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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.
- 00:04 Hi, I'm Ray Sheen.
- 00:05 An important issue to be aware of with fractional
- 00:10 factorial DOEs is the issue of confounding effects.
- 00:14 Confounding effects are based upon the selection of sample configurations when
- 00:18 doing a fractional factorial DOE.
- 00:20 The accuracy of the statistical analysis is dependent upon which
- 00:24 sample configuration are selected.
- 00:26 There are two important criteria to consider.
- 00:29 First is that the selected runs are balanced.
- 00:32 That means that in the runs,
- 00:34 each factor has as many high level settings as low level settings.
- 00:38 Next, the selection must be orthogonal.
- 00:40 And that means that, each control factor can be analyzed independently.
- 00:44 I'll show the formula for that on another slide, but
- 00:47 it means that the sum of the interactions for the factor will add to 0.
- 00:51 The selection of the tests or runs that make up the test plan matrix for
- 00:56 a fractional factorial DOE should only include the runs that create a balanced
- 01:00 and orthogonal test matrix.
- 01:02 Now trying to manually decide which set of runs will do this can be laborious and
- 01:07 lead to an error in run configuration.
- 01:09 So I recommend that when doing a fractional factorial DOE,
- 01:13 use a DOE software application which can make the selection for you and
- 01:17 ensure that your test matrix is balanced and orthogonal.
- 01:21 Let's look at an example of balanced and orthogonal.
- 01:24 This matrix represents a one-half fractional DOE with
- 01:29 three two-level control factors.
- 01:32 A full factor DOE would require 8 runs, and
- 01:35 we have organized these using the Yates method.
- 01:39 But as a one-half fractional factorial DOE, we will only need 4 runs,
- 01:43 in this case runs 1, 4, 6, and 7.
- 01:46 A matrix of those four runs will be balanced.
- 01:49 For each of the control factors, there are two runs where the level is high and
- 01:53 two runs with level is low.
- 01:55 As long as we have the same number of high and low level runs,
- 01:59 the testing is balanced.
- 02:01 Orthogonal is a little more complex.
- 02:03 Let's first look at the factor A and B interactions.
- 02:07 Run 1 is +1 times +1, so that is a +1.
- 02:11 Run 4 is -1 times -1, so that is also a +1.
- 02:15 Together, they add to 2.
- 02:18 Now, run 6 is -1 times +1, which is a -1.
- 02:22 We combine that with our previous sum of 2 and
- 02:26 we're back to just a sum of 1.
- 02:29 Then run 7 is +1 times -1, another -1, and
- 02:34 combining that with our previous sum of +1 brings us back to 0.
- 02:39 You can follow the same process with B*C and A*C.
- 02:44 You'll find that they also result in a sum of 0, so the test matrix is orthogonal.
- 02:51 Now let me introduce the term confounding effects.
- 02:54 A disadvantage of fractional factorial DOE is that it introduces confounding effects,
- 03:00 sometimes called aliasing effects.
- 03:03 I will use both terms interchangeably in our discussion.
- 03:07 Aliasing occurs when a primary effect, or an interaction effect,
- 03:11 have the exact same combination of high and low factor settings.
- 03:16 Look at our example, factor A and the combination of integration effect for
- 03:21 B and C, have exactly the same settings.
- 03:24 We see the same problem with factor B and the integration of A and C.
- 03:28 And the problem exist for factor C and the integration of A and B.
- 03:33 So let's look at the first example.
- 03:36 If that portion of the analysis data has a high coefficient,
- 03:40 it's not clear if this due to factor A or the combination of B and C.
- 03:44 That is what we mean by confounding or aliasing.
- 03:48 We also note that the combination of A, B and C was not orthogonal.
- 03:52 The sum of the product of the four runs is not zero,
- 03:55 rather it is a positive number, so we can't use that coefficient.
- 03:59 The confounding issue is summarized by the use of a resolution number.
- 04:04 As we can see, fractional factorial DOE studies cannot analyze all
- 04:08 the interaction effects because only a fraction of the runs are completed as
- 04:13 compared to a full factorial DOE.
- 04:15 The ability or power to analyze those interaction effects is expressed
- 04:20 using a resolution number, this number is a Roman numeral.
- 04:24 And the higher the value, the more interactions the study can analyze.
- 04:29 As you can see in the matrix, the number of runs associated with a full
- 04:33 factorial DOE is labeled with the term full in green.
- 04:37 Then a resolution number is shown based upon the number of runs and factors for
- 04:42 fractional factorial.
- 04:44 A level three resolution will only be able to consider the primary effects.
- 04:49 All interaction effects are confounded.
- 04:51 This was this case for the example I illustrated on the previous slide.
- 04:56 And you can see that the 3-factor four runs is a red Roman numeral III.
- 05:00 The Roman numeral IV indicates that some of the two level interactions can
- 05:05 be analyzed.
- 05:06 And this resolution is shown in yellow.
- 05:09 The Roman numeral V is main affects and
- 05:12 two level interactions and some three level interactions.
- 05:16 It's now shown as green.
- 05:18 The higher Roman numerals will have even higher levels of interactions effects
- 05:22 that are analyzed.
- 05:23 I mentioned several times that you must be careful in the selection of the run
- 05:29 configuration for a fractional factorial DOE and now you know why.
- 05:34 The selection must be balanced and orthogonal and
- 05:38 the resolution will tell you what level of confounding or aliasing occurs.
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