lect14 - IE 410: Design of Experiments Lecture 14: Balanced...

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IE 410: Design of Experiments Lecture 14: Balanced Incomplete Block Designs Example: Recall the tire experiment. Suppose we wanted to test 5 brands of tires. Then we can't fit all 5 brands in each block (car).

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Treatment (Ti re Brand) A B C D E - - - - - - - - - - - - - - X X X X Car 1 X X X X Car 2 X X X X Car 3 X X X X Car 4 X X X X Car 5
A design in which not all treatments appear in blocks is an INCOMPLETE BLOCK DESIGN Well the design above doesn't look too bad. How about 6 brands?

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Treatment (Ti re Brand) A B C D E F - - - - - - - - - - - - - - - - - X X X X Car 1 X X X X Car 2 X X X X Car 3 This looks worse.
How about 4 brands of motorcycle tires? Tire Brand A B C D - - - - - - - - - - - X X Bike 1 X X Bike 2 DESIGN A X X Bike 3 X X Bike 4

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Is this a good Design?
T i re Brand A B C D - - - - - - - - - - - X X Bike 1 X X Bike 2 DESIGN B X X Bike 3 X X Bike 4 How about this design?

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Analysis of Design A Design A is clearly No Good! For one thing, the contrast: μ 1 + μ 2 = μ 3 + μ 4 is completely confounded with blocks!
Suppose the model is given by: Y ij = μ + τ i + β j Suppose we got the following data (assume for the moment that there is no error at all)

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Treatment (Ti re Brand) A B C D - - - - - - - - - - - - - - 1 - 1 Bike 1 101 99 Bike 2 1 - 1 Bike 3 100 99 Bike 4
Check to see if the following parameters fit the data perfectly: T(1) = 1 T(2) = -1 T(3) = 1 T(4) = -1 B(1) = 0 B(2) = 100 B(3) = 0 B(4) = 100 m = 0

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Now Check to see if the following different parameters fit the data perfectly: T(1) = 1 T(2) = -1 T(3) = 101 T(4) = 99 B(1) = 0 B(2) = 0 B(3) = 0 B(4) = 0 m = 0
So is there a big difference between treatments? We can't tell. What can we tell from these data?

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We can see that the difference between treatment means 1 and 2 is 2, and the same for treatment means 3 and 4. We cannot say anything about the following comparisons: 1 vs 3 1 vs 4 2 vs 3 2 vs 4
That is, we can only compare treatments WHEN THEY APPEAR TOGETHER IN THE SAME BLOCK!!!

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important, then we would want: 1. Each pair of treatments to appear together in a block the same number of times. 2. We also want to observe each treatment
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This note was uploaded on 08/06/2008 for the course IE 410 taught by Professor Storer during the Fall '04 term at Lehigh University .

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lect14 - IE 410: Design of Experiments Lecture 14: Balanced...

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