jack’s gone to the dogs in alaska february 25, 2005
TRANSCRIPT
Jack’s gone to the dogs in AlaskaJack’s gone to the dogs in Alaska
February 25, 2005February 25, 2005
Alaskan Wedding Feast
Marvelous Marvin father of the Groom
Analyses of Lattice SquaresAnalyses of Lattice Squares
YYijkijk = = + r + rii + b + baajj + t + taa
k k ++ eeijkijk
See Table 5 & 6, See Table 5 & 6, Page 105 & 106Page 105 & 106
Analyses of Lattice SquaresAnalyses 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.
Analyses of Lattice SquaresAnalyses 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.
Analyses of Lattice SquaresAnalyses 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
Lattice Square ANOVA - d.f.Lattice Square ANOVA - d.f.Source df
Reps k 4
Trt(unadj) k2 – 1 15
Block(adj) k2 – 1 15
Intra-Block Error (k-1)(k2-1) 45
Trt (adj) k2 – 1 15
Effective Error (k-1)(k2-1) 45
Total k2(k+1)-1 79
Analyses of Lattice SquaresAnalyses 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
Analyses of Lattice SquaresAnalyses of Lattice Squares
Compute the total SS as:
Total SS = xij2 – CF
[1472+1522+…+2252] – 2,361,906
= 58,856
Analyses of Lattice SquaresAnalyses of Lattice Squares
Compute the replicate block SS as:
Replicate SS = R2/k2 – CF
[25952+27292+…+29252]/16 – 2,361,906
= 5,946
Analyses of Lattice SquaresAnalyses 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
Analyses of Lattice SquaresAnalyses 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
Analyses of Lattice SquaresAnalyses 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
Lattice Square ANOVALattice Square ANOVA
Source df SS MS
Reps 4 5,946 1,486
T(unadj) 15 26,995 1,800
Blk(adj) 15 11,382 759
Intra block error 45 14,533 323
Calculate Mean Squares for block(adj) and IBE.
Analyses of Lattice SquaresAnalyses of Lattice Squares
Compute adjusted treatment totals (T’) as:
T’i = Ti + Wi
= [Blk(adj) MS-IBE MS]/[k2 Blk(adj) MS]
Analyses of Lattice SquaresAnalyses 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]
Analyses of Lattice SquaresAnalyses 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]
Analyses of Lattice SquaresAnalyses 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]
Analyses of Lattice SquaresAnalyses 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]
Analyses of Lattice SquaresAnalyses of Lattice Squares
Compute adjusted treatment means (M’) as:
M’ = T’/[k+1]
Analyses of Lattice SquaresAnalyses 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
Analyses of Lattice SquaresAnalyses of Lattice Squares
Compute effective error MS as:
EE MS = (Intra-block error MS)(1+k)
323[1 + 4(0.0359)]
369
Lattice Square ANOVALattice Square ANOVA
Source df SS MS F
Reps 4 5,946
T(unadj) 15 26,995
Blk(adj) 15 11,382
Intra error 45 14,533
T(adj) 15 24,030
Eff. Error 45 16,605
Lattice Square ANOVALattice Square ANOVA
Source df SS MS F
Reps 4 5,946 1,486 4.03 *
T(unadj) 15 26,995 1,800 -
Blk(adj) 15 11,382 759 2.35 ns
Intra error 45 14,533 323 -
T(adj) 15 24,030 1,602 4.34 **
Eff. Error 45 16,605 369 -
Efficiency of Lattice DesignEfficiency of Lattice Design
100 x [Blk(adj)SS+Intra error SS]/k(k100 x [Blk(adj)SS+Intra error SS]/k(k2-1)EMS-1)EMS
100 [11,382 + 14,533]/4(16)369100 [11,382 + 14,533]/4(16)369
117%117%
I II III IV VI II III IV V
I II IIII II III
IV VIV V
Lattice Square ANOVALattice Square ANOVA
Source df SS MS F
Reps 4 5,946 1,486 4.03 *
T(unadj) 15 26,995 1,800 -
Blk(adj) 15 11,382 759 2.35 ns
Intra error 45 14,533 323 -
T(adj) 15 24,030 1,602 4.34 **
Eff. Error 45 16,605 369 -
RCB ANOVARCB ANOVA
Source df SS MS F
Reps 4 5,946 1,486 3.44 *
T(unadj) 15 26,995 1,800 4.25 **
Error 60 25,915 432 -
Lattice Square ANOVALattice Square ANOVA
Source df SS MS F
Reps 4 5,946 1,486 4.03 *
T(unadj) 15 26,995 1,800 -
Blk(adj) 15 1,382 92 0.17 ns
Intra error 45 24,533 545 -
T(adj) 15 24,030 1,602 2.71 *
Eff. Error 45 26,605 591 -
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 ANOVALattice Square ANOVA
Comparison of RankingsComparison of Rankings
0
2
4
6
8
10
12
14
16
0 5 10 15
RCB Rank
Lat
tice
Ran
k
ANOVA of Factorial DesignsANOVA of Factorial Designs
Factorial AOV ExampleFactorial AOV Example
Spring barley ‘Malter’Three seeding rates (low, Medium and
High).Six nitrogen levels (90, 100, 110, 120,
130, 140 units).Three replicatesPage 107 of class notes
Factorial AOV ExampleFactorial AOV Example
CF = (297.0)CF = (297.0)22/54 = 3676.6/54 = 3676.6
TSS = [8.19TSS = [8.1922 + 8.37 + 8.3722 + … + 4.15 + … + 4.1522]-CF ]-CF = 4612.56= 4612.56
Rep SS = [98.6Rep SS = [98.622 + 99.1 + 99.122 + 99.3 + 99.322]/18-CF ]/18-CF = 0.01= 0.01
Factorial AOV ExampleFactorial AOV Example
Seed rate
Nitrigen level
90 100 110 120 130 140 Total
High 12.8 13.7 15.4 18.0 19.6 24.9 104.4
Med. 12.7 12.9 14.1 16.1 19.2 23.0 98.0
Low 12.2 12.9 13.6 15.7 18.9 21.2 94.6
Total 37.7 39.5 43.1 49.8 57.8 69.2 297.0
Seed rate SS = [104.4Seed rate SS = [104.422 + 98.0 + 98.022 + 94.6 + 94.622]/18 – CF ]/18 – CF = 2.75= 2.75
Factorial AOV ExampleFactorial AOV Example
Seed rate
Nitrigen level
90 100 110 120 130 140 Total
High 12.8 13.7 15.4 18.0 19.6 24.9 104.4
Med. 12.7 12.9 14.1 16.1 19.2 23.0 98.0
Low 12.2 12.9 13.6 15.7 18.9 21.2 94.6
Total 37.7 39.5 43.1 49.8 57.8 69.2 297.0
N rate SS = [37.7N rate SS = [37.722 + 39.5 + 39.522 + …+ 69.2 + …+ 69.222]/9 – CF = ]/9 – CF = 2.752.75
Factorial AOV ExampleFactorial AOV Example
Seed rate
Nitrigen level
90 100 110 120 130 140 Total
High 12.8 13.7 15.4 18.0 19.6 24.9 104.4
Med. 12.7 12.9 14.1 16.1 19.2 23.0 98.0
Low 12.2 12.9 13.6 15.7 18.9 21.2 94.6
Total 37.7 39.5 43.1 49.8 57.8 69.2 297.0
Seed x N SS = [12.8Seed x N SS = [12.822 + 13.7 + 13.722 + …+ 21.2 + …+ 21.222]/3 – CF ]/3 – CF
- Seed rate SS – Nitrogen SS = 1.33- Seed rate SS – Nitrogen SS = 1.33
Factorial AOV ExampleFactorial AOV Example
Error SS=TSS–Seed SS–N SS–NxS SS–Rep SSError SS=TSS–Seed SS–N SS–NxS SS–Rep SS
Factorial AOV ExampleFactorial AOV Example
Source df SS MS F
Reps 2 0.01 0.005 ns
Seed Density 2 2.75 1.375 33.9 ***
Nitrogen 5 81.56 16.312 401.9***
S x N 10 1.33 0.133 3.28***
Error 34 1.38 0.041
Total 53 87.03
Factorial AOV ExampleFactorial AOV Example
CV = CV = // x 100 x 100
= = 0.041/5.50 = 3.38%0.041/5.50 = 3.38%
RR22 = [TSS-ESS]/TSS = [TSS-ESS]/TSS
= [87.03-1.38]/87.03 = 96.2%= [87.03-1.38]/87.03 = 96.2%
Factorial AOV ExampleFactorial AOV Example
Source df SS MS F
Reps x Seed Rate 4 0.2268 0.0567 1.63 ns
Rep x N rate 10 0.4528 0.0453 1.30 ns
Rep x Seed x N 20 0.6936 0.0347
Factorial AOV ExampleFactorial AOV Example
Seed rate
Nitrigen level
90 100 110 120 130 140 Total
High 4.28 4.45 5.14 6.00 6.53 8.30 5.80
Med. 4.23 4.30 4.70 5.36 6.41 7.67 5.44
Low 4.07 4.30 4.53 5.24 6.31 7.08 5.26
Total 4.19 4.39 4.79 5.53 6.42 7.68 5.50sed[within] = sed[within] = (2(222/3) = 0.165/3) = 0.165
sed[Seed rate] = sed[Seed rate] = (2(222/18) = 0.067/18) = 0.067
sed[N rate] = sed[N rate] = (2(222/9) = 0.095/9) = 0.095
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
90 100 110 120 130 140
Nitrogen applied
Yie
ld (
ton
/acre)
high med low
Factorial AOV ExampleFactorial AOV Example
Factorial AOV ExampleFactorial AOV Example
Source df SS MS F
Reps 2 0.01 0.005 ns
Seed Density 2 2.75 1.375 33.9 ***
Nitrogen 5 81.56 16.312 401.9***
S x N 10 1.33 0.133 3.28***
Error 34 1.38 0.041
Total 53 87.03
Factorial AOV ExampleFactorial AOV Example
Source df SS MS F
Reps x Seed Rate 4 0.2268 0.0567 1.63 ns
Rep x N rate 10 0.4528 0.0453 1.30 ns
Rep x Seed x N 20 0.6936 0.0347
Split-plot AOVSplit-plot AOV
Source df SS MS F
Reps 2 0.01 0.005
Seed Density 2 2.75 1.375
Reps x Seed 4 0.2268 0.0567
Nitrogen 5 81.56 16.312
S x N 10 1.33 0.133
Rep x N rate 10 0.4528 0.0453
Rep x Seed x N 20 0.6936 0.0347
Total 53 87.03
Split-plot AOVSplit-plot AOV
Source df SS MS F
Reps 2 0.01 0.005 ns
Seed Density 2 2.75 1.375 24.2 ***
Error (1) 4 0.2268 0.057 -
Nitrogen 5 81.56 16.312 426.9***
S x N 10 1.33 0.133 3.5***
Error (2) 30 1.1464 0.038 -
Total 53 87.03
Strip-plot AOVStrip-plot AOV
Source df SS MS F
Reps 2 0.01 0.005
Seed Density 2 2.75 1.375
Reps x Seed 4 0.2268 0.0567
Nitrogen 5 81.56 16.312
Rep x N rate 10 0.4528 0.0453
S x N 10 1.33 0.133
Rep x Seed x N 20 0.6936 0.0347
Total 53 87.03
Strip-plot AOVStrip-plot AOV
Source df SS MS F
Reps 2 0.01 0.005 ns
Seed Density 2 2.75 1.375 24.2 ***
Error 1 (Seed) 4 0.2268 0.0567 -
Nitrogen 5 81.56 16.312 360.1***
Error 2 (N) 10 0.4528 0.0453 -
S x N 10 1.33 0.133 3.83***
Error 3 (SxN) 20 0.6936 0.0347 -
Total 53 87.03
Fixed and Random Fixed and Random EffectsEffects
Expected Mean SquaresExpected Mean Squares
Dependant on whether factor Dependant on whether factor effects are Fixed or Random.effects are Fixed or Random.
Necessary to determine which Necessary to determine which F-tests are appropriate and F-tests are appropriate and which are not.which are not.
Setting Expected Mean SquaresSetting Expected Mean Squares
The expected mean square The expected mean square for a source of variation (say X) for a source of variation (say X) contains.contains.
the error term.a term in 2
x. (or S2x )
a variance term for other selected interactions involving the factor X.
Coefficients for EMSCoefficients 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.
Expected Mean SquaresExpected Mean Squares
Which interactions to include in an Which interactions to include in an EMS?EMS?
All the letter (i.e. A, B, C, …) All the letter (i.e. A, B, C, …) appear in X.appear in X.
All the other letters in the All the other letters in the interaction (except those in X) are interaction (except those in X) are Random Effects.Random Effects.
A and B Fixed EffectsA and B Fixed Effects
Source d.f. EMSq
A (a) a-1 2e + rbS2
A
B (b) b-1 2e + raS2
B
A x B (a-1)(b-1) 2e + rS2
AB
Error r(a-1)(b-1) 2e
A and B Random EffectsA and B Random Effects
Source d.f. EMSq
A (a) a-1 2e + r2
AB + rb2A
B (b) b-1 2e + r2
AB + ra2B
A x B (a-1)(b-1) 2e + r2
AB
Error r(a-1)(b-1) 2e
A Fixed and B RandomA Fixed and B Random
Source d.f. EMSq
A (a) a-1 2e + r2
AB + rbS2A
B (b) b-1 2e + ra2
B
A x B (a-1)(b-1) 2e + r2
AB
Error r(a-1)(b-1) 2e
A, B, and C are FixedA, B, and C are FixedSource d.f. EMSq
A (a) a-1 2e + rbcS2
A
B (b) b-1 2e + racS2
B
C (c) c-1 2e + rabS2
C
A x B (a-1)(b-1) 2e + rcS2
AB
A x C (a-1)(c-1) 2e + rbS2
AC
B x C (b-1)(c-1) 2e + raS2
BC
A x B x C (a-1)(b-1)(c-1) 2e + rS2
ABC
Error r(a-1)(b-1)(c-1) 2e
A, B, and C are RandomA, B, and C are RandomSource d.f. EMSq
A (a) a-1 2e
B (b) b-1 2e
C (c) c-1 2e
A x B (a-1)(b-1) 2e
A x C (a-1)(c-1) 2e
B x C (b-1)(c-1) 2e
A x B x C (a-1)(b-1)(c-1) 2e
Error r(a-1)(b-1)(c-1) 2e
A, B, and C are RandomA, B, and C are RandomSource d.f. EMSq
A (a) a-1 2e + rbc2
A
B (b) b-1 2e + rac2
B
C (c) c-1 2e + rab2
C
A x B (a-1)(b-1) 2e + rc2
AB
A x C (a-1)(c-1) 2e + rb2
AC
B x C (b-1)(c-1) 2e + ra2
BC
A x B x C (a-1)(b-1)(c-1) 2e + r2
ABC
Error r(a-1)(b-1)(c-1) 2e
A, B, and C are RandomA, B, and C are RandomSource d.f. EMSq
A (a) a-1 2e + rbc2
A
B (b) b-1 2e + rac2
B
C (c) c-1 2e + rab2
C
A x B (a-1)(b-1) 2e + r2
ABC + rc2AB
A x C (a-1)(c-1) 2e + r2
ABC + rb2AC
B x C (b-1)(c-1) 2e + r2
ABC + ra2BC
A x B x C (a-1)(b-1)(c-1) 2e + r2
ABC
Error r(a-1)(b-1)(c-1) 2e
A, B, and C are RandomA, B, and C are RandomSource d.f. EMSq
A (a) a-1 2e+r2
ABC + rc2AB + rb2
AC+ rbc2A
B (b) b-1 2e+r2
ABC + rc2AB + ra2
BC + rac2B
C (c) c-1 2e+r2
ABC + rb2AC + ra2
BC + rab2C
A x B (a-1)(b-1) 2e + r2
ABC + rc2AB
A x C (a-1)(c-1) 2e + r2
ABC + rb2AC
B x C (b-1)(c-1) 2e + r2
ABC + ra2BC
A x B x C (a-1)(b-1)(c-1) 2e + r2
ABC
Error r(a-1)(b-1)(c-1) 2e
A Fixed, B and C are RandomA Fixed, B and C are RandomSource d.f. EMSq
A (a) a-1 2e
B (b) b-1 2e
C (c) c-1 2e
A x B (a-1)(b-1) 2e
A x C (a-1)(c-1) 2e
B x C (b-1)(c-1) 2e
A x B x C (a-1)(b-1)(c-1) 2e
Error r(a-1)(b-1)(c-1) 2e
A Fixed, B and C are RandomA Fixed, B and C are RandomSource d.f. EMSq
A (a) a-1 2e + rbc2
A
B (b) b-1 2e + rac2
B
C (c) c-1 2e + rab2
C
A x B (a-1)(b-1) 2e + rc2
AB
A x C (a-1)(c-1) 2e + rb2
AC
B x C (b-1)(c-1) 2e + ra2
BC
A x B x C (a-1)(b-1)(c-1) 2e + r2
ABC
Error r(a-1)(b-1)(c-1) 2e
A Fixed, B and C are RandomA Fixed, B and C are RandomSource d.f. EMSq
A (a) a-1 2e + rbc2
A
B (b) b-1 2e + rac2
B
C (c) c-1 2e + rab2
C
A x B (a-1)(b-1) 2e + r2
ABC + rc2AB
A x C (a-1)(c-1) 2e + r2
ABC + rb2AC
B x C (b-1)(c-1) 2e + ra2
BC
A x B x C (a-1)(b-1)(c-1) 2e + r2
ABC
Error r(a-1)(b-1)(c-1) 2e
A Fixed, B and C are RandomA Fixed, B and C are Random
Source d.f. EMSq
A (a) a-1 2e+r2
ABC + rc2AB + rb2
AC+ rbcS2A
B (b) b-1 2e+ra2
BC + rac2B
C (c) c-1 2e+ra2
BC + rab2C
A x B (a-1)(b-1) 2e + r2
ABC + rc2AB
A x C (a-1)(c-1) 2e + r2
ABC + rb2AC
B x C (b-1)(c-1) 2e + ra2
BC
A x B x C (a-1)(b-1)(c-1) 2e + r2
ABC
Error r(a-1)(b-1)(c-1) 2e
Analysis of Split-plots and Analysis of Split-plots and Strip-plots and nested designsStrip-plots and nested designs
Multiple ComparisonsMultiple Comparisons