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## Quick reference

### Confounding Effects

Confounding effects applies to Fractional Factorial DOE studies. By reducing the experimental runs to a fraction of those used in Full Factorial DOE, the ability to analyze some interactions effects is curtailed and this is known as confounding.

### When to use

All Fractional Factorial DOE studies have some level of confounding. The design of the study must determine whether the confounding level is acceptable. Typically, high levels of confounding can be tolerated in the screening phase, but not in the refining or optimizing phase.

### Instructions

Confounding, or as it is often called, aliasing, describes the level of confidence that can be placed in the analysis of any interaction effects within a DOE study. To understand confounding, the concepts of balanced and orthogonal must be explained first.

#### Balanced

A balanced Fractional Factorial DOE is one where for each factor, the same number of sample test runs are conducted with that factor at its high level and at its low level. The balanced matrix ensures that equal weighting is given to both the high and low values for each factor.

In the example below, the ½ Fractional Factorial DOE is using runs 1, 4, 6, and 7. In these runs, each of the factors will have two high level runs and two low level runs.

#### Orthogonal

An orthogonal Fractional Factorial DOE is one where each of the factors can be analyzed independent of the other factors. This can be determined by analyzing the test configuration matrix when it is setup with the plus one and minus one values for high and low settings (rather than the actual factor values). In this matrix, the product of each two-factor interaction is calculated. (AxB, AxC, AxD, BxC, etc.) If the sum of all those values for a each interaction is zero, the matrix is orthogonal. In the example shown above, multiplying AxB for runs 1, 4, 6, and 7 yields answers of 1, 1, -1, and -1. Which sum to zero. Similar calculations for AxC and BxC show that this matrix is orthogonal.

#### Confounding (Aliasing)

Confounding occurs when the configuration of high and low factor setting within a test matrix for a control factor is identical to the configurations for an interaction effect. This will occur for at least some interactions in every Fractional Factorial DOE. Depending upon the number of factors and the fraction used, the confounding may be at 2-factor interaction effects, 3-factor interactions effects or even higher numbers of factor interaction effects. The level at which confounding starts is referred to as a resolution number. The higher the resolution number, the higher the number of factor involved in an interaction before confounding starts. So, for instance, Resolution of III means that all two factor interactions are confounded. Resolution IV only has some two factor interactions that are confounded. Resolution V does not have any confounding for two factor interactions and only some of the three factor interactions are confounded. The table below shows the Resolution number for different fractional levels and numbers of factors.

### Hints & tips

- Statistical software, such as Minitab, will generate a fractional factorial test matrix that is balanced and orthogonal.
- Low resolution is OK for screening phases, but when doing refining and optimizing, strive for a high resolution, or even Full Factorial test matrices.
- Confounding or aliasing is of greater concern with complex systems or those systems where some factors cannot be easily controlled so a secondary factor is being used as a surrogate.

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