# note12 - Chapter 4 Paired comparisons and Blocked Designs 4...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the 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 t-test – 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 F-test 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

{[ snackBarMessage ]}

### Page1 / 9

note12 - Chapter 4 Paired comparisons and Blocked Designs 4...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online