Randomized Complete Block Designs Incomplete Block Designs. Block Designs. 1 Randomized Complete Block Designs. 2 Incomplete Block Designs

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Derrick Dennis
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1 Block Designs Randomized Complete Block Designs 1 Randomized Complete Block Designs 2 0 / 18

2 1 Randomized Complete Block Designs 2 1 / 18

3 Randomized Complete Block Design RCBD is the most widely used experimental design More efficient than the 1-factor design What is new? Random or fixed effects Correlated observations 2 / 18

4 Biochemical Experiment Serum levels after four medical treatments. Only four people can be treated per day, one for each medication. Day Treat. I II III IV / 18

5 Block Design Randomized Complete Block Designs Subjects Randomisation Block 1 Block 2... Block J Group 1 Group 2 Group Group I 4 / 18

6 Block Randomisation R: sample(rep(1:8,4)), sample(4) or sample(32) Subjects Day Treatment I II III IV / 18

7 Serum levels by Treatment Serumlevel I II III IV Treatment Mean: / 18

8 Serum levels by Day Serumlevel Tag Mean: / 18

9 Randomized Complete Block Design Each treatment in each block equally often. Model: Y ij = µ + A i + b j + ɛ ij i = 1,..., I; j = 1,..., J b j : Effect of block j Fixed-Effects Model: Ai = 0, b j = 0, ɛ ij N (0, σ 2 ) Mixed Model: Ai = 0, b j N (0, σ 2 b ), ɛ ij N (0, σ 2 e) all b j and ɛ ij independent. 8 / 18

10 Anova table Randomized Complete Block Designs y ij y.. = y i. y }{{.. + y }.j y.. + y ij y i. y.j + y.. }{{}}{{} deviation of the deviation of the residual treatment mean block mean SS tot = SS treat + SS blocks + SS res Source SS df MS F Blocks 47.3 J 1 = Treatments 27.9 I 1 = Residual 35.3 (I 1)(J 1) = Total N 1 = 31 9 / 18

11 Expected mean squares Fixed-effects model E(MS res ) = σ 2 E(MS treat ) = σ 2 + J A 2 i /(I 1) E(MS block ) = σ 2 + I bj 2 /(J 1) Mixed-effects model E(MS res ) = σe 2 E(MS treat ) = σe 2 + J A 2 i /(I 1) E(MS block ) = σe 2 + Iσb 2 10 / 18

12 F Tests Randomized Complete Block Designs Fixed-effects Model H 0 : A i = 0 i, F = MStreat MS res H 0 : b j = 0 j, F = MS blocks MS res F I 1,(I 1)(J 1) F J 1,(I 1)(J 1) Mixed Model H 0 : A i = 0 i, F = MStreat MS res H 0 : σb 2 = 0 F = MS blocks MS res as above Blocks are usually not tested MS blocks MS res : Blocking good MS blocks MS res : Blocking not necessary 11 / 18

13 Biochemical example Source SS df MS F P value Blocks Treatments Residual Total Question: What happens without blocking? Hint: Consider SS tot = SS treat + SS blocks + SS res 12 / 18

14 1 Randomized Complete Block Designs 2 13 / 18

15 Test of 7 different Tyres Cars x x x x 2 x x x x 3 x x x x Tyres 4 x x x x 5 x x x x 6 x x x x 7 x x x x Blocks Treatments Small block size, larger number of treatments Non-orthogonal designs 14 / 18

16 Balanced incomplete block design n treatments, block size k, (k < n) Any two treatments occur together the same number of times (λ times) First Solution: ( n k) blocks, a different combination of treatments in each block. n = 7, k = 4 : ( 7 4 ) = = 35 cars Search for smaller designs 15 / 18

17 Necessary conditions for a BIBD b blocks, each treatment occurs r times nr = bk (1) r(k 1) = λ(n 1) (2) (1) number of observations (2) number of treatment pairs for a fixed treatment Design is called symmetric if n = b. 16 / 18

18 Construction of BIBD Problem: Given k and n, how large are r,b, and λ? Conditions (1) and (2) are necessary but not sufficient. Several methods of construction exist. There are tables of BIBD with small sizes (Cochran & Cox 1992). Partially balanced block designs (PBIB) if some treatment comparisons are less important. 17 / 18

19 Analysis of BIBD Randomized Complete Block Designs Statistical model: Y ij = µ + β j + T i + ɛ ij where T i is the treatment effect, β j the block effect. Block and treatment factor are not orthogonal, because not all combinations appear. Calculate first block sum of squares, then adjusted treatment sum of squares. 18 / 18

Chapter 4 Experiments with Blocking Factors

Chapter 4 Experiments with Blocking Factors 許湘伶 Design and Analysis of Experiments (Douglas C. Montgomery) hsuhl (NUK) DAE Chap. 4 1 / 54 The Randomized Complete Block Design (RCBD; 隨機化完全集區設計 ) 1 Variability