This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 4: Paired comparisons and Blocked Designs 4. A Permutation Test for a Randomized Complete Block Design • Randomized complete block design (RCBD) – blocking: when the experimental units to which treatments are to be are not homogeneous, or when the conditions under which the experiment is to be conducted cannot be held constant throughout – a RCBD is a generalization of a paired ttest – a typical randomization for a RCBD (4 blocks & 3 treatments) Block 1 Block 2 Block 3 Block 4 Trt2 Trt1 Trt3 Trt1 Trt3 Trt2 Trt2 Trt3 Trt1 Trt3 Trt2 Trt1 – features of RCBD ∗ experimental units are divided into blocks in such a say that units or experimental conditions within blocks are homogeneous (block homogeneity) ∗ blocks have the same number of experimental units as there are treatments (completeness) ∗ the treatments are randomly assigned to experimental units within blocks (randomness) – blocking factors: weights of the animals, medical profile , time, loca tion, etc. • F Statistic for RCBD – data display ( k treatments & b blocks) blocks Treatments 1 2 . . . b Means 1 X 11 X 12 . . . X 1 b ¯ X 1 . 2 X 21 X 22 . . . X 2 b ¯ X 2 . . . . . . . . . . . . . . . . . . . k X k 1 X k 2 . . . X kb ¯ X k. Means ¯ X . 1 ¯ X . 2 . . . ¯ X .b ¯ X 1 – the model X ij = μ + τ i + β j + ǫ ij where for i = 1 , . . ., k and j = 1 , . . ., b μ = an overall effect τ i = the i th treatment effect β j = the j th block effect ǫ ij = iid random variables with median 0 – Hypothesis H : τ 1 = τ 2 = ··· = τ k H a : at lest one of τ i negationslash = τ j – when ǫ ij ∼ N (0 , σ 2 ) F = b ∑ k i =1 ( ¯ X i. − ¯ X ) 2 / ( k − 1) ∑ k i =1 ∑ b j =1 ( X ij − ¯ X i. − ¯ X .j + ¯ X ) 2 / [( k − 1)( b − 1)] ∼ F k 1 , ( k 1)( b 1) – Expected Mean Squares (EMS) for RCB ANOVA Source df EMS Blocks b − 1 σ 2 ǫ + kσ 2 β Treatments k − 1 σ 2 ǫ + b k 1 ∑ k i =1 ( τ i − ¯ τ ) 2 T × B ( k − 1)( b − 1) σ 2 ǫ • Permutation Ftest for RCBD – when unwilling to assume the normality of ǫ ij – testing steps (a) compute F obs for the original data (b) permute the observations within each of the blocks, for all the blocks: ( k !) b possibilities ⇒ compute F l (c) p − value s p − value = 1 ( k !) b ( k !) b summationdisplay l =1 I ( F l ≥ F obs ) with all possible permutations hatwider p − value = 1 R R summationdisplay...
View
Full Document
 Spring '08
 Staff
 Variance, complete block design, randomized complete block

Click to edit the document details