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
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

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

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

7. 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