Lecture 11: Nested and Split-Plot Designs

Transcription

1 Lecture 11: Nested and Split-Plot Designs Montgomery, Chapter 14 1 Lecture 11 Page 1

2 Crossed vs Nested Factors Factors A (a levels)and B (b levels) are considered crossed if Every combinations of A and B (ab of them) occurs; An example: Factor A Factor B xx xx xx xx 2 xx xx xx xx 3 xx xx xx xx A B x x x x x x x x x x x x x x x x x x x x x x x x 2 Lecture 11 Page 2

3 Factor B is considered nested under A (a levels) if Levels of B are similar for different levels of A; Levels of B are not identical for different levels of A. An example (aka two-stage nested design or hierarchical design): A B x x x x x x x x x x x x x x x x x x x x x x x x It is a balanced nested design because of an equal number of levels of B within each level of A and an equal number of replicates. 3 Lecture 11 Page 3

4 Material Purity Experiment A company buys raw material in batches from three different suppliers. Purity of raw material varies considerably, causing problems in manufactured product. Q: is the variability in purity attributable to difference between the suppliers? Four batches of raw material are selected at random from each supplier, three determinations of purity are made on each batch. The data, after coding by subtracting 93 are given below. Supplier 1 Supplier 2 Supplier 3 Batches y ij y i Lecture 11 Page 4

5 Other Examples for Nested Factors 1 Drug company interested in stability of product Two manufacturing sites Three batches from each site Ten tablets from each batch 2 Stratified random sampling procedure Randomly sample five states Randomly select three counties Randomly select two towns Randomly select five households 5 Lecture 11 Page 5

6 Statistical Model Two factor nested model y ijk = µ+τ i +β j(i) +ǫ k(ij) i = 1,2,...,a j = 1,2,...,b k = 1,2,...,n Bracket notation represents nesting factor Here factor B (levelj) is nested under factor A (leveli) Cannot include interaction Factors may be random or fixed Can use EMS algorithm to derive tests 6 Lecture 11 Page 6

7 Sum of Squares Decomposition y ijk = ȳ... +(ȳ i.. ȳ... )+(ȳ ij. ȳ i.. )+(y ijk ȳ ij. ). a b n (y ijk ȳ... ) 2 = bn i=1 j=1 k=1 a (ȳ i.. ȳ... ) 2 +n i=1 a b (ȳ ij. ȳ i.. ) 2 i=1 j=1 + a b n (y ijk ȳ ij. ) 2 i=1 j=1 k=1 SS T = SS A +SS B(A) +SS E 7 Lecture 11 Page 7

8 Analysis of Variance Table Source of Sum of Degrees of Mean F 0 Variation Squares Freedom Square A SS A a 1 MS A B(A) SS B(A) a(b 1) MS B(A) Error SS E ab(n 1) MS E Total SS T abn 1 SS T = y 2 ijk y 2.../abn SS A = 1 SS B(A) = 1 n bn y 2 i.. y.../abn 2 y 2 ij. 1 bn y 2 i.. SS E = y 2 ijk 1 n y 2 ij. Use EMS to define proper tests 8 Lecture 11 Page 8

9 Two-Factor Nested Model with Fixed Effects y ijk = µ+τ i +β j(i) +ǫ k(ij) where (1) a i=1 τ i = 0, (2) b j=1 β j(i) = 0 for eachi. F F R a b n term i j k EMS τ i 0 b n σ 2 + bnστ2 i a 1 β j(i) 1 0 n σ 2 + nσσβ2 j(i) a(b 1) ǫ k(ij) σ 2 Estimates: ˆτ i = ȳ i.. ȳ... ; ˆβj(i) = ȳ ij. ȳ i... Tests: MS A /MS E forτ i = 0; MS B(A) /MS E forβ j(i) = 0. 9 Lecture 11 Page 9

10 Two-Factor Nested Model with Random Effects y ijk = µ+τ i +β j(i) +ǫ k(ij) whereτ i N(0,σ 2 τ) andβ j(i) N(0,σ 2 β). R R R a b n term i j k EMS τ i 1 b n σ 2 +nσβ 2 +bnστ 2 β j(i) 1 1 n σ 2 +nσβ 2 ǫ k(ij) σ 2 Estimates: ˆσ τ 2 = (MS A MS B(A) )/nb; ˆσ β 2 = (MS B(A) MS E )/n. tests: MS A /MS B(A) forστ 2 = 0; MS B(A) /MS E forσβ 2 = Lecture 11 Page 10

11 Two-Factor Nested Model with Mixed Effects y ijk = µ+τ i +β j(i) +ǫ k(ij) where a i=1 τ i = 0, andβ j(i) N(0,σ 2 β). F R R a b n term i j k EMS τ i 0 b n σ 2 +nσ 2 β + bnστ2 i a 1 β j(i) 1 1 n σ 2 +nσ 2 β ǫ k(ij) σ 2 Estimates: ˆτ i = ȳ i.. ȳ... ; ˆσ 2 β = (MS B(A) MS E )/n. Tests: MS A /MS B(A) forτ i = 0; MS B(A) /MS E forσ 2 β = Lecture 11 Page 11

12 SAS Code for Purity Experiment option nocenter ps=40 ls=72; data purity; input supp batch datalines; ; proc mixed method=type1; class supp batch; model resp=; random supp batch(supp); proc mixed method=type1; class supp batch; model resp=supp; random batch(supp); run; quit; 12 Lecture 11 Page 12

13 Both Suppliers and Batches are Random Effects Sum of Source DF Squares Mean Square supp batch(supp) Residual Source Expected Mean Square Error Term supp Var(Residual) + 3Var(batch(supp)) + 12 Var(supp) MS(batch(supp)) batch(supp) Var(Residual) + 3Var(batch(supp)) MS(Residual) Residual Var(Residual). Source DF F Value Pr > F supp batch(supp) Residual... Covariance Parameter Estimates Cov Parm Estimate supp batch(supp) Residual Lecture 11 Page 13

14 Suppliers are Fixed Effects and Batches are Random Sum of Source DF Squares Mean Square supp batch(supp) Residual Source Expected Mean Square Error Term supp Var(Residual) + 3 Var(batch(supp)) + Q(supp) MS(batch(supp)) batch(supp) Var(Residual) + 3 Var(batch(supp)) MS(Residual) Residual Var(Residual) Source DF F Value Pr > F supp batch(supp) Residual... Covariance Parameter Estimates Cov Parm Estimate batch(supp) Residual Lecture 11 Page 14

15 Results Summary When Suppliers are Fixed Effects Estimates: ˆτ 1 = ȳ 1.. ȳ... = 28/36 ˆτ 2 = ȳ 2.. ȳ... = 1/36 ˆτ 3 = ȳ 3.. ȳ... = 29/36 Hypothesis test ˆσ 2 β = MS B(A) MS E n ˆσ 2 = MS E = 2.64 = = 1.71 H 0 : τ 1 = τ 2 = τ 3 = 0: F 0 =.97, P-value = , AcceptH 0 H 0 : σβ 2 = 0: F 0 = 2.94, P-value = , RejectH 0 Suppliers are not different, variability due to batches. 15 Lecture 11 Page 15

16 Staggered Nested Designs Other Scenarios for Nested Factors Generalm-Stage Nested Designs y ijkl = µ+τ i +β j(i) +γ k(ij) +ǫ l(ijk) Designs with Both Nested and Factorial Factors y ijkl = µ+τ i +β j +γ k(j) +(τβ) ij +(τγ) ik(j) +ǫ l(ijk) Sections 14.2, 14.3 in Montgomery. 16 Lecture 11 Page 16

17 Split-Plot Designs Example 1: Study six corn varieties and four fertilizers and yield is the response. Three replicates are needed. Method 1: completely randomized full factorial design, 24 level combinations of variety and fertilizer are applied to 24*3=72 pieces of land (each to three). Method 2: Select three fields of large area. Each field is divided into four areas (four whole-plots), four fertilizers are randomly assigned to the four whole-plots. Each area is further divided into six subareas (sub-plots), and the six varieties are randomly planted in these sub-plots. This leads to a split-plot design: a. whole-plot (treatment) factor: fertilizer b. sub-plot (treatment) factor: corn variety 17 Lecture 11 Page 17

18 Example 2: A paper manufacturer is investigating three different pulp preparation methods and four different cooking temperatures for the pulp and study their effect on the tensile strength of the paper. Three replicates are needed. Because the pilot plant is only capable of making 12 runs per day, so the experimenter decides to run one replicate on each of the three days and to consider the days as blocks. On any day, a batch of pulp is produced by one of the the three methods (a whole-plot). Then the batch is divided into four samples (four sub-plots), and each sample is cooked at one of the four temperatures. Then a second batch of pulp is made up using another of the three methods. This second batch is also divided into four samples that are tested at the four temperatures. The process is then repeated for the third method. The data is given below. whole-plot factor: preparation method sub-plot factor: cooking temperature 18 Lecture 11 Page 18

19 Day 1 Day 2 Day 3 Temp\Method Lecture 11 Page 19

20 Split-Plot Structure factors are crossed (different than nested) randomization restriction (different than completely randomized) split-plot can be considered as two superimposed blocked designs: A: whole-plot factor(a); B: sub-plot factor (b), r replicates RCBD A : number of trt: a, number of blk: r. RCBD B : number of trt: b, number of blk: ra. for whole-plots, subdivision to smaller sub-plots are ignored. For sub-plots, whole-plots considered blocks. More power for main subplot effect and interaction Should use design only for practical reasons Randomized factorial design more powerful if feasible 20 Lecture 11 Page 20

21 A Typical Data Layout Block 1 Block 2 Block 3 WP-Factor A SP-Factor B 1 y 111 y y y 112 y y y 113 y y y 114 y y 334 y ijk i denotes blocki j denotes thejth level of the whole-plot factora k denotes thekth level of the sub-plot factorb 21 Lecture 11 Page 21

22 Statistical Model I (Restricted Mixed Model) y ijk = µ+r i +α j +(rα) ij +β k +(rβ) ik +(αβ) jk +(rαβ) ijk +ǫ ijk i = 1,2,...,r; j = 1,2,...,a; k = 1,2,...,b r i : block effects (random),r i iid N(0,σ 2 r) α j : whole-plot factor (A) main effects (fixed) (rα) ij : whole-plot error (random), a j=1 (rα) ij = 0,(rα) ij N(0, b 1 b σ2 rα) β k : sub-plot factor (B) main effects (fixed) (rβ) ik : block-b interaction (random), b k=1 (rβ) ik = 0, (rβ) ik N(0, b 1 b σ2 rβ) (αβ) jk Interaction between A and B (fixed) (rαβ) ijk : sub-plot error (random),(rαβ) ijk N(0, (a 1)(b 1) ab a j=1 (rαβ) ijk = 0, b k=1 (rαβ) ijk = 0 ǫ ijk : random error,ǫ ijk iid N(0,σ 2 ) σ 2 rαβ) 22 Lecture 11 Page 22

23 Sum of Squares SS r = ab i (ȳ i.. ȳ... ) 2, df=r-1. SS A = rb j (ȳ.j. ȳ... ) 2, df=a-1. SS ra = b i,j (ȳ ij. ȳ i.. ȳ.j. +ȳ... ) 2, df=(r-1)(a-1) SS B = ar k (ȳ..k ȳ... ) 2, df=(b-1) SS rb = a i,k (ȳ i.k ȳ i.. ȳ..k +ȳ... ) 2 df=(r-1)(b-1) SS AB = r j,k (ȳ.jk ȳ.j. ȳ..k +ȳ... ) 2 df=(a-1)(b-1) SS rab = i,j,k (y ijk ȳ ij. ȳ i.k ȳ.jk +ȳ i.. +ȳ.j. +ȳ..k ȳ... ) 2, df=(r-1)(a-1)(b-1). SS E =? 23 Lecture 11 Page 23

24 Expected Mean Squares (Restricted Mixed Model) r a b R F F term i j k E(M S) r i 1 a b σ 2 +abσ 2 r whole plot α j r 0 b σ 2 +bσrα 2 + rbσα2 j a 1 (rα) ij 1 0 b σ 2 +bσrα 2 (whole plot error) β k r a 0 σ 2 +aσ 2 rβ + raσβ2 k b 1 (rβ) ik 1 a 0 σ 2 +aσ 2 rβ subplot (αβ) jk r 0 0 σ 2 +σ 2 rαβ + rσσ(αβ)2 jk (a 1)(b 1) (rαβ) ijk σ 2 +σ 2 rαβ (subplot error) ǫ ijk σ 2 (not estimable) 24 Lecture 11 Page 24

25 Estimates and Tests of Fixed Effects ˆα j = ȳ.j. ȳ... forj = 1,2,...,a ˆβ k = ȳ..k ȳ... fork = 1,2,...,b ˆ (αβ) jk = ȳ.jk ȳ.j. ȳ..k +ȳ.... TestH 0 : α j = 0,F 0 = MS A /MS ra TestH 0 : β k = 0,F 0 = MS B /MS rb TestH 0 : (αβ) jk = 0,F 0 = MS AB /MS rab. 25 Lecture 11 Page 25

26 SAS Code data paper; input block method temp datalines; ; proc glm data=paper; class block method temp; model resp=block method block*method temp block*temp method*temp block*method*temp; random block block*method block*temp block*method*temp; test h=method e=block*method; test h=temp e=block*temp; test h=method*temp e=block*method*temp; run; quit; 26 Lecture 11 Page 26

27 SAS Output Sum of Source DF Squares Mean Square F Value Pr > F Model Error CoTotal Source DF Type III SS Mean Square F Value Pr > F block method block*method temp block*temp method*temp blo*meth*tmp Lecture 11 Page 27

28 Tests Using the Type III MS for block*method as Error Term Source DF Type III SS Mean Square F Value Pr > F method Tests Using the Type III MS for block*temp as Error Term Source DF Type III SS Mean Square F Value Pr > F temp Tests Using the Type III MS for block*method*temp as E.Term Source DF Type III SS Mean Square F Value Pr > F method*temp Lecture 11 Page 28

29 Statistical Model II (Restricted Mixed Model) y ijk = µ+r i +α j +(rα) ij +β k +(αβ) jk +ǫ ijk r i : block effects (random),r i iid N(0,σ 2 r) α j : whole-plot factor (A) main effects (fixed) (rα) ij : whole plot error, a j=1 (rα) ij = 0,(rα) ij N(0, a 1 a σ2 rα) β k : sub-plot factor (B) main effects (fixed) (αβ) jk : A and B interaction (fixed) ǫ ijk : sub-plot error,ǫ ijk iid N(0,σ 2 ǫ) Blocks B and blocks AB interactions have been pooled withǫ ijk to form the subplot error It is entirely satisfactory if blocks B and blocks AB interactions are negligible 29 Lecture 11 Page 29

30 Expected Mean Square (Restricted Model) Term E(M S) r i α j (A) σ 2 ǫ +abσ 2 r σ 2 ǫ +bσ 2 rα + rbσα2 j a 1 (rα) ij σ 2 ǫ +bσ 2 rα (whole plot error) β k (B) (αβ) jk (AB) σ 2 ǫ + raσβ2 k b 1 σ 2 ǫ + rσσ(αβ)2 jk (a 1)(b 1) ǫ ijk σ 2 ǫ (subplot error) 30 Lecture 11 Page 30

31 General Split-Plot Designs Can have>one whole-plot factor and>one subplot factor with various blocking schemes. Split-plot design consists of two superimposed blocked design Whole Plot CRD, RCBD, Factorial D, BIBD, etc. Subplot RCBD, BIBD, Factorial Design, etc. Analysis of Covariance Covariate linear with response in subplot and whole plot 31 Lecture 11 Page 31

32 Other Variations Split-split-plot design 1. randomization restriction can occur at any number of levels within the experiment 2. two-level: split-split-plot design Strip-split-plot design ( or Criss cross design, or Split-block design) Example: we want to compare the yield of a certain crop under different systems of soil preparation (A :a 1,a 2,a 3,a 4 ) and different density of seeding (B: b 1,b 2,b 3,b 4,b 5 ). Both operations (tilling and seeding) are done mechanically and it is impossible to perform both on small pieces of land. The arrangement shown below (strip-split-plot design) is then replicatedr times, each time using different randomizations foraandb. a 4 b 1 a 4 b 4 a 4 b 2 a 4 b 3 a 4 b 5 a 1 b 1 a 1 b 4 a 1 b 2 a 1 b 3 a 1 b 5 a 2 b 1 a 2 b 4 a 2 b 2 a 2 b 3 a 2 b 5 a 3 b 1 a 3 b 4 a 3 b 2 a 3 b 3 a 3 b 5 For statistical models and analyses, refer to other books. 32 Lecture 11 Page 32