604. 34 50. 36 1446. 23 overall 15 67. 53 604. 34 P-Value 0. 0053

Example: Paxil, Graphical Representation 0 mg 20 mg 40 mg