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Example: a 2 5 design, each block holds eight runs. Needs four blocks. Select ADE and BCE to be confounded. Two def

Confounding the 2 k Factorial Design in Four Blocks. Example: a 2 5 design, each block holds eight runs. Needs four blocks. Select ADE and BCE to be confounded. Two defining contrasts L 1 = x 1 + x 4 + x 5 L 2 = x 2 + x 3 + x 5 ( L 1 , L 2 ) = (0,0), (0,1), (1,0), (1,1).

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Example: a 2 5 design, each block holds eight runs. Needs four blocks. Select ADE and BCE to be confounded. Two def

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  1. Confounding the 2k Factorial Design in Four Blocks • Example: a 25 design, each block holds eight runs. Needs four blocks. Select ADE and BCE to be confounded. Two defining contrasts • L1 = x1 + x4 + x5 • L2 = x2 + x3 + x5 • (L1, L2) = (0,0), (0,1), (1,0), (1,1)

  2. Confounding in the 2k Factorial Design • Four blocks: 3 dof • ADE: 1 dof • BCE: 1 dof • An additional effect must be confounded: generalized interaction of ADE and BCE (ADE)(BCE) = ABCD • dof (blocking) = dof (effects confounded)

  3. Confounding the 2k Factorial Design in 2p Blocks • pindependent effects to be confounded • Each block contains 2k-p runs • Blocks may be created by the p defining contrasts L1, L2, …, Lp • Number of generalized effects to be confounded: 2p – p - 1

  4. Partial Confounding • Estimate of error: • Prior estimate of error • Assuming certain interactions to be negligible • Replicating the design • If an effect is confounded in all replicates – completely confounded • If an effect is confounded in some replicates, but not all – partial confounding

  5. Interaction sums of squares: only data from the replicates in which an interaction is not confounded are used

  6. Example 7-3: A 23 Design with Partial Confounding • Factors: carbonation, pressure, and line speed • Response: fill height • Each batch of syrup only large enough to test four treatment combinations • Two replicates • ABC is confounded in replicate I, AB confounded in replicate II

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