A robust parameter design, introduced by Genichi Taguchi, is an experimental design used to exploit the interaction between control and uncontrollable noise variables by robustification—finding the settings of the control factors that minimize response variation from uncontrollable factors.Brewer, K., Carraway, L., and Ingram, D. (2010) "Forward Selection as a Candidate for Constructing Nonregular Robust Parameter Designs." Arkansas State University. Control variables are variables of which the experimenter has full control. Noise variables lie on the other side of the spectrum. While these variables may be easily controlled in an experimental setting, outside of the experimental world they are very hard, if not impossible, to control. Robust parameter designs use a naming convention similar to that of FFDs. A 2(m1+m2)-(p1-p2) is a 2-level design where m1 is the number of control factors, m2 is the number of noise factors, p1 is the level of fractionation for control factors, and p2 is the level of fractionation for noise factors.
Consider an RPD cake-baking example from Montgomery (2005), where an experimenter wants to improve the quality of cake.Montgomery, D. (2005), Design and Analysis of Experiments. 6th ed. Wiley. While the cake manufacturer can control the amount of flour, amount of sugar, amount of baking powder, and coloring content of the cake, other factors are uncontrollable, such as oven temperature and bake time. The manufacturer can print instructions for a bake time of 20 minutes but in the real world has no control over consumer baking habits. Variations in the quality of the cake can arise from baking at 325° instead of 350° or from leaving the cake in the oven for a slightly too short or too long period of time. Robust parameter designs seek to minimize the effects of noise factors on quality. For this example, the manufacturer hopes to minimize the effects in fluctuation of bake time on cake quality, and in doing this the optimal settings for the control factors are required.
RPDs are primarily used in a simulation setting where uncontrollable noise variables are generally easily controlled. Whereas in the real world, noise factors are difficult to control; in an experimental setting, control over these factors is easily maintained. For the cake-baking example, the experimenter can fluctuate bake-time and oven-temperature to understand the effects of such fluctuation that may occur when control is no longer in his/her hands.
Robust parameter designs are very similar to fractional factorial designs (FFDs) in that the optimal design can be found using Hadamard matrices, principles of effect hierarchy and factor sparsity are maintained, and aliasing is present when full RPDs are fractionated. Much like FFDs, RPDs are screening designs and can provide a linear model of the system at hand. What is meant by effect hierarchy for FFDs is that higher-order interactions tend to have a negligible effect on the response.Wu, C.F.J. and Hamada, M. (2000), Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley. As stated in Carraway, main effects are most likely to have an effect on the response, then two-factor interactions, then three-factor interactions, and so on.Carraway, L. (2008). "Investigating the Use of Computational Algorithms for Constructing Non-Regular Robust Parameter Designs," Masters Thesis, Arkansas State University. The concept of effect sparsity is that not all factors will have an effect on the response. These principles are the foundation for fractionating Hadamard matrices. By fractionating, experimenters can form conclusions in fewer runs and with fewer resources. Oftentimes, RPDs are used at the early stages of an experiment. Because two-level RPDs assume linearity among factor effects, other methods may be used to model curvature after the number of factors has been reduced.
Construction
Hadamard matrices are square matrices consisting of only + and −. If a Hadamard matrix is normalized and fractionated, a design pattern is obtained. However, not all designs are equal, which means that some designs are better than others, and specific design criteria are used to determine which design is best. After obtaining a design pattern, experimenters generally know to which setting each factor should be set. Each row, in the pattern, indicates a run, and each column indicates a factor. For the partial design pattern shown left, the experimenter has identified seven factors that may have an effect on the response and hopes to gain insight as to which factors have an effect in eight runs. In the first run, factors 1, 4, 5, and 6 are set to high levels while factors 2, 3, and 7 are set to low levels. Low levels and high levels are settings typically defined by the subject matter expert. These values are extremes but not so extreme that the response is pushed into non-smooth regions. After each run, results are obtained; and by fluctuating multiple factors in single runs instead of using the OFAT method, interactions between variables may be estimated as well as the individual factor effects. If two factors interact, then the effect one factor has on the response is different depending on the settings of another factor.
Fractionating Hadamard matrices appropriately is very time-consuming. Consider a 24-run design accommodating six factors. The number of Hadamard designs from each Hadamard matrix is 23 choose 6; that is 100,947 designs from each 24×24 Hadamard matrix. Since there are 60 Hadamard matrices of that size, the total number of designs to compare is 6,056,820. Leoppky, Bingham, and Sitter (2006) used complete search methodology and have listed the best RPDs for 12, 16, and 20 runs. Because complete search work is so exhaustive, the best designs for larger run sizes are often not readily available. In that case, other statistical methods may be used to fractionate a Hadamard matrix in such a way that allows only a tolerable amount of aliasing. Efficient algorithms such as forward selection and backward elimination have been produced for FFDs, but due to the complexity of aliasing introduced by distinguishing control and noise variables, these methods have not yet been proven effective for RPDs.Ingram, D. (2000), "The construction of generalized minimum aberration designs by efficient algorithm". Dissertation, University of Memphis.Ingram, D. and Tang, B. (2001), Efficient Computational Algorithms for Searching for Good Designs According to the Generalized Minimum Aberration Criterion, American Journal of Mathematical and Management Sciences, 21 325–344.Ingram, D. And Tang, B. (2005), Construction of minimum G-aberration Designs via Efficient Computational Algorithms, Journal of Quality Technology, 37 101-114.
History and design criteria To fully understand the design criteria, an understanding of history and fractional factorial designs is necessary. FFDs seek to understand which factors have an effect on a response and seek to optimize the response by finding the appropriate factor settings. Unlike RPDs, FFDs do not distinguish between control and noise variables.
Resolution and minimum aberration In 2003, Bingham and SitterBingham, D. and Sitter, R.R. (2003), "Fractional Factorial Split-Plot Designs for Robust Parameter Experiments", Technometrics, 45 80–89. defined maximum resolution and minimum aberration for two-level fractional factorial designs. Resolution determines the worst amount of aliasing present, and aberration determines how much of that worst-case aliasing is present in the design. Resolution III designs alias main effects with two-factor interactions. Resolution IV designs alias main effects with three-factor interactions. Resolution V designs alias main effects with four-factor interactions. As the resolution increases, the level of aliasing becomes less serious because higher order interactions tend to have negligible effects on the response. Resolution measures regular designs; that is, effects are either fully aliased or not aliased at all. Consider the following statement, "Factor A is aliased with the two-factor interaction of factors BC." This means that if the two-factor interaction BC has an effect on the response, then the estimation of factor A's effect on the response is contaminated because factor A's effect cannot be distinguished from BC's effect. Clearly a resolution V design is preferred over a resolution IV design.
Designs of the same resolution are not always equal, and the knowledge of which type of aliasing is the worst involved is not enough to know which design is better. Instead further investigation of how much of the worst-case aliasing is needed. This idea is known as minimum aberration. Better designs contain the least amount of the worst-case aliasing. If designs D1 and D2 are both resolution V designs, but D1 has more instances of main effects aliased with 4-factor interactions, then D2 is the better design. D2 is the better design because there is a larger quantity of well-estimated effects.
Generalized resolution and generalized minimum aberration Fontana, Pistone, and Rogantin Fontana, R. Pistone, G. and Rogantin, M.P. (2000), "Classification of Two-Level Factorial Fractions", Journal of Statistical Planning and Inference, 87 149–172. had created an indicator function for two-level fractional factorial designs, and in 2003 Ye expanded the indicator function for regular and nonregular designs.Ye, K.Q. (2003), "Indicator Functions and Its Application in Two Level Factorial Designs", Annals of Statistics, 31 984–994. In doing this, Ye established generalized resolution and generalized minimum aberration. Whereas regular designs are designs with run size equaling a power of two; nonregular designs can be any multiple of four. In nonregular designs, effects can be fully aliased, partially aliased, or not aliased at all. Generalized minimum aberration and generalized resolution take this partial aliasing into account.
Formally, Ye (2003) distinguishes between regular and nonregular designs and states that any polynomial function can be written as Where:
If then the design is regular; otherwise partial aliasing exists.
While Ye developed this indicator function, Bingham and Sitter were working on clarification of resolution and aberration for robust parameter designs. In 2006, Leoppky, Bingham, and Sitter published the extended word-length pattern and indicator function for robust parameter designs. Because RPDs are concerned about minimizing process variation due to noise factors, the priority of effects changes from the hierarchy of effects of FFDs. Main effects are still the first priority, and two-factor interactions are still the second priority; but if any interactions have a control-by-noise (CN) interaction, then that interaction is increased by 0.5 on the priority scale. For example, a CCN three-factor interaction would be a priority 3 in a FFD because three-factor interactions are the third priority, two-factor interactions are the second priority, and main effects are the first priority. However, since RPDs are concerned about noise variables, the CCN interaction is a priority 2.5 effect. The CN interaction bumps the priority up by 0.5; so the traditional priority 3 minus the 0.5 for the CN interaction results in a 2.5 priority. A full table of priorities can be found in Leoppky, Bingham, and Sitter (2006).Loeppky, J. L., Bingham, D. and Sitter R.R, (2006), Constructing Non-Regular Robust Parameter Designs, Journal of Statistical Planning and Inference, 136 3710-3729.
Design comparison Further investigation of the principles introduced will provide a deeper understanding of design comparison.
For regular fractional factorial designs, the word length will determine what types of aliasing are present. For example, the word "2367" can be broken into aliasing structures as follows: