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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 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 = + 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 F-test 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