note11 - Chapter 4 Paired comparisons and Blocked Designs...

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Unformatted text preview: Chapter 4: Paired comparisons and Blocked Designs • Paring and Blocking are experimental design techniques that enable a researcher to detect differences among treatments more easily in environments that have a lot of variability among experimental units. 1. Paired-Comparison Permutation Test • Example: Two methods of obtaining dietary information. Student 1 2 3 4 5 24-Hour Recall 1530 2130 2940 1960 2270 Survey 1290 2250 2430 1900 2120 Difference 240-120 510 60 150 1 Permutation Student 1 Student 2 Student 3 Student 4 Student 5 Mean of Differences 1 240 120 510 60 150 216 2 240 120 510 60-150 156 3 240 120 510-60 150 192 4 240 120-510 60 150 12 5 * 240-120 510 60 150 168 6-240 120 510 60 150 120 7 240 120 510-60-150 132 8 240 120-510 60-150-48 9 240-120 510 60-150 108 10-240 120 510 60-150 60 11 240 120-510-60 150-12 12 240-120 510-60 150 144 13-240 120 510-60 150 96 14 240-120-510 60 150-36 15-240 120-510 60 150-84 16-240-120 510 60 150 72 17 240 120-510-60-150-72 18 240-120 510-60-150 84 19 240-120-510 60-150-96 20 240-120-510-60 150-60 21-240 120 510-60-150 36 22-240 120-510 60-150-144 23-240 120-510-60 150-108 24-240-120 510 60-150 12 25-240-120 510-60 150 48 26-240-120-510 60 150-132 27 240-120-510-60-150-120 28-240 120-510-60-150-168 29-240-120 510-60-150-12 30-240-120-510 60-150-192 31-240-120-510-60 150-156 32-240-120-510-60-150-216 2 • Hypotheses H : F ( x ) = 1 − F ( − x ) H a : F ( x ) ≤ 1 − F ( − x ) H : F ( x ) = 1 − F ( − x ) H a : F ( x ) ≥ 1 − F ( − x ) H : F ( x ) = 1 − F ( − x ) H a : F ( x ) negationslash = 1 − F ( − x ) where G(x) is a distribution that is symmetric about 0. • The procedure for a paired-comparison permutation test (a) Compute the differences, D i ’s, for the n pairs of data ⇒ ¯ D obs (b) the 2 n possible assignments of plus and minus signs to the | D i | ’s ⇒ ¯ D b (c) p − value s: P upper = 1 2 n B summationdisplay b =1 I ( ¯ D b ≥ ¯ D obs ) P lower = 1 2 n B summationdisplay b =1 I ( ¯ D b ≤ ¯ D obs ) P two- sided = 2 ∗ 1 2 n B summationdisplay b =1 I ( ¯ D b ≤ ¯ D obs ) • Alternative Forms of the Statistic S + or S- braceleftbig ¯ D = ( S + + S- ) /n bracerightbig 3 • Randomly selected permutations (a) Compute the differences, D i ’s, for the n pairs of data (b) generate n U i ’s from U i = braceleftbigg − 1 with probabillity 0 . 5 1 with probabillity 0 . 5 OR generate n V i ’s from V i = braceleftbigg with probabillity 0 . 5 1 with probabillity 0 . 5 (c) A randomly selected mean of difference ¯ D = ∑ n i =1 U i | D i | n OR a randomly selected value of S + S + = n summationdisplay i =1 V i | D i | (d) repeat R times (1000 to 5000), then obatin an approximate p − value • Large-sample approximations – E ( U i ) = 0 and var ( U i ) = 1, thus E ( ¯ D ) = 0 var ( ¯ D ) = 1 n 2 n summationdisplay i =1 | D i | 2 Z = ¯ D radicalBig var ( ¯ D ) ∼ N (0 , 1) – E ( V i ) = 1 / 2 and var ( V i ) = 1 / 4, thus...
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This note was uploaded on 07/22/2011 for the course STA 4502 taught by Professor Staff during the Spring '08 term at University of Florida.

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note11 - Chapter 4 Paired comparisons and Blocked Designs...

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