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

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Presentation on theme: "Jack’s gone to the dogs in Alaska February 25, 2005."— Presentation transcript:

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

2 Alaskan Wedding Feast Marvelous Marvin father of the Groom

3 Analyses of Lattice Squares Y ijk =  + r i + b a j + t a k + e ijk 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 B t values which is the sum of all block totals that contain the i th 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 B t values which is the sum of all block totals that contain the i th treatment.

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

7 Lattice Square ANOVA - d.f. Sourcedf Repsk4 Trt(unadj)k 2 – 115 Block(adj)k 2 – 115 Intra-Block Error(k-1)(k 2 -1)45 Trt (adj)k 2 – 115 Effective Error(k-1)(k 2 -1)45 Totalk 2 (k+1)-179

8 Analyses of Lattice Squares  Compute the total correction factor as: CF = (∑x ij ) 2 /n CF = G 2 /[(k 2 )(k+1)] (13,746) 2 /(16)(5) 2,361,906

9 Analyses of Lattice Squares  Compute the total SS as: Total SS =  x ij 2 – CF [147 2 +152 2 +…+225 2 ] – 2,361,906 = 58,856

10 Analyses of Lattice Squares  Compute the replicate block SS as: Replicate SS =  R 2 /k 2 – CF [2595 2 +2729 2 +…+2925 2 ]/16 – 2,361,906 = 5,946

11 Analyses of Lattice Squares  Compute the unadjusted treatment SS as: Treatment (unadj) SS =  T 2 /(k+1)–CF [809 2 +794 2 +…+866 2 ]/5 – 2,361,906 = 26,995

12 Analyses of Lattice Squares  Compute the adjusted block SS as: Block (adj) SS =  W 2 /k 3 (k+1) – CF [809 2 +794 2 +…+866 2 ]/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 SourcedfSSMS Reps45,9461,486 T(unadj)1526,9951,800 Blk(adj)1511,382759 Intra block error4514,533323  Calculate Mean Squares for block(adj) and IBE.

15 Analyses of Lattice Squares  Compute adjusted treatment totals (T’) as: T ’ i = T i +  W i  = [Blk(adj) MS-IBE MS]/[k 2 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]/[k 2 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]/[k 2 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]/[k 2 Blk(adj) MS]

19 Analyses of Lattice Squares  Compute adjusted treatment totals (T’) as:  T ’ 5 = T 5 +  W 5  T ’ 5 = 816 + 0.0359 x (-45) = 814 T’ = T +  W  = [Blk(adj) MS-IBE MS]/[k 2 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 [829 2 +805 2 +…+839 2 ]/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 SourcedfSSMSF Reps45,946 T(unadj)1526,995 Blk(adj)1511,382 Intra error4514,533 T(adj)1524,030 Eff. Error4516,605

24 Lattice Square ANOVA SourcedfSSMSF Reps45,9461,4864.03 * T(unadj)1526,9951,800- Blk(adj)1511,3827592.35 ns Intra error4514,533323- T(adj)1524,0301,6024.34 ** Eff. Error4516,605369-

25 Efficiency of Lattice Design 100 x [Blk(adj)SS+Intra error SS]/k(k-1)EMS 100 x [Blk(adj)SS+Intra error SS]/k(k 2 -1)EMS 100 [11,382 + 14,533]/4(16)369 117% I II III IV V I II III I II III IV V

26 Lattice Square ANOVA SourcedfSSMSF Reps45,9461,4864.03 * T(unadj)1526,9951,800- Blk(adj)1511,3827592.35 ns Intra error4514,533323- T(adj)1524,0301,6024.34 ** Eff. Error4516,605369-

27 RCB ANOVA SourcedfSSMSF Reps45,9461,4863.44 * T(unadj)1526,9951,8004.25 ** Error6025,915432-

28 Lattice Square ANOVA SourcedfSSMSF Reps45,9461,4864.03 * T(unadj)1526,9951,800- Blk(adj)151,382920.17 ns Intra error4524,533545- T(adj)1524,0301,6022.71 * Eff. Error4526,605591-

29  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. Lattice Square ANOVA

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 = [8.19 2 + 8.37 2 + … + 4.15 2 ]-CF = 4612.56 Rep SS = [98.6 2 + 99.1 2 + 99.3 2 ]/18-CF = 0.01

34 Factorial AOV Example Seed rate Nitrigen level 90100110120130140Total High12.813.715.418.019.624.9104.4 Med.12.712.914.116.119.223.098.0 Low12.212.913.615.718.921.294.6 Total37.739.543.149.857.869.2297.0 Seed rate SS = [104.4 2 + 98.0 2 + 94.6 2 ]/18 – CF = 2.75

35 Factorial AOV Example Seed rate Nitrigen level 90100110120130140Total High12.813.715.418.019.624.9104.4 Med.12.712.914.116.119.223.098.0 Low12.212.913.615.718.921.294.6 Total37.739.543.149.857.869.2297.0 N rate SS = [37.7 2 + 39.5 2 + …+ 69.2 2 ]/9 – CF = 2.75

36 Factorial AOV Example Seed rate Nitrigen level 90100110120130140Total High12.813.715.418.019.624.9104.4 Med.12.712.914.116.119.223.098.0 Low12.212.913.615.718.921.294.6 Total37.739.543.149.857.869.2297.0 Seed x N SS = [12.8 2 + 13.7 2 + …+ 21.2 2 ]/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 SourcedfSSMSF Reps20.010.005ns Seed Density22.751.37533.9 *** Nitrogen581.5616.312401.9 *** S x N101.330.1333.28 *** Error341.380.041 Total5387.03

39 Factorial AOV Example CV =  /  x 100 =  0.041/5.50 = 3.38% R 2 = [TSS-ESS]/TSS = [87.03-1.38]/87.03 = 96.2%

40 Factorial AOV Example SourcedfSSMSF Reps x Seed Rate40.22680.05671.63 ns Rep x N rate100.45280.04531.30 ns Rep x Seed x N200.69360.0347

41 Factorial AOV Example Seed rate Nitrigen level 90100110120130140Total High4.284.455.146.006.538.305.80 Med.4.234.304.705.366.417.675.44 Low4.074.304.535.246.317.085.26 Total4.194.394.795.536.427.685.50 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 SourcedfSSMSF Reps20.010.005ns Seed Density22.751.37533.9 *** Nitrogen581.5616.312401.9 *** S x N101.330.1333.28 *** Error341.380.041 Total5387.03

44 Factorial AOV Example SourcedfSSMSF Reps x Seed Rate40.22680.05671.63 ns Rep x N rate100.45280.04531.30 ns Rep x Seed x N200.69360.0347

45 Split-plot AOV SourcedfSSMSF Reps20.010.005 Seed Density22.751.375 Reps x Seed40.22680.0567 Nitrogen581.5616.312 S x N101.330.133 Rep x N rate100.45280.0453 Rep x Seed x N200.69360.0347 Total5387.03

46 Split-plot AOV SourcedfSSMSF Reps20.010.005ns Seed Density22.751.37524.2 *** Error (1)40.22680.057- Nitrogen581.5616.312426.9 *** S x N101.330.1333.5 *** Error (2)301.14640.038- Total5387.03

47 Strip-plot AOV SourcedfSSMSF Reps20.010.005 Seed Density22.751.375 Reps x Seed40.22680.0567 Nitrogen581.5616.312 Rep x N rate100.45280.0453 S x N101.330.133 Rep x Seed x N200.69360.0347 Total5387.03

48 Strip-plot AOV SourcedfSSMSF Reps20.010.005ns Seed Density22.751.37524.2 *** Error 1 (Seed)40.22680.0567- Nitrogen581.5616.312360.1 *** Error 2 (N)100.45280.0453- S x N101.330.1333.83 *** Error 3 (SxN)200.69360.0347- Total5387.03

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.

51 Setting Expected Mean Squares  The expected mean square for a source of variation (say X) contains.  the error term.  a term in  2 x. (or S 2 x )  a variance term for other selected interactions involving the factor X.

52 Coefficients for EMS Coefficient for error mean square is always 1 Coefficient of other expected mean squares is n times the product of factors levels that do not appear in the factor name.

53 Expected Mean Squares  Which interactions to include in an EMS?  All the letter (i.e. A, B, C, …) appear in X.  All the other letters in the interaction (except those in X) are Random Effects.

54 A and B Fixed Effects

55 A and B Random Effects

56 A Fixed and B Random

57 A, B, and C are Fixed

58 A, B, and C are Random

59

60

61

62 A Fixed, B and C are Random

63

64

65

66 Analysis of Split-plots and Strip-plots and nested designs Multiple Comparisons

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