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Blocking and confounding system for Two-level factorials: Blocking, Confounding, Combining Blocking and Confounding

Blocking and confounding are two important concepts in the design of experiments, especially in the context of two-level factorial designs.
Both are used to manage and control the variability in experimental results, but they serve different purposes and are applied in different ways.

A two-level factorial design involves experiments where each factor (independent variable) is set at two levels, often denoted as -1 (low) and +1 (high).
For example, if there are three factors, the design would involve 2^3 = 8 experimental runs, covering all combinations of the levels of the factors.

Blocking

Blocking is a technique used to reduce or eliminate the impact of confounding variables by grouping similar experimental units into blocks.
Each block is a subset of the experimental runs that are more homogeneous.
The goal of blocking is to isolate the variability due to the blocking factor, allowing for a more accurate estimation of the effects of the primary factors of interest.

Identify the Blocking Factor: Choose a factor that you believe may affect the outcome but is not of primary interest (e.g., batches of raw material, shifts, or different days).
Create Blocks: Divide the experimental runs into blocks such that each block contains similar conditions regarding the blocking factor.
Run Experiments within Blocks: Conduct the experimental runs within each block, ensuring each block is treated as a mini experiment.

Consider an experiment with two factors, A and B, each at two levels. Suppose there is a potential confounding factor, like the time of day.
We could create two blocks, one for the morning and one for the afternoon, and within each block, we would run all combinations of A and B.

Confounding occurs when the effects of two or more factors (or interactions between factors) are mixed up in such a way that their individual contributions cannot be separately identified.
In factorial designs, especially when we have many factors, it is often necessary to confound some interactions to reduce the number of runs required.

Identify High-Order Interactions: In many practical situations, higher-order interactions (e.g., three-factor or four-factor interactions) are less important and can be confounded with blocks or other factors.
Alias Structure: Create an alias structure where certain interactions are deliberately confounded with each other. This involves defining how different effects are linked or aliased.
Fractional Factorial Designs: Use fractional factorial designs to run a subset of the total possible runs, accepting that some interactions will be confounded with others. For instance, in a 2k design, you might use a 2k−p design, where p is the number of confounded interactions.

In a 2^3 factorial design with factors A, B, and C, suppose you decide to run a half-fractional design (2^3−1 = 4 runs instead of 8).
You might confound the ABC interaction with the mean effect, resulting in the aliasing of A with BC, B with AC, and C with AB.
This means that you cannot distinguish between the main effect of A and the interaction effect of BC, and so on.

In practice, blocking and confounding can be used together to manage complex experimental designs.

Define Blocks: First, decide on the blocking factor and create the blocks.
Determine Confounding: Within each block, determine which interactions or main effects can be confounded. Typically, this involves confounding higher-order interactions.
Run Experiments: Conduct the experiments within the structure of the blocks and confounding scheme.

Suppose you have three factors A, B, and C, and a blocking factor such as day of the week. You might:

Create two blocks, one for each day.
Use a fractional factorial design within each block, confounding some interactions (like ABC) with the block effect.

This approach reduces the number of runs needed while controlling for variability due to the blocking factor.