Randomized Block Designs: Comparison of Treatments with Blocks

1. Chapter 13 Complete Block Designs

2. Randomized Block Design (RBD) • g > 2 Treatments (groups) to be compared • r Blocks of homogeneous units aresampled. Blocks can be individual subjects. Blocks are made up of t subunits • Subunits within a block receive one treatment. When subjects are blocks, receive treatments in random order. • Outcome when Treatment i is assigned to Block j is labeled Yij • Effect of Trt i is labeled ai (Typically Fixed) • Effect of Block j is labeled bj (Typically Random) • Random error term is labeled eij • Efficiency gain from removing block-to-block variability from experimental error

3. Randomized Complete Block Designs • Model: • Test for differences among treatment effects: • H0: a1 = ... = ag= 0 (m1= ... = mg ) • HA: Not all ai = 0 (Not all mg are equal) Typically not interested in measuring block effects (although sometimes wish to estimate their variance in the population of blocks). Using Block designs increases efficiency in making inferences on treatment effects

4. RBD - ANOVA F-Test (Normal Data) • Data Structure: (g Treatments, r Subjects (Blocks)) • Mean for Treatment i: • Mean for Subject (Block) j: • Overall Mean: • Overall sample size: N = rg • ANOVA:Treatment, Block, and Error Sums of Squares

5. RBD - ANOVA F-Test (Normal Data) • ANOVA Table: • H0: a1 = ... = ag= 0 (m1= ... = mg ) • HA: Not all ai = 0 (Not all mi are equal)

6. Pairwise Comparison of Treatment Means • Tukey’s Method- with n = (r-1)(g-1) • Bonferroni’s Method - with n = (r-1)(g-1), C=g(g-1)/2

7. Expected Mean Squares / Relative Efficiency • Expected Mean Squares: As with CRD, the Expected Mean Squares for Treatment and Error are functions of the sample sizes (r, the number of blocks), the true treatment effects (a1,…,ag) and the variance of the random error terms (s2) • By assigning all treatments to units within blocks, error variance is (much) smaller for RBD than CRD (which combines block variation&random error into error term) • Relative Efficiency of RBD to CRD (how many times as many replicates would be needed for CRD to have as precise of estimates of treatment means as RBD does):

8. Example - Caffeine and Endurance • Treatments: g=4 Doses of Caffeine: 0, 5, 9, 13 mg • Blocks: r=9 Well-conditioned cyclists • Response: yij=Minutes to exhaustion for cyclist j @ dose i • Data:

10. Example - Caffeine and Endurance

11. Example - Caffeine and Endurance

12. Example - Caffeine and Endurance

13. Example - Caffeine and Endurance • Would have needed 3.79 times as many cyclists per dose to have the same precision on the estimates of mean endurance time. • 9(3.79)  35 cyclists per dose • 4(35) = 140 total cyclists

14. Latin Square Design • Design used to compare g treatments when there are two sources of extraneous variation (types of blocks), each observed at g levels • Best suited for analyses when g 10 • Classic Example: Car Tire Comparison • Treatments: 4 Brands of tires (A,B,C,D) • Extraneous Source 1: Car (1,2,3,4) • Extraneous Source 2: Position (Driver Front, Passenger Front, Driver Rear, Passenger Rear)

15. Latin Square Design - Model • Model (g treatments, rows, columns, N=g2) :

16. Latin Square Design - ANOVA & F-Test • H0: a1 = … = ag = 0 Ha: Not all ai = 0 • TS: Fobs = MST/MSE = (SST/(g-1))/(SSE/((g-1)(g-2))) • RR: Fobs Fa, g-1, (g-1)(g-2)

17. Pairwise Comparison of Treatment Means • Tukey’s Method- with n = (g-1)(g-2) • Bonferroni’s Method - with n = (g-1)(g-2), C=g(g-1)/2

18. Expected Mean Squares / Relative Efficiency • Expected Mean Squares: As with CRD, the Expected Mean Squares for Treatment and Error are functions of the sample sizes (g, the number of blocks), the true treatment effects (a1,…,ag) and the variance of the random error terms (s2) • By assigning all treatments to units within blocks, error variance is (much) smaller for LS than CRD (which combines block variation&random error into error term) • Relative Efficiency of LS to CRD (how many times as many replicates would be needed for CRD to have as precise of estimates of treatment means as LS does):

19. Replicated Latin Squares • To Increase Power (and Error degrees of freedom), experimenters often will use multiple (m>1) gxg latin squares for their design. There are 3 possible model structures: • Model 1: Separate Row and Column blocks used in each square • Model 2: Common Row, but separate Column blocks used in each square • Model 3: Common Row and Column blocks used in each square

20. Model 1 – Separate Row and Column Blocks

21. Model 2 – Common Row, Separate Column Blocks

22. Model 3 – Common Row and Column Blocks

23. Designs Balanced for Carry-Over Effects • Subjects receive g treatments, one in each of g time periods • Treatments are balanced with equal number of replicates per time period (across subjects) • Design balanced such that each treatment follows each other treatment equal number of times and appears in the first position equal number of times. • Carryover effect that observation in Period 1 receives is 0

24. Example 13.12 – Milk Yield

25. Example 13.2 – Factor/Carryover Coding Note: Trt1,Trt2 and Res1,Res2 are coded so thatTrt and Res Effects sum to Zero (“Trt3 = -Trt1-Trt2” and “Res3 = -Res1-Res2”, with no Res effects in Period 1)