Design Your Experiments: Fractional Designs


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Design Your Experiments Part VII: Fractional Designs

by Kevin Kilty

Half and partial fraction designs

Experiments can be very expensive in terms of money and time. One way to help lower the cost of experimentation is to use a half- or fractional factorial design.

The main effects postulate states that single-factor effects tend to dominate two-factor interactions, and these in turn dominate three-factor interactions. Fractional design makes use of this postulate by sacrificing high order interactions to economize experiments. A half-fraction design will take one of the high order interactions (the defining interaction for the design) and limit it to factor levels that are either +1 or -1, but not both. This will reduce the number of potential experiments by one-half. Further reductions are possible if a person should choose to eliminate the highest order interaction completely, and begin making half-fraction designs of the next order down.

However, a person cannot organize a reduction in cost and effort of experiments without consequences. Eventually the reduction of experimental design has a detrimental effect on identifying which factors or interactions control a process. Reducing the completeness of an experimental design leaves particular interactions invisible to later analysis. They appear exactly like other interactions that we can still observe. People often say that the lost interactions are aliased, although the term alias refers to many other hidden factors, such as the summation of frequency components that results from digitizing a continuous signal.

Aliasing of interactions and factors

I will provide one example of a factional factorial design and the aliased structure which results from it. In addition to all of its other usefulness, the table of contrasts can show aliasing of factors. Let me use the example of a 2³ factorial design, which is a full design for experiments involving 3 factors at two levels each. The table of contrasts (minus the term for the constant) is:

Hypothetical Table of Contrasts

A    B    C   AB  AC   CB   ABC
-------------------------------
-1  -1   -1   1    1   1    -1  
 1  -1   -1  -1   -1   1     1  
-1   1   -1  -1    1  -1     1  
 1   1   -1   1   -1  -1    -1  
-1  -1    1   1   -1  -1     1  
 1  -1    1  -1    1  -1    -1  
-1   1    1  -1   -1   1    -1  
 1   1    1   1    1   1     1  
-------------------------------

I've included each factor at the -1 (low) level and the (+1) high level only. The eight rows in this table shows the full 2³ design. I'll now reduce the design to a half fraction by suppressing all experiments which result in the -1 level for the highest order interaction (ABC). Consider the table of contrasts which results from this suppression.


Reduced Experiment Table of Contrasts
A    B    C   AB  AC   CB   ABC
-------------------------------
 1  -1   -1  -1   -1   1     1  
-1   1   -1  -1    1  -1     1  
-1  -1    1   1   -1  -1     1  
 1   1    1   1    1   1     1  
-------------------------------

Notice that the column for the factor A, is identical to the column for the CB interaction. There is no means by which this experiment design can separate these two effects-the A factor or the CB interaction. These two are aliased in this reduce design. How is it that a design with one factor aliased or confounded with a higher order interaction is useful? The answer is that I have some independent information which tells me that either the factor A is significant or the CB interaction, but not both. Therefore, even though the two are aliased I can decide which of them my experimental data determines.

This idea of using the table of contrasts to identify which interactions and factors are aliased will work with any reduced experiment design.

Resolution

Resolution might mean one of many different things of interest to experimental design. In terms of formal experimental design, resolution refers to the lowest order interaction that is aliased with a single design factor. For example, in the previous section I used a three-factor interaction to define the reduced experiment design. This design left a two-term interaction (CB) aliased with a primary experiment factor (A). It is the simplest reduced design available and has the lowest order resolution. It is certainly adequate for experiments intended for screening which factors control a process. However, if a scientist believes that higher order interactions are important, then the scientist has to make use of a design with higher resolution. For example, a higher resolution design is one in which the primary factors are not aliased with any two-factor interaction, but only with a three-factor interaction. This design allows a person to unambiguously identify all primary and two-factor interactions, and completely measure them.

Composite designs

Experiments should proceed sequentially, with each successive experiment planned according to what the last one showed. This saves time and resources. In particular it saves resources for the final most resource intensive experiments. The idea is to use a mixture, a composite, of low resolution and high resolution experiments.

For example, I can begin to investigate some phenomenon with simple experiments of low resolution. This will illuminate a restricted region of response space. Response space is an N+1-dimensional space in which one of the dimensions is the objective, be it cost, profit, yield or whatever, and the remaining N dimensions are factors. Response is a surface in this space, and the surface might be non-linear and complex.

From the results of this experiment I may decide to embark on one of several lines of inquiry. I may use the results to decide on an initial direction for a path to explore in response space. Generally I will do this when I am trying to optimize an objective. This direction is an initial one only, and I will probably vary my search direction as I proceed, guided by the results of further low-resolution experiments. I can make this a path of steepest descent, for instance.

Occasionally I can utilize higher resolution experiments to test the adequacy of my guiding model and to more nearly locate the optimum points for operating a process. Such a process of mixing low resolution and high resolution experiments may not locate the global optimum process, but only a local minimum, and that minimum may be correct only to the extent to which the model I am using is adequate.

Another use of the initial low-resolution experiment is to help plan for a more complete investigation of this same portion of response space. These experiments don't have to be limited to just two levels per factor. I could use many different levels per factor to illuminate non-linear relationships of a factor and an objective. A second order experiment is one in which I plan to estimate non-linear response of the objective to the various factors. It will involve finding the most appropriate coefficients to terms such as (x_a)². In order to do this I must use more than two levels for factors. One method is to use composite central design, which I spoke of earlier, but another possibility is to use design of each factor in various fractional levels. The specific choice depends on convenience, or perhaps on constraints about what levels are possible.

There are other formal methods of experimental design. I do not plan to discuss anything more about experimental design until the end of this series, where I hope to have a couple of realistic, and complete, design procedures to present. However, if any readers have a hankering for more information about design, let me refer you to a couple of references. The first is an inexpensive Dover edition of E. Bright Wilson's classic text titled Introduction to Scientific Research. Wilson presents factorial design in a different way, using a different mnemonic than the table of contrasts. He also covers a much greater range of topics than I. Another more modern and expensive book is G.G. Vining's book Introduction to Engineering Statistics. The unfortunate thing about Vining's book beyond its price is that he concentrates on engineering issues, which are somewhat different than scientific issues.

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