Video 3. Blocking and Confounding Explained in 2^k Factorial Design of Experiments DOE Using MS Excel.

Let’s expand the 2³ design into 2⁴ design with four factors/variables (Table 6). A total of 16 experimental units are required to complete a full replication of the 2⁴ design. Assume that only four samples/experimental units can be produced from one batch of the raw materials. Therefore, four batches (blocks) are required to complete the 16 experimental units. The total number of effects confounded with block is demonstrated in Video 4.

Video 4. How Many Confounded Treatments are There in a Multiple Confounded Effects Design of Experiments DOE.

If four blocks are used, a total of three (4-1) degrees of freedom are lost due to the blocks. Therefore, three effects will be indistinguishable from the block effects, meaning that three effects will be confounded within the blocks. Generally higher-order interactions are confounded. As there is only one four-way interaction, two three-way interactions would be the next potential candidates for the confounding.

Table 6. The 2⁴ Factorial Design of Experiments

Any three-way interactions can be chosen for the purpose of confounding. Assume that the ABC and ABD interactions are confounded with the blocks. It must be noted that their general interaction, CD (ABC*ABD = A²B²CD = CD) is also confounded with the block. Simply the confounded effects could be treated as factors and each treatment combination of these two confounded effects shall be assigned to a separate block as provided in Table 7. Only two higher interaction effects are required to produce four treatment combinations, similar to a 2² base full factorial design. Each of these treatment combinations is assigned one particular block (Table 7). The complete assignments of four blocks of the two confounding effects and their generalized interactions can be found in Table 8.

Table 7. Block Assignment Technique for Confounding Two Effects in -1/+1 Coding System

Table 8. Assignment of Four Blocks for Two Confounding Effects using -1/+1 Coding System