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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 = + i + j + ij where for i = 1 , . . ., k and j = 1 , . . ., b = an overall effect i = the i th treatment effect j = the j th block effect ij = iid random variables with median 0 Hypothesis H : 1 = 2 = = k H a : at lest one of i negationslash = j when ij N (0 , 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 2 + k 2 Treatments k 1 2 + b k 1 k i =1 ( i ) 2 T B ( k 1)( b 1) 2 Permutation Ftest for RCBD when unwilling to assume the normality of 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...
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 Spring '08
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