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# Balanced Incomplete Block Designs.pdf

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School
University of Alberta
Department
Statistics
Course
STAT368
Professor
Douglas Wiens
Semester
Winter

Description
95 13. Balanced Incomplete Block Designs In the RCBD, C=Complete means that each block contains each treatment.E.g. each coupon is sub- jected to each of the 4 tips. Suppose that a coupon is only large enough that 3 tips can be usThen the blocks would be incompleteOne way to run the experiment is to randomly assign 3 tips to each block, perhaps requiring that each tip appears 3 times in total.There is a more e cient way.An incom- plete block design is balanced if any two treatments appear in the same block an equal number of times. This is then a Balanced Incomplete Block Design. Hardness testing BIBD design and data. Coupon Tip 1 2 3 4 1 9 3 9 4  10 0 28 7 1000 2  9 3 9 8 9 9 29 0 1667 3 9 2 9 4 9 5  28 1 4333 4 9 7  10 0 10 2 29 9 7000 28 2 28 1 29 3 30 1 96 Notation: = # of treatments, = # of blocks. = 4 = 4. = # of treatments per block. = 3. = # of times each treatment appears in the entire experiment. = 3. = = = # of observations. = 12. = # of times each pair of treatments appears to- gether. = 2. It can be shown that = ( 1) 1 Since these parameters have to be integers, BIBD de- signs dont exist for all choices of . There are tables available of the ones that do exist. 97 Model is as for a RCBD: = + + + , with both sums of e ects va³ishinA´ usual, the P 2 total sum of squares = ¯ on 1 P ³ ´2 d.f. and the SS for Blocks is = =1 ¯ ¯ on 1 d.f. The treatment SS depends on the ad- justed total for thtreatment 1X = =1 where = 1 if treatmentappears in blockand = 0 otherwise. So P is the total of the =1 block totals, counting only those blocks that contain treatment : 1 1 = 28 7 (28 2 + 28 1 + 30 1) 1000 3 1 2 = 29 0 3 (28 1 + 29 3 + 30 1) 1667 1 3 = 28 1 (28 2 + 28 1 + 29 3) 4333 3 1 4 = 29 9 3 (28 2 + 29 3 + 30 1) = 7000
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