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

Alaskan Wedding Feast Marvelous Marvin father of the Groom

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  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.

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.

Analyses of Lattice Squares  Treatment 5 is in block 2, 5, 10, 15, and 20, so B 5 = =  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

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

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

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

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

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

Analyses of Lattice Squares  Compute the adjusted block SS as: Block (adj) SS =  W 2 /k 3 (k+1) – CF [ … ]/320 – 2,361,906 = 11,382

Analyses of Lattice Squares  Compute the intra-block error SS as: IB error SS = TSS–Rep SS–Treat(unadj) SS–Blk(adj) SS 58, , , ,382 = 14,533

Lattice Square ANOVA SourcedfSSMS Reps45,9461,486 T(unadj)1526,9951,800 Blk(adj)1511, Intra block error4514,  Calculate Mean Squares for block(adj) and IBE.

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]

Analyses of Lattice Squares  Compute adjusted treatment totals (T’) as:   = [ ]/(16)(759) = T’ = T +  W  = [Blk(adj) MS-IBE MS]/[k 2 Blk(adj) MS]

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]

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]

Analyses of Lattice Squares  Compute adjusted treatment totals (T’) as:  T ’ 5 = T 5 +  W 5  T ’ 5 = x (-45) = 814 T’ = T +  W  = [Blk(adj) MS-IBE MS]/[k 2 Blk(adj) MS]

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

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

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

Lattice Square ANOVA SourcedfSSMSF Reps45,946 T(unadj)1526,995 Blk(adj)1511,382 Intra error4514,533 T(adj)1524,030 Eff. Error4516,605

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

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, ,533]/4(16) % I II III IV V I II III I II III IV V

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

RCB ANOVA SourcedfSSMSF Reps45,9461, * T(unadj)1526,9951, ** Error6025,

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

 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

Comparison of Rankings

ANOVA of Factorial Designs

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

Factorial AOV Example CF = (297.0) 2 /54 = TSS = [ … ]-CF = Rep SS = [ ]/18-CF = 0.01

Factorial AOV Example Seed rate Nitrigen level Total High Med Low Total Seed rate SS = [ ]/18 – CF = 2.75

Factorial AOV Example Seed rate Nitrigen level Total High Med Low Total N rate SS = [ … ]/9 – CF = 2.75

Factorial AOV Example Seed rate Nitrigen level Total High Med Low Total Seed x N SS = [ … ]/3 – CF - Seed rate SS – Nitrogen SS = 1.33

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

Factorial AOV Example SourcedfSSMSF Reps ns Seed Density *** Nitrogen *** S x N *** Error Total

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

Factorial AOV Example SourcedfSSMSF Reps x Seed Rate ns Rep x N rate ns Rep x Seed x N

Factorial AOV Example Seed rate Nitrigen level Total High Med Low Total sed[within] =  (2  2 /3) = sed[Seed rate] =  (2  2 /18) = sed[N rate] =  (2  2 /9) = 0.095

Factorial AOV Example

SourcedfSSMSF Reps ns Seed Density *** Nitrogen *** S x N *** Error Total

Factorial AOV Example SourcedfSSMSF Reps x Seed Rate ns Rep x N rate ns Rep x Seed x N

Split-plot AOV SourcedfSSMSF Reps Seed Density Reps x Seed Nitrogen S x N Rep x N rate Rep x Seed x N Total

Split-plot AOV SourcedfSSMSF Reps ns Seed Density *** Error (1) Nitrogen *** S x N *** Error (2) Total

Strip-plot AOV SourcedfSSMSF Reps Seed Density Reps x Seed Nitrogen Rep x N rate S x N Rep x Seed x N Total

Strip-plot AOV SourcedfSSMSF Reps ns Seed Density *** Error 1 (Seed) Nitrogen *** Error 2 (N) S x N *** Error 3 (SxN) Total

Fixed and Random Effects

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

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.

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.

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.

A and B Fixed Effects

A and B Random Effects

A Fixed and B Random

A, B, and C are Fixed

A, B, and C are Random

A Fixed, B and C are Random

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