ii. More than one per Trt*Block -> EXPLORATORY MODEL



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1. What is the experimental unit? WHAT IS RANDOMIZED. Are we taking multiple measures of the SAME e.u.? a. Subsamples AVERAGE FIRST! b. Repeated measures in time 3. Are there blocking variables? a. No -> CRD b. One->RCBD i. One observation per T*Block -> TUKEY ADDITIVITY TEST ii. More than one per Trt*Block -> EXPLORATORY MODEL c. More than one blocking variable i. Only one Trt per Blocks combination -> LATIN SQUARE ii. All Trt in each blocks combination -> TWO BLOCKING VARIABLES 4. More than one Treatment? a. Not all combinations b. All levels of one Factor at all levels of the other Factor->FACTORIAL i. Missing data LSMEANS & TYPE IIISS ii. Random factors MIXED or RANDOM MODELS Random /test iii. Splits SPLIT PLOT, STRIP PLOT, SPLIT-SPLIT PLOT c. Significant Interaction -> SIMPLE EFFECTS d. Not significant interactions -> MAIN EFFECTS 5. Covariable LSMEANS & TYPE IIISS If too complex transform y into Z and then do normal ANOVA 1

ANCOVA for a 3x Factorial arranged as an RCBD Data ABFact; Input Block A B X Y @@; Z = Y Beta * (X - Xmean); Cards;... ; Beta from solution Xmean from X ANOVA Individual ANOVAs Class Block A B; Model X Y = Block A B A*B; ANCOVA (Type III SS & LS means) If split? Class Block A B; Model Y = Block A B A*B X / solution; LSMeans A; LSMeans B; Contrast 'A linear' A -1 0 1; Run; Quit; If A*B interaction NS main effects, if significant simple effects 1. To test normality of residuals ANOVA on Z Class Block A B; Model Z = Block A B A*B; Output out = ABFactPRz p = Pred r = Res; Output can be in ANCOVA Proc Univariate Data = ABFactPRz normal; Var Res;. To test homogeneity of variances Create a Treatment ID (TRT) with 6 levels, one for each A-B combination (if interaction *) or separate GLM and Levene tests for A and B (if interaction NS) Class TRT; Model Z = TRT; Means TRT / hovtest = Levene; (or separate Levene test for A and B if there is no interaction) 3. To test homogeneity of slopes Class Block TRT; Model Y = Block TRT X TRT*X; or in no interaction different Proc GLM Model Y = Block A X A*X; and Model Y = Block B X B*X;

4. To partition the A*B interaction Proc GLM order=data; BE CAREFUL WITH DARTA ORDER! Class Block TRT; Model Y = Block TRT X; LSMeans TRT; Contrast 'A Linear' TRT -1 0 1-1 0 1; Contrast 'A Quadratic' TRT 1-1 1-1; Contrast 'B' TRT 1 1 1-1 -1-1; Contrast 'A Lin x B' TRT -1 0 1 1 0-1; Contrast 'A Quad x B' TRT 1-1 -1-1; 5. To analyze a mixed model or split plot with covariable Example: Factor A random, Factor B fixed Model Y = Block A B A*B X; Random Block A A*B / test; ANOVA without X in the model Source Type III Expected Mean Square Block A B A*B Var(Error) + 6 Var(Block) Var(Error) + 4 Var(A*B) + 8 Var(A) Var(Error) + 4 Var(A*B) + Q(B) Var(Error) + 4 Var(A*B) Test h = A e = A*B; Contrast 'A Linear' A -1 0 1 /e = A*B; With X in the model the coefficients are adjusted by the covariable and the test statement is required. Contrasts and mean comparisons require Proc Mixed Alternative: Transform Y into Z using the covariable and then used the simple model and error terms in Z (from the mixed model or from the split plot). Model Z = Block A B A*B; with the EMS as above. Test h = A e = A*B; Contrast 'A Linear' A -1 0 1 /e = A*B; 3

Mixed Model: Blocks nested within Random Locations Five varieties of barley are tested in three locations selected at random across the Sacramento Valley. In each location, the experiment is organized as an RCBD with four blocks. The intention is to: a. Characterize the variability across the valley. b. Characterize the stability of variety performance across the valley. c. Recommend a variety(ies) for the valley. Design: Response Variable: Experimental Unit: Class Variable 1 n Block or Treatment Number of Levels Fixed or Random Description Class Location Block Variety; Model Yield = Location Block(Location) Variety Location*Variety; Random Location Block(Location) Location*Variety / test; Source Type III Expected Mean Square F. Location Block(Location) Variety Location*Variety b=4 v=5 Error b v bv (MS L +MS E )/(MS BL(L) +MS L*V ) LOC* VAR Error v BLOCK ( LOC) Error b LOC* VAR Error b LOC * VAR BLOCK ( LOC) LOC MS BL(L) / MS E Q(var) MS V / MS L*V MS L*V / MS E Proc VarComp Method = Type1; Class Location Block Variety; Model Y = Variety Location Block(Location) Location*Variety / Fixed = 1; FIXED=n: specifies that the first n effects in the MODEL statement are fixed effects. The remaining effects are assumed to be random. By default, PROC VARCOMP assumes that all effects are random in the model. Variance Component Estimate σ (Location) 60.4536 σ (Location*Variety) 1.5485 σ (Block(Location)) 15.16845 σ (Error) 150.8798 4

Three-way factorials with one split (NOT a split-split-plot) Example 1: RCBD with BLOCK 1 BLOCK Main plot = Factorial treatment structure (A*B) Subplot = Factor C a-b1 a1-b a1-b1 a-b c3 c1 c c c c c1 c3 c1 c3 c3 c1 a1-b1 a-b a-b1 a1-b c1 c c1 c3 c3 c1 c c c c3 c3 c1 Class Block A B C; Model Y = Block A B A*B Block*A*B C A*C B*C A*B*C; Test h = A e = Block*A*B; Test h = B e = Block*A*B; Test h = A*B e = Block*A*B; If A and B would have had three levels, for example: Means A / Tukey e = Block*A*B; or Contrast 'B Linear' B 1 0-1 / e = Block*A*B; Alternative: open up the A*B factorial, expressing the combinations as levels of some single factor TRT: Class Block TRT C; Model Y = Block TRT Block*TRT C C*TRT; Test h = TRT e = Block*TRT; Means TRT / Tukey e = Block*TRT; # A1B1 A1B AB1 AB: Contrast 'B ' TRT 1-1 1-1 / e = Block*TRT; Contrast 'A ' TRT 1 1-1 -1 / e = Block*TRT; Contrast 'A*B ' TRT 1-1 -1 1 / e = Block*TRT; If you do not have the A and B factors, then you need contrasts to separate the effects of A, B and A*B 5

Example : RCBD with Main plot = Factor A ( levels) Subplot = x3 factorial treatment structure (B*C) BLOCK 1 BLOCK a a1 b1-c3 b-c1 b-c b1-c b-c b1-c b-c1 b1-c3 b1-c1 b-c3 b-c3 b1-c1 a1 a b1-c1 b1-c b-c1 b1-c3 b-c3 b-c1 b1-c b-c b-c b1-c3 b-c3 b1-c1 Class Block A B C; Model Y = Block A Block*A B C B*C A*B A*C A*B*C; Test h = A e = Block*A; Means A / Tukey e = Block*A; Contrast 'A Linear' A 1 0-1 / e = Block*A; If A had three levels, for example Alternatively, open up the B*C factorial, expressing the combinations as levels of some single factor TRT: Class Block A TRT; Model Y = Block A Block*A TRT A*TRT; Test h = A e = Block*A; TRT= B C B*C and A*TRT= A*B A*C A*B*C The model is simpler but then you need contrasts to solve: B C B*C A*B A*C A*B*C 6

Split-split-plot with last split in time Example: In each of 3 randomly picked clinics used as block, select 8 diabetic patients from each sex and then randomly assign within each sex the 8 patients to treatments A, B, C and D (two patients per treatment). Glucose level in the blood of the patients is monitored at 1, 4 and 36 hours. Level one Clinic SEX Clinic*SEX= Error A Level two Trt SEX*Trt Clinic*Trt + Clinic*SEX*Trt= Error B Level three: use conservative degrees of freedom: different F threshold (df/(3-1) time SEX*time Trt*time SEX*Trt*time Clinic*time + Clinic*SEX*time + Clinic*Trt*time + Clinic*SEX*Trt*time= Error C When you cannot randomly assign one treatment but instead you need to select first one category (sex, breed, etc.) and then within that class randomize the other treatment, this needs to be treated as a split in which the first category is the main plot. Model Glucose= Clinic Sex Clinic*Sex Trt Sex*Trt Clinic*Sex*Trt Time Sex*time Trt*Time Sex*Trt*Time; Test h=sex e= Clinic*Sex; Test h=trt e= Clinic*Sex*Trt; Test h=sex*trt e= Clinic*Sex*Trt; Means Trt / tukey e= Clinic*Sex*Trt; 7

SPLIT PLOT A Error A CRD RCBD Latin Square Rows a-1 Blocks r-1 Columns a-1 a-1 A a-1 A a-1 a(r-1) Error A (r-1)(a-1) Error A (a-1)(a-) Total ra-1 Total ra-1 Total ra-1 Factor B A x B Error B Total b-1 (a-1)(b-1) a(r-1)(b-1) rab-1 Factor B A x B Error B Total b-1 (a-1)(b-1) a(r-1)(b-1) rab-1 Factor B A x B Error B Total b-1 (a-1)(b-1) a(r-1)(b-1) rab-1 CRD RCBD LS proc glm; proc glm; proc glm; class rep A B; class bl A B; class row col A B; model y= A rep*a model y=bl A bl*a model y=col row A col*row*a B A*B; B A*B; B A*B; test h=a e=rep*a; test h=a e=bl*a; test h=a e= col*row*a; STRIP PLOT proc GLM; class block A B; model yield = block A A*block B B*block B*A; test h=a e=a*block; test h=b e=b*block; Replicated LS (= rows & columns) proc glm; class sqr row col A B; model y=sqr col row A sqr*col*row*a B A*B; test h=a e= sqr*col*row*a; SPLIT SPLIT PLOT proc glm; class Block a b c; model response= Block a Block*a b a*b Block*b*a c a*c b*c a*b*c; test h=a e= Block*a; test h=b e= Block*b*a; test h=a*b e= Block*b*a; 8

Strip Plot or Split Block EMS with Random statement proc glm; class block A B; model yield=block A block*a B block*b A*B; random block block*a block*b /Test; Source block A block*a B block*b A*B Type III Expected Mean Square Var(Error) + 4 Var(block*B) + 4 Var(block*A)+ 8 Var(block) Var(Error) + 4 Var(block*A) + Q(A,A*B) Var(Error) + 4 Var(block*A) Var(Error) + 4 Var(block*B) + Q(B,A*B) Var(Error) + 4 Var(block*B) Var(Error) + Q(A*B) test h=a e=block*a; test h=b e=block*b; Split Plot EMS with Random statement proc glm; class block A B; model yield=block A block*a B A*B; random block block*a /Test; Source block A block*a B A*B Type III Expected Mean Square Var(Error) + 4 Var(block*A) + 8 Var(block) Var(Error) + 4 Var(block*A) + Q(A,A*B) Var(Error) + 4 Var(block*A) Var(Error) + Q(B,A*B) Var(Error) + Q(A*B) test h=a e=block*a; 9

A 4x3x3 factorial experiment, organized as an RCBD (fixed model) Class Block A B C; Model Y = Block A B C; Two -way interactions significant: If A*B and A*C are significant, but B*C is NS: Proc Sort; Proc Sort; By A; By B C; Class Block B C; Class Block A; Model Y = Block B C; Model Y = Block A; Means B / Tukey; Means A / Tukey; Means C / Tukey; Contrast 'Linear A' A -3 1 1 1; Contrast 'Linear B' B 1 0-1; By B C; By A; One -way interaction significant: If A*B is significant, but A*C and B*C are NS: It is valid to analyze the main effect of C: e.g. Contrast 'Linear C' C 1 0-1; Means C / Tukey; But for A and B: 1. Describe the effect of A within each level of B (across all C levels). Describe the effect of B within each level of A (across all C levels) Proc Sort; Proc Sort; By B; By A; Class Block A; Class Block B; Model Y = Block A; Model Y = Block B; By B; By A; If C is random include C and A*C in the model and use A*C as error term If all three -way interactions are significant: 1. Describe the effect of A within each combination of B and C. Describe the effect of B within each combination of A and C 3. Describe the effect of C within each combination of A and B 10

Two nested and 1 crossed factors Objective: extend conclusions about fire vs. not fire in Central Valley Grasslands. Variable: diversity of nematodes in the soil types of Environments A: Fires between 1950-1990 B: No Fires 4 random locations with previous Fire and 4 within no Fire. 1 3 4 blocks 1 treatments T: Grazed and Not Grazed 1 data ex99_3; input Fire $ L block N Diversity@@; cards; YES 1 1 1 3.7 YES 1 1 3.9 YES 1 1 3.9 YES 1 3.7 YES 1 1 3.3 YES 1 3.1 YES 1 3.5 YES 3.7 YES 3 1 1 3.4 YES 3 1 3. YES 3 1 4.0 YES 3 4.3 YES 4 1 1 3.6 YES 4 1 3.5 YES 4 1 3.8 YES 4 3.5 NO 1 1 1 4.1 NO 1 1 4.4 NO 1 1 4.0 NO 1 4. NO 1 1 3.6 NO 1 3.7 NO 1 3.9 NO 3.4 NO 3 1 1 3.8 NO 3 1 3.9 NO 3 1 3.7 NO 3 3.1 NO 4 1 1 3.5 NO 4 1 3. NO 4 1 3.9 NO 4 3.8 ; class FIRE L T Block; model Diversity= FIRE L(FIRE) T T*FIRE T*L(FIRE) Block(L FIRE); random L(FIRE) Block(L FIRE) T*L(FIRE); test h= T e= T*L(FIRE); test h= T*FIRE e= T*L(FIRE); test h= FIRE e= L(FIRE); run; quit; Source Type III Expected Mean Square Fire Var(Error) + Var(block(L Fire)) + Var(T*L(Fire)) + 4 Var(L(Fire)) + Q(Fire) L(Fire) Var(Error) + Var(block(L Fire)) + Var(T*L(Fire)) + 4 Var(L(Fire)) T Var(Error) + Var(T*L(Fire)) + Q(T) Fire*T Var(Error) + Var(T*L(Fire)) + Q(Fire*T) T*L(Fire) Var(Error) + Var(T*L(Fire)) block(l Fire) Var(Error) + Var(block(L 11

RCBD with 1 observation The error is automatically the Trt*Block If Trt NS -> Tukey test to test multiplicative effects: if yes transform data RCBD with more than 1 observation Trt1 Trt Trt3 Block 1 X X X X X X Block X X X X X X To test the block*trt we include it in the model proc GLM; class block Trt ; model yield = block Trt block*trt; random block Trt*block ; run; quit; Source block Trt block*trt Type III Expected Mean Square Var(Error) + 4 Var(block*Trt) + 1 Var(block) Var(Error) + 4 Var(block*Trt) + Q(Trt) Var(Error) + 4 Var(block*Trt) A) If Block*Trt is NOT significant: eliminate it from the model This is equivalent to declare block and block*trt as random) B) If Block*Trt interaction is significant: we have a problem. If it becomes NS by transformation, it was a multiplicative effect If the interaction is still significant indicates that the effect of the Trt is different in each block! It does not make much sense to describe the Trt effects by block (this is a general situation for any random factor). If you still want to make a conclusion for Trt that is valid across all blocks, the correct error term is the interaction. See the result of the random statement. So the advice will be the same: eliminate Block*Trt from the model Remember: the significance of the interaction may be pointing to an interesting biological or ecological problem! 1

RCBD: Power Calculation [ST&D 41] (ST&D 07) Percent oil in flax seed harvested from plants inoculated with rust disease at different stages. Treatment Block 1 Block Block 3 Block 4 Mean τ i Seedling 34.4 35.9 36.0 34.1 35.100 0.188 Early bloom 33.3 31.9 34.9 37.1 34.300 1.51 Late bloom 34.4 34.0 34.5 33.1 34.000.351 Full bloom 36.8 36.6 37.0 36.4 36.700 1.361 Ripening 36.3 34.9 35.9 37. 36.075 0.93 Control 36.4 37.3 37.7 36.7 37.05.5 Mean 35.67 35.100 36.000 35.767 35.533 Σ = 7.940 β i 0.071 0.188 0.18 0.054 Σ = 0.531 Source df SS MS F Treatment 5 31.758 6.35 4.790 Block 3 3.187 1.06 0.801 Error 15 19.888 1.36 To calculate the power of an RCBD, use Pearson and Hartley's power function charts (1953, Biometrika 38:11-130). To begin, calculate : r MSE t i 4 1.36 7.940.00 6 Looking in the power charts for df t = 5, df e = 15, and = 0.05, we find a power of ~ 90%. Other models: In a split plot use the correct MSE depending on the question 13

Power analysis in SAS is also straightforward: Data Flax; Do Trtmt = 1 to 6; Do Block = 1 to 4; Input Oil @@; Output; End; End; Cards; 34.4 35.9 36.0 34.1 33.3 31.9 34.9 37.1 34.4 34.0 34.5 33.1 36.8 36.6 37.0 36.4 36.3 34.9 35.9 37. 36.4 37.3 37.7 36.7 ; Proc GLM Data = Flax; Class Trtmt Block; Model Oil = Trtmt Block; Proc GLMPower Data = Flax; Class Trtmt Block; Model Oil = Trtmt Block; Power Stddev = 1.15147 The Root MSE from ANOVA table Alpha = 0.05 Specify the alpha ntotal = 4 Specify n, the No. of EUs in the exp. Run; Quit; Power =.; A period here tells SAS to calculate power The output: Fixed Scenario Elements Dependent Variable Oil Alpha 0.05 Error Standard Deviation 1.15147 Total Sample Size 4 Error Degrees of Freedom 15 Computed Power Test Index Source DF Power 1 Trtmt 5 0.904 14

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