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Analyses of Lattice Squares in Alaska: Comprehensive Study

Explore detailed analyses and calculations of lattice squares in a comprehensive study conducted in Alaska, including block totals, treatment values, and design efficiency.

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Analyses of Lattice Squares in Alaska: Comprehensive Study

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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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