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