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Pick a Design, Any Design

by C.J. Keller and Richard Scranton

If you're like most people, you've heard of Designed Experiments, but have found the inside of your eyelids more pleasant to look at than the books devoted to the topic. This is surely unfortunate, given the wealth of information you could uncover about your process using these techniques. The problem with most of these texts is that they have to choose between a comprehensive coverage with design construction and evaluation of many design types or a simplistic coverage of only one approach. The latter books tend to leave the impression that there's only one way to skin the proverbial cat.

Often times, there are many designs that can be used to estimate the effects of process parameters (factors) on a process outcome (response). A design's suitability for a given application is limited by the number of runs in the design (i.e. number of conditions you run in the experiment). At most, you can only estimate the effect of the mean, n-1 factors and factor interactions in a design, where n is the number of runs. So if you assume there are no interactions between the factors (sometimes called a screening design), a properly conceived 8-run design can estimate the effect of 7 factors on a response. However, not all designs are created equally, so there are larger designs that can only estimate the same 7 factors. In some cases additional runs are added to improve estimates of error or the precision of estimates.

So what is the difference between these designs? Many times, it's just the name. Take for example one of Taguchi's well known L8 designs, shown below with an 8-run Fractional Factorial and an 8-run Plackett-Burman design. Looking in the upper half of these tables, the designs appear to be different. They have, in fact, a common core of three columns (labeled a, b and c), which are easily identified by rearranging the rows as shown in the bottom half of the tables. While the non-core columns in the three formats do not appear to be always identical, the difference is only in which level of the factor is run, resulting in the design estimating the negative sense of the factor (i.e. rather than estimating the ab interaction, it estimates minus ab). The row order is mathematically inconsequential, especially given that run order should be randomized for the experiment.

Each core column can be used to represent a process parameter. The effects of the three core factors (factors a, b and c) may be independently estimated with the eight-run design. The non-core columns may be used, for example, to estimate the possible interactions of the three core factors (three 2-factor interactions (ab, ac, be) and one 3-factor interaction (abc)). If an assumption is made that no factor interacts with any other (a screening design), then up to seven factors may be estimated by this same design. The rows at the bottom of the Fractional Factorial table show an alternative assignment for 3 factors (1 of 168 such assignments) and typical assignments for 4, 5 and 7 factors.

The 8-run design is often used to estimate the effect of four factors. If no 3-factor interaction is expected (generally a good assumption), the fourth factor d may be represented by the column otherwise used for the abc interaction. That design is the second of the Taguchi L8 linear graphs. Note that an estimate of the bc interaction is now confounded with the ad interaction; that is, they cannot be independently estimated.

If five factors are to be estimated with an 8-run design, with no interactions required, then any one of the 2-factor interactions may be chosen as the alias (replaced factor) for the fifth factor. However, it is usually better to choose both the fourth and fifth factors from the set of 2-factor interactions to get the least confounding in the design. For example, if abc and ac were chosen for the fourth and fifth factors, the interaction of b and abc (the alias of the fourth factor) would be ac. If however ab and ac were chosen for the fourth and fifth factors, of the ten possible 2-factor interactions of any main factor, three would be with the unused combination abc.

The assignment of factors to columns is not unique. However, once the main factors have been assigned, the columns which estimate their interactions (and aliasing) are determined. The last row of the bottom section of the fractional factorial design shows a different allocation of columns to factors. In general, for larger number of runs, some column assignments are more efficient than others, i.e., more parameters or interactions may be independently estimated.

If this still seems confusing, don't sweat it. In DOE-PC IV the emphasis is on describing the goals of the design in terms of required estimates for factors and interactions. The program takes care of the details of assigning factors to columns so that all the required main and interaction effects may be independently estimated. For instance, the preceding design requirement for {a,b,c,d,e,f,g} or for {a,b,c,d,ab,ac,bc} or for {a,b,c,d,ab,ac,ad} will result in an 8-run design as above. A requirement for {a,b,c,d,ab,ac,ad,bc,bd,cd} will result in a 12-run design .

When a total of 7 factors and interactions (and the mean) are to be estimated with an 8-run design, the design is said to be saturated. Often these designs are D-optimal (D-efficiency = 1). As such, they're not too useful because additional runs are required to estimate error or precision. In DOE-PC IV this can be conveniently accomplished by Action/Change/Add. Adding at least two runs will improve the average variance of coefficient, make the minimal detectable effect calculation feasible and decrease the D-efficiency. The result is a more practical and useful experimental design.