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lect11 - IE 410 Design of Experiments Chapter 4 Block...

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IE 410 Design of Experiments Chapter 4: Block Designs, Latin Squares and Others Lecture 11: Randomized Complete Block Designs
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Well we have finally reached the point where we move from Analysis of experiments to Design of experiments. Less math and more concepts
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Randomized Complete Block Designs (RCBD’s) In the RCBD we are still interested in determining the effects of a single factor on the response. As always we assume that the variation observed in the response variable Y has two causes:
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1. Y varies depending of which treatment it's in 2. Y varies as a result of "experimental error" which is the sum total of all other causes of variation in Y
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RCBD's are designed to help reduce the experimental error by accounting for some important exogenous variables. To control the effects of exogenous variable we have 3 things we can do:
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1. Randomize the experiment to “balance out” the effects 2. Run a highly controlled experiment so that the exogenous variable is constant 3. Run a block design experiment
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Example: Hardness Testing The hardness of a material can be tested by pushing a pointed tip into the material with constant force, and recording how far the tip penetrates.
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Suppose we have 4 different types of pointed tips. We want to see if they yield the same results or not (i.e. do they penetrate the same amount or not). Perhaps the tips are from different manufacturers, and we want to see if we get the same results from each.
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Suppose we choose n=4 so that each tip will be pushed into a piece of metal n=4 times, and the response, Y = penetration depth will be recorded. Clearly the piece of metal that the tip is pushed into may effect our results in a significant way. Here are three ways to deal with it.
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1. Get 16 pieces of scrap metal, 1 for each observation. Randomly assign scraps to observations. 2. Get one single piece of metal. Cut it up into 16 pieces. 3. Get 4 pieces of metal. Cut each into 4 pieces. Assign like this:
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Treatment 1 2 3 4 --------------------------- | X X X X | Piece 1 = Block 1 --------------------------- --------------------------- | X X X X | Piece 2 = Block 2 --------------------------- --------------------------- | X X X X | Piece 3 = Block 3 --------------------------- --------------------------- | X X X X | Piece 4 = Block 4 ---------------------------
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And not like this:
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Treatment 1 2 3 4 ------------------------- | X X | Piece 1 = Block 1 | X X | ------------------------ ------------------------ | X X | Piece 2 = Block 2 | X X | ------------------------ ------------------------ | X X | Piece 3 = Block 3 | X X | ------------------------ ------------------------ | X X | Piece 4 = Block 4 | X X | ------------------------
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We believed that the exogenous variable "piece of metal" will cause the error variance to be larger than we want it. (if we completely randomize it) By running the experiment in "blocks" we can "block" out this effect.
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Example 2. Test 3 cures for baldness Let a=3 and n=4 Method 1. Get 12 (balding) subjects.
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