General Blocking and Confounding Scheme

The method using either -1/+1 or 0/1 coding systems described above can be generalized for the 2k design in 2p blocks (p= number of independent confounded effects). Table 16 shows the number of factors, number of blocks, the number of samples can be taken from a block, the block generator, and the confounded interactions with the blocks. A few guidelines to choose the effects to generate the blocks are provided below.

  1. As there is less interest in the higher-order interaction terms, they should be the potential candidates for the confounding effects to reduce the primary information loss. The goal is to keep as many lower-order interactions as possible unconfounded.
  2. The effects must be independent, meaning that no generalized intersections are chosen for the block generator. For an example, if AB and AC are chosen as the generators, their generalized interaction BC (AB*AC = A2BC =BC) cannot be a block generator.
  3. The degrees of freedom for the block is 2p -1. Therefore, the total number of effects confounded with blocks are 2p -1, including the block generator effects which are also already confounded with blocks. For an example, if three effects are confounded to produce eight blocks, the total degrees of freedom for the block is 7 (=8-1). Therefore, total 7 effects are confounded with the blocks. No information for these seven effects will be obtained/distinguished from the blocks.
  4. Of course, k>p.

Table 16. Suggested Blocking and Confounding for 2k Design in 2p Blocks