Latin square designs are special block designs with two blocking factors and only one treatment per block instead of every treatment per block.

Transcription

1 G. Latin Square Designs Latin square designs are special block designs with two blocking factors and only one treatment per block instead of every treatment per block. 500

2 CLASSIC AG EXAMPLE: A researcher wants to determine the optimal seeding rate for a new variety of wheat: 30, 80, 130, 180, or 230 pounds of seed per acre. The experimental plot of land available has an irrigation source along one edge and a slope perpendicular to the irrigation flow. 501

3 irrigation source A B C D E B C D E A C D E A B D E A B C E A B C D - slope -> where the five seeding rates are randomly assigned to the five letters A, B, C, D, E. How often does each treatment appear? 502

4 A Latin square design does not have to correspond to a physical layout. EXAMPLE: In a study of a new chemotherapy treatment for breast cancer, researchers wanted to control for the effects of age and BMI. They believe the responses of younger patients will be more like each other than those of older patients, and likewise that the responses of heavier patients will be more like each other than those lighter patients. 503

5 Age (years) [40,50) [50,60) [60,70) 70+ <20 A B C D BMI [20,25) B C D A [25,30) C D A B 30+ D A B C 504

6 A standard Latin square has the treatment levels (A, B, etc.) written alphabetically in the first row and the first column. The remaining cells are filled in by incrementing the letters by one within each row and column. A B C D B C D 505

7 Therefore, what restrictions are needed for an experiment to be able to use a Latin square design? 506

8 Randomization Randomization is a bit complex because there are multiple possible Latin squares. For example, for t = 4, A B C D B C D A C D A B D A B C A B C D B A D C C D B A D C A B 507

9 For t = 3,4,5: 1. Choose a standard Latin square at random. 2. Randomly permute (re-order) all rows but the first. 3. Randomly permute all columns. 4. Randomly assign treatments to the letters A, B, C, etc. 508

10 For t 6: 1. Choose a standard Latin square not at random. 2. Randomly permute all rows. 3. Randomly permute all columns. 4. Randomly assign treatments to the letters A, B, C, etc. 509

11 Advantages of a Latin square design: 510

12 Disadvantages of a Latin square design: 511

13 More disadvantages of a Latin square design: 512

14 More disadvantages of a Latin square design: 513

15 Model Y ij = µ + ρ i + γ j + τ k + e ij e ij iid N(0, σ 2 e ) i = 1,..., t, j = 1,..., t, k = 1,..., t with row effect ρ i, column effect γ j, and treatment effect τ k. We can have any combination of fixed or random for each of these, adding constraints as needed for fixed effects and random effects independent of each other. 514

16 Why is there no k subscript on Y ij? 515

17 Deviations: With only one observation per cell, no interactions are estimable: Y ij Ȳ.. }{{} total = (Ȳ i. Ȳ.. ) }{{} row +(Ȳ.j Ȳ.. ) }{{} column +(Ȳ k Ȳ.. ) }{{} treatment + (Y ij Ȳ i. Ȳ.j Ȳ k + 2Ȳ.. ) }{{} error where the error deviation comes from subtraction. 516

18 ANOVA table: Source df SS Rows t 1 t i (Ȳ i. Ȳ.. ) 2 Columns t 1 t j (Ȳ.j Ȳ.. ) 2 Treatment t 1 t k (Ȳ k Ȳ.. ) 2 Error (t 1)(t 2) by subtraction Total t 2 1 i j (Y ij Ȳ.. ) 2 517

19 With no replication, df Error is quite small. For this design to be effective, we need SS(Rows) and SS(Columns) to be large. 518

20 Source E[MS] F Rows Columns Treatment σ 2 e + t t 1 k (τ k ) 2 Error σ 2 e Total Rows, columns, and treatments can be fixed or random as needed, which dictate the appropriate E[MS]. 519

21 Was blocking effective? We can compare the efficiency of the Latin square design to what we would have seen with a CRD or with various CBDs: Efficiency relative to a CRD: RE = MSRows + MSColumns + (t 1)MSError (t + 1)MSError 520

22 Efficiency relative to a CBD using the row blocks only: RE = MSColumns + (t 1)MSError t MSError Efficiency relative to a CBD using the column blocks only: RE = MSRows + (t 1)MSError t MSError Each of these could be used with the df correction: (df Error(LS) + 1)(df Error(other) + 3) (df Error(LS) + 3)(df Error(other) + 1) RE 521

23 Extensions The Latin square design can be extended to include: replicates within square subsampling within square 522

24 replicate squares - with no blocking factor in common across sqaures - with one blocking factor in common across squares - with both blocking factors in common across squares 523

25 H. Latin Squares with Subsampling Subsampling can be done within each cell of a Latin square. Y ijl = µ + ρ i + γ j + τ k + e ij + δ ijl e ij iid N(0, σe 2 ) δ ijl iid N(0, σd 2 ) i = 1,..., t, j = 1,..., t, k = 1,..., t, l = 1,..., n with any combination of fixed or random, adding constraints as needed for fixed effects and random effects independent of each other. 524

26 ANOVA table: Source df SS Rows t 1 tn i (Ȳ i.. Ȳ... ) 2 Columns t 1 tn j (Ȳ.j. Ȳ... ) 2 Treatment t 1 tn k (Ȳ k Ȳ... ) 2 Error (t 1)(t 2) by subtraction Subsampling t 2 (n 1) i j l(y ijl Ȳ ij ) 2 Total nt 2 1 i j l (Y ijl Ȳ... ) 2 525

27 Source E[MS] F Rows Columns Treatment Error σ 2 d + nσ2 e + tn t 1 σ 2 d + nσ2 e k (τ k ) 2 Subsampling σ 2 d Total Rows, columns, and treatments can be fixed or random as needed, which dictate the appropriate E[MS]. 526

28 I. Replicated Latin Squares Often Latin square designs are replicated in their entirety to get more error df. Two possibilities are:...a Latin rectangle: A B C D A B C D B C D A B C D A C D A B C D A B D A B C D A B C where the row blocks are identical across the two squares. 527

29 ...or replicated Latin squares: A B C D B C D A C D A B D A B C A B C D B A D C C D B A D C A B where neither the row blocks nor the column blocks are identical across the two squares. 528

30 For a Latin rectangle, randomization could be done: separately for each square (thus we have 4 columns nested within each of 2 squares) across all columns at once (thus we have 8 columns). Your analysis should match the randomization! 529

31 For replicated Latin squares, randomization is done separately for each square we have row(square) and column(square) effects (nesting within square). 530

32 Replicated Latin Squares Model Y ijl = µ + ρ i(l) + γ j(l) + τ k + κ l + e ijl e ijl iid N(0, σe 2 ) i = 1,..., t, j = 1,..., t, k = 1,..., t, l = 1,..., s with any combination of fixed or random for each of these, adding constraints as needed for fixed effects and random effects independent of each other. A square by treatment interaction (τκ) kl could be considered as well. 531

33 ANOVA table: Source df SS Squares s 1 t 2 l(ȳ..l Ȳ... ) 2 Rows(Square) s(t 1) t i l(ȳ i.l Ȳ..l ) 2 Columns(Square) s(t 1) t j l(ȳ.jl Ȳ..l ) 2 Treatment t 1 st k(ȳ k Ȳ... ) 2 Error (t 1)(t 2) by subtraction Total st 2 1 i j l(y ijl Ȳ... ) 2 532

34 Source E[MS] F Square Rows(Square) Columns(Square) Treatment Error σ 2 e + st t 1 σ 2 e k (τ k ) 2 Total Rows, columns, and treatments can be fixed or random as needed, which dictate the appropriate E[MS]. 533

35 Latin Rectangle Model 1 Y ij = µ + ρ i + γ j + τ k + e ij e ij iid N(0, σ 2 e ) i = 1,..., t, j = 1,..., st, k = 1,..., t with any combination of fixed or random for each of these, adding constraints as needed for fixed effects and random effects independent of each other. 534

36 ANOVA table: Source df SS Rows t 1 st i (Ȳ i. Ȳ.. ) 2 Columns st 1 t j (Ȳ.j Ȳ.. ) 2 Treatment t 1 st k (Ȳ k Ȳ.. ) 2 Error (t 1)(st 2) by subtraction Total st 2 1 i j (Y ij Ȳ.. ) 2 535

37 Source E[MS] F Rows Columns Treatment σ 2 e + st t 1 k (τ k ) 2 Error σ 2 e Total Rows, columns, and treatments can be fixed or random as needed, which dictate the appropriate E[MS]. 536

38 Latin Rectangle Model 2 Y ijl = µ + ρ i + γ j(l) + τ k + κ l + e ijl e ijl iid N(0, σ 2 e ) i = 1,..., t, j = 1,..., t, k = 1,..., t, l = 1,..., s with any combination of fixed or random for each of these, adding constraints as needed for fixed effects and random effects independent of each other. 537

39 ANOVA table: Source df SS Squares s 1 t 2 l(ȳ..l Ȳ... ) 2 Rows t 1 st i(ȳ i.. Ȳ... ) 2 Columns(Square) s(t 1) t j l(ȳ.jl Ȳ..l ) 2 Treatment t 1 st k(ȳ k Ȳ... ) 2 Error (t 1)(st 2) by subtraction Total st 2 1 i j l(y ijl Ȳ... ) 2 538

40 Source E[MS] F Rows Columns Treatment σ 2 e + st t 1 k (τ k ) 2 Error σ 2 e Total Rows, columns, and treatments can be fixed or random as needed, which dictate the appropriate E[MS]. 539

41 How do we get from the Latin rectangle Model 2 ANOVA table to the Latin rectangle Model 1 ANOVA table? 540