UMBRELLA TESTS for one-way layouts Test whether or not medians of treatment groups increase to that of a particular treatment group and then decrease afterwards Two versions: Given peak and unknown peak Initializing random-number generator at 12345678 Number of permutations: 10000 FIRST DATA SET: 5 treatment groups with 50 total observations: #1 (n=10): 308 308 12 206 368 264 88 240 118 264 #2 (n=10): 126 252 262 258 136 204 170 276 314 308 #3 (n=10): 218 364 300 322 46 180 278 454 70 188 #4 (n=10): 358 366 394 252 430 274 216 284 350 290 #5 (n=10): 332 358 356 178 262 158 176 278 240 304 Ord, Values, Group, Midranks, Tiegroups: 1. 308 1 36.0 2 2. 308 1 36.0 2 3. 12 1 1.0 2 4. 206 1 15.0 2 5. 368 1 47.0 2 6. 264 1 25.5 3 7. 88 1 4.0 2 8. 240 1 18.5 0 9. 118 1 5.0 0 10. 264 1 25.5 0 11. 126 2 6.0 0 12. 252 2 20.5 0 13. 262 2 23.5 0 14. 258 2 22.0 0 15. 136 2 7.0 0 16. 204 2 14.0 0 17. 170 2 9.0 0 18. 276 2 28.0 0 19. 314 2 38.0 0 20. 308 2 36.0 0 21. 218 3 17.0 0 22. 364 3 45.0 0 23. 300 3 33.0 0 24. 322 3 39.0 0 25. 46 3 2.0 0 26. 180 3 12.0 0 27. 278 3 29.5 0 28. 454 3 50.0 0 29. 70 3 3.0 0 30. 188 3 13.0 0 31. 358 4 43.5 0 32. 366 4 46.0 0 33. 394 4 48.0 0 34. 252 4 20.5 0 35. 430 4 49.0 0 36. 274 4 27.0 0 37. 216 4 16.0 0 38. 284 4 31.0 0 39. 350 4 41.0 0 40. 290 4 32.0 0 41. 332 5 40.0 0 42. 358 5 43.5 0 43. 356 5 42.0 0 44. 178 5 11.0 0 45. 262 5 23.5 0 46. 158 5 8.0 0 47. 176 5 10.0 0 48. 278 5 29.5 0 49. 240 5 18.5 0 50. 304 5 34.0 0 Sample means and rank means for treatment groups: #1 (n=10): DataAv=217.6 RankAv= 21.35 #2 (n=10): DataAv=230.6 RankAv= 20.40 #3 (n=10): DataAv=242.0 RankAv= 24.35 #4 (n=10): DataAv=321.4 RankAv= 35.40 #5 (n=10): DataAv=264.2 RankAv= 26.00 THE UMBRELLA TEST for KNOWN p=4 (Mack-Wolfe): Single-umbrella test: Is the data significant for a `peak` in treatment-group medians at p=4? First Step: Matrix of Mann-Whitney differences between treatment groups: 1. 0.0 51.0 54.0 77.0 59.5 2. 49.0 0.0 56.0 81.5 64.5 3. 46.0 44.0 0.0 69.0 52.5 4. 23.0 18.5 31.0 0.0 28.5 5. 40.5 35.5 47.5 71.5 0.0 Example of an Ap statistic: Ap(4) = U(1,2)+U(1,3)+U(1,4)+U(2,3)+U(2,4)+U(3,4) + U(5,4) The umbrella statistics Ap with normal approximations Astar: (P-values are one-sided normal.) p=1 Ap=406.5 Mean=500.0 Var=3416.67 Astar=-1.600 P=0.94516 p=2 Ap=299.0 Mean=350.0 Var=2391.67 Astar=-1.043 P=0.85149 p=3 Ap=311.0 Mean=300.0 Var=2050.00 Astar= 0.243 P=0.40402 p=4 Ap=460.0 Mean=350.0 Var=2391.67 Astar= 2.249 P=0.01225 p=5 Ap=593.5 Mean=500.0 Var=3416.67 Astar= 1.600 P=0.05484 Simulating the P-value of Ap=460 given p=4 using 10000 permutations: Number greater-or-equal to old Ap(p) and number of permuations: Nge=130 nsims=10000 95% Conf.Int. for one-sided P-value bracketing estimate: 0.0108 0.0130 0.0152 THE UMBRELLA TEST with UNKNOWN p (Chen-Wolfe): Multiple-umbrella test: Is there a `peak` in treatment-group medians at SOME p, and if so, at which treatment group? Note that the one-sided P-values in the table above are NOT multiple-comparison corrected. Max_Astar=2.24927 is attained at p=4 Simulating the P-value of Max_Astar=2.24927 using 10000 permutations: Number greater-or-equal to Max_Astar and number of permuations: 546 10000 95% Conf.Int. for one-sided P-value bracketing estimate: 0.0501 0.0546 0.0591