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)

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)

Confounding the 2k Factorial Design in 2p Blocks p independent 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

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

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

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