Video 1. What is Blocking and Confounding in Design of Experiments DOE Explained With Application Examples

In an ideal situation, a completely randomized full factorial with multiple numerous replications would make a lot of statistical theoretical sense, including reducing the confidence interval, the higher power of the findings, and so on. In fact, completely randomized design has been considered the most efficient over the years. However, when some obvious known nuisance factors/variables are present to introduce variation in the responses, blocking technique has been utilized to handle this kind of situations better for the Randomized Complete Block Design in the earlier Module 3. As the 2^k design is primarily used to screen factors/variables, often a very large number of experimental units are required to complete even one full replication. For an example, 2⁶ design with six variables requires 64 experimental units to complete one full replication. In the 2^k design of experiment, blocking technique is used when enough homogenous experimental units are unavailable.

Ideally, experiments should be run by using completely randomized experimental units. However, often, there is not enough experimental units from one homogenous sample. For example, if there are not enough raw materials to produce all the experimental units for all replications, blocking is utilized to control the nuisance effects of the experimental units coming from, possibly non homogenous batches. Different batches do not necessarily mean non-homogeneity all the time. However, keeping track of the batch numbers as blocks (the statistical term) would provide an opportunity, if in case there is non-homogeneity from batch to batch. Therefore, a block is defined by a homogenous large unit, including, raw materials, areas, places, plants, animals, humans, etc. where samples or experimental units drawn are considered identical twins, but independent.

Let’s start with the basic 2² factorial design to introduce the effective use of blocking into the 2^k design (Table 1). Let’s assume that we need at least three replications for this particular experiment. If one batch can produce enough raw materials for only four samples (experimental units), only one replication can be made from one batch. Therefore, three batches will be required to complete the three full replications for the 2² basic factorial design (Table 2).

Table 1. The Basic 2² Factorial Design of Experiments

Table 2. The Basic 2² Factorial Design of Experiments with three replications in blocks

2. What is Confounding in 2^k Factorial Design of Experiments?

Let’s expand the 2² basic factorial design to a 2³ design for three variables in Table 3. Eight experimental units will be required to complete a full replication of the 2³ design. However, if only four samples or experimental units can be produced from one batch, there are not enough samples or experimental units to complete the full replication of the 2³ factorial design. Therefore, two batches/blocks are required to complete the full replication of the 2³ factorial design. Therefore, one (2-1 = 1) degrees of freedom is lost due to the two blocks. One effect won’t be possible to estimate from this experiment. Generally, higher-order interaction terms are practically meaningless. Therefore, the higher-order interactions are sacrificed when there are not enough experimental units are available. For an example, in a 2³-factorial design of experiment, the three-way interaction (ABC interaction) is sacrificed by confounding with the block, meaning that it won’t be possible to distinguish the effect of ABC interaction from the block effect. To make the ABC interaction indistinguishable from the block, all the “positive terms” and the “negative terms” in the contrast of ABC is run in separate blocks. The indistinguishable effects are known as confounded effects. Of course, the run orders are selected randomly within the block, rather than the standard order, as in any other experimental designs. The entire systematic process of making some effects indistinguishable by blocking is known as confounding.

Table 3. The 2³ Factorial Design of Experiments

3. How to Confound an Effect within the Block Using -1/+1 Coding System?

Video 2 demonstrates the process of confounding and blocking using the -1/+1 coding system.

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

Using the -1/+1 coding system, the confounding can be performed as described in the following steps.

1. Produce the signs -1/+1 for the effect of ABC interaction column by simply multiplying the main effect columns of A, B, and C.

2. Multiply the effect ABC effect column with the treatment combination column to find the contrast of ABC.

3. Run all the positive treatment combinations in one block and all negative ones in another block.

Table 4. The 2³ Factorial Design of Experiments with two blocks

With this +1/-1 coding system, it is not really necessary to produce the contrast. Simply, all the positive signs go into one block, and all negative signs go into another block. However, visually, it is easier to understand why the effect of the ABC interaction cannot be determined from this experiment. Figure 1 shows the graphical representation of the 2³ design with ABC confounded with block. As all the “highs” of ABC are run in block #1, and all the “lows” of ABC are run in block #2, it won’t be possible to distinguish whether the effect of ABC is due to the block or by its own. Therefore, the name confounding is appropriately justified for the ABC interaction effect in this design.

Equation 1

Figure 1. Graphical Representation of Blocking and Confounding in a 2³ Design with ABC Confounded with Block.

4. How to Replicate with Confounding an Effect with the Block?

To replicate the design, simply copy and paste the design in Table 4 as many times as necessary. Randomization is only required for the runs within a block. The experiment should be run in random order rather than the standard order. Simply, the random number generator function RAND() in MS Excel can be used to develop the run order for the experiment. Table 5 shows three replications of the 2³ design with two blocks for each full replication.

Table 5. The 2³ Factorial Design of Experiments with Two Blocks and Three Replications

5. How to Confound Two Effects with Four Blocks Using the -1/+1 Coding System?

Video 3 demonstrates the process of confounding two effects with four blocks using the -1/+1 coding system.

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

6. How to Confound Three Effects with Eight Blocks Using the -1/+1 Coding System?

Video 5 demonstrates the confounding procedure for three effects with eight blocks using the -1/+1 coding system.

Video 5. How to Confound Three or More Effects in Eight or More Blocks in 2K Design of Experiments Explained.

If the experiment is designed for five variables, which requires 32 experimental units to complete a full replication. However, if there are only four samples/experimental units can be prepared from one batch, a total of 8 numbers of batches is required to produce 32 samples (4X8) to complete one full replication. Three higher-order interaction terms are needed to be confounded with 8 blocks because three factors (higher-order interaction terms in this case) produce a total of eight treatment combinations like a 2³ full factorial design (Table 3). Any three higher-order interaction terms can be chosen for the purpose of confounding with 8 blocks. Table 9 shows confounding technique for three higher-order interaction ABCD, ABCE, and ABDE. Their generalized interactions of ABCD*ABCE=DE, ABCD*ABDE=CE, ABCE*ABDE=CD, & ABCD*ABCE*ABDE=ABE are also confounded with blocks, resulting in a total of 7 effects, including three higher-order interactions and their generalized interactions of a total of 4, cannot be distinguished from the block effects. The seven indistinguishable effects can also be understood by the seven degrees of freedom for the block effects (7=8-1).

Table 9. Block Assignment for Three Confounded Effects to Produce 8 Blocks in -1/+1 Coding System.

The complete assignment of the 8 blocks for three confounding effects can be found in Table 10. Similarly, any desired number of confounding and their associated blocks can be developed.

Table 10. Assignment of Eight Blocks for Three Confounding Effects in a 2⁵ Factorial Design of Experiments

Confounding and Blocking Using Linear Combination Method and 0/1 Coding System

Video 6 demonstrates the linear combination method for blocking and confounding using the 0/1 coding system.

Video 6. Blocking and Confounding Explained in 2K Factorial Design of Experiments DOE Using MS Excel

Using the 0/1 coding system, the linear combinations of the defining relation of all factors are used as in Equation 2.

Equation 2

For a 2³ factorial design,

therefore, the Equation 2 becomes Equation 3 for confounding the ABC effect.

Equation 3

To determine the associated block for a treatment combination, simply following steps can be performed provided in Table 11.

1. Add the codes for the factors as in the L column.

2. As the summation values bounce between 0 and 1 coding (known as modulus 2 system or mod 2), value 2 becomes 0, 3 becomes 1, 4 becomes 0, 5 becomes 1 and so on.

3. Finally, assign the treatment combination to one block for all same codes, such as block # 1 for the 0 codes while block #2 for the 1 codes.

Table 11. Assignment of Two Blocks for One Confounding Effect Using 0/1 Coding System

8. Confounding Two Effects with Four Blocks Using 0/1 Coding System?

Recall the -1/+1 coding system to confound two higher-order interactions, including ABC and ABD when four blocks are necessary to produce enough samples to complete a full replication (Table 7 & Table 8). Using the 0/1 coding system the block assignments for the confounded effects are provided in Table 12.

Table 12. Block Assignment Technique for Confounding Two Effects in 0/1 Coding System

The two linear combinations for the ABC and the ABD effects are provided in Equation 4 and Equation 5, respectively. Table 13 shows the block information for each treatment combination using the 0/1 coding system utilizing the linear combination method.

Equation 4

Equation 5

Table 13. Assignment of Four Blocks for Two Confounding Effects using 0/1 Coding System

9. How to Confound Three Effects with Eight Blocks Using the o/1 Coding System?

Recall the confounding scheme for three effects with eight blocks in a 2⁵ design using the -1/+1 coding system provided in Table 9 & Table 10. Table 9 for the -1/+1 coding system can be comparable to the Table 14 for the 0/1 coding system. The linear combinations for the three confounding effects ABCD, ABCE, and ABDE are provided in Equation 6, Equation 7, & Equation 8, respectively.

Equation 6

Equation 7

Equation 8

Table 14. Block Assignment for Three Confounded Effects to Produce 8 Blocks Using the 0/1 Coding System.

Table 15. Assignment of Eight Blocks for Three Confounding Effects in a 2⁵ Factorial Design of Experiments Using the 0/1 Coding System

Video 7. What is Complete vs Partial Confounding in 2k Design of Experiments DOE, and The Appropriate Use.

If the replications are possible with confounding and blocking experiments, the confounding can be performed either completely or partially depending on the interest of the research questions or hypothesis. For an example, the ABC interaction is completely confounded with blocks in Figure 2 (Kempthorne 1952; Yates 1978; Montgomery 2013). In this situation, the three-way ABC interaction is not an interest of the experiment. In this design, no information can be retrieved for the ABC interaction. However, all the main effects and the second-order interaction can be obtained 100%.

However, if some information is useful for the ABC interaction, it could be partially confounded as in Figure 3. In this situation, the ABC, AB, AC, and BC are confounded with blocks in the replication I, II, III, and IV, respectively. Therefore, 3/4^th (75%) information can be retrieved for each of the interaction terms. For an example, the AB interaction effect can be obtained from replication I, III, and IV. This confounding process is known as partial confounding (Yates 1978; Hinkelmann and Kempthorne 2005; Montgomery 2013). Nevertheless, three-way interaction ABC effect is rarely a practical interest. Therefore, complete confounding of higher-order interactions for the interest of the lower-order interactions would be preferable.

Figure 2. Complete Confounding: ABC Interaction Confounded with Blocks in All Four Replications

Figure 3. Partial Confounding: ABC, AB, AC, and BC are Confounded with Blocks in Replication I, II, III, and IV, respectively

11.1 ANOVA Table for a Partially Confounded 23 Design