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Jack’s gone to the dogs in Alaska February 25, 2005

Jack’s gone to the dogs in Alaska February 25, 2005. Marvelous Marvin father of the Groom. Alaskan Wedding Feast. Analyses of Lattice Squares. Y ijk =  + r i + b a j + t a k + e ijk. See Table 5 & 6, Page 105 & 106. Analyses of Lattice Squares.

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Jack’s gone to the dogs in Alaska February 25, 2005

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  1. Jack’s gone to the dogs in AlaskaFebruary 25, 2005

  2. Marvelous Marvin father of the Groom Alaskan Wedding Feast

  3. Analyses of Lattice Squares Yijk =  + ri + baj + tak +eijk See Table 5 & 6, Page 105 & 106

  4. Analyses of Lattice Squares • Calculate sub-block totals (b) and replicate totals (R). • Calculate the treatment totals (T) and the grand total (G). • For each treatment, calculate the Bt values which is the sum of all block totals that contain the ith treatment.

  5. Analyses of Lattice Squares • Calculate sub-block totals (b) and replicate totals (R). • Calculate the treatment totals (T) and the grand total (G). • For each treatment, calculate the Bt values which is the sum of all block totals that contain the ith treatment.

  6. Analyses of Lattice Squares • Treatment 5 is in block 2, 5, 10, 15, and 20, so B5 = 616+639+654+675+827 = 3411. • Note that the sum of the Bt values is G x k, where k is the block size. • For each treatment calculate: W = kT – (k+1)Bt + G W5 = 4(816)-(5)(3,411)+13,746 = -45

  7. Lattice Square ANOVA - d.f.

  8. Analyses of Lattice Squares • Compute the total correction factor as: CF = (∑xij)2/n CF = G2/[(k2)(k+1)] (13,746)2/(16)(5) 2,361,906

  9. Analyses of Lattice Squares • Compute the total SS as: Total SS = xij2 – CF [1472+1522+…+2252] – 2,361,906 = 58,856

  10. Analyses of Lattice Squares • Compute the replicate block SS as: Replicate SS = R2/k2 – CF [25952+27292+…+29252]/16 – 2,361,906 = 5,946

  11. Analyses of Lattice Squares • Compute the unadjusted treatment SS as: Treatment (unadj) SS = T2/(k+1)–CF [8092+7942+…+8662]/5 – 2,361,906 = 26,995

  12. Analyses of Lattice Squares • Compute the adjusted block SS as: Block (adj) SS = W2/k3(k+1) – CF [8092+7942+…+8662]/320 – 2,361,906 = 11,382

  13. Analyses of Lattice Squares • Compute the intra-block error SS as: IB error SS = TSS–Rep SS–Treat(unadj) SS–Blk(adj) SS 58,856 - 5,946 - 26,995 - 11,382 = 14,533

  14. Lattice Square ANOVA • Calculate Mean Squares for block(adj) and IBE.

  15. Analyses of Lattice Squares • Compute adjusted treatment totals (T’) as: T’i = Ti + Wi  = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

  16. Analyses of Lattice Squares • Compute adjusted treatment totals (T’) as: •  = [759-323]/(16)(759) = 0.0359 T’ = T + W  = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

  17. Analyses of Lattice Squares • Compute adjusted treatment totals (T’) as: • Note if IBE MS > Blk(adj) MS, then =zero. So no adjustment. T’ = T + W  = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

  18. Analyses of Lattice Squares • Compute adjusted treatment totals (T’) as: • Note also greatest adjustment when Blk(adj) MS large and IBE MS is small. T’ = T + W  = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

  19. Analyses of Lattice Squares • Compute adjusted treatment totals (T’) as: • T’5 = T5 + W5 • T’5 = 816 + 0.0359 x (-45) = 814 T’ = T + W  = [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]

  20. Analyses of Lattice Squares • Compute adjusted treatment means (M’) as: M’ = T’/[k+1]

  21. Analyses of Lattice Squares • Compute adjusted treatment SS as: Treat (adj) SS = T’2/(k+1) – CF [8292+8052+…+8392]/5 – 2,361,906 = 24,030

  22. Analyses of Lattice Squares • Compute effective error MS as: EE MS = (Intra-block error MS)(1+k) 323[1 + 4(0.0359)] 369

  23. Lattice Square ANOVA

  24. Lattice Square ANOVA

  25. Efficiency of Lattice Design 100 x [Blk(adj)SS+Intra error SS]/k(k2-1)EMS 100 [11,382 + 14,533]/4(16)369 117% I II III IV V I II III IV V

  26. Lattice Square ANOVA

  27. RCB ANOVA

  28. Lattice Square ANOVA

  29. Lattice Square ANOVA • CV Lattice = 11.2%; CV RCB = 12.1%. • Range Lattice 119 to 197; Range RCB 116 to 199. • Variation between treatments is small compared to environmental error or variation.

  30. Comparison of Rankings

  31. ANOVA of Factorial Designs

  32. Factorial AOV Example • Spring barley ‘Malter’ • Three seeding rates (low, Medium and High). • Six nitrogen levels (90, 100, 110, 120, 130, 140 units). • Three replicates • Page 107 of class notes

  33. Factorial AOV Example CF = (297.0)2/54 = 3676.6 TSS = [8.192 + 8.372 + … + 4.152]-CF = 4612.56 Rep SS = [98.62 + 99.12 + 99.32]/18-CF = 0.01

  34. Factorial AOV Example Seed rate SS = [104.42 + 98.02 + 94.62]/18 – CF = 2.75

  35. Factorial AOV Example N rate SS = [37.72 + 39.52 + …+ 69.22]/9 – CF = 2.75

  36. Factorial AOV Example Seed x N SS = [12.82 + 13.72 + …+ 21.22]/3 – CF - Seed rate SS – Nitrogen SS = 1.33

  37. Factorial AOV Example Error SS=TSS–Seed SS–N SS–NxS SS–Rep SS

  38. Factorial AOV Example

  39. Factorial AOV Example CV = / x 100 = 0.041/5.50 = 3.38% R2 = [TSS-ESS]/TSS = [87.03-1.38]/87.03 = 96.2%

  40. Factorial AOV Example

  41. Factorial AOV Example sed[within] = (22/3) = 0.165 sed[Seed rate] = (22/18) = 0.067 sed[N rate] = (22/9) = 0.095

  42. Factorial AOV Example

  43. Factorial AOV Example

  44. Factorial AOV Example

  45. Split-plot AOV

  46. Split-plot AOV

  47. Strip-plot AOV

  48. Strip-plot AOV

  49. Fixed and Random Effects

  50. Expected Mean Squares • Dependant on whether factor effects are Fixed or Random. • Necessary to determine which F-tests are appropriate and which are not.

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