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Split-Plot Designs: What, Why, and How Split-Plot Designs: What, Why, and How BRADLEY JONES SAS Institute, Cary, NC 27513 CHRISTOPHER J. NACHTSHEIM Carlson School of Management, University of Minnesota, Minneapolis, MN 55455 The past decade has seen rapid advances in the development of new methods for the design and analysis of split-plot experiments. Unfortunately, the value of these designs for industrial experimentation has not been fully appreciated. In this paper, we review recent developments and provide guidelines for the use of split-plot designs in industrial applications. Key Words: Block Designs; Completely Randomized Designs; Factorial Designs; Fractional Factorial De- signs; Optimal Designs; Random-Effects Model; Random-Run-Order Designs; Response Surface Designs; Robust Parameter Designs. Introduction ments previously thought to be completely random- ized experiments also exhibit split-plot structure. All industrial experiments are split-plot experiments. This surprising result adds creedence to the Daniel proclamation and has motivated a great deal of pi- HIS provocative remark has been attributed to oneering work in the design and analysis of split- T the famous industrial statistician, Cuthbert plot experiments. We note in particular the identi- Daniel, by Box et al. (2005) in their well-known text fication of minimum-aberration split-plot fractional on the design of experiments. Split-Plot experiments factorial designs by Bingham and Sitter and cowork- were invented by Fisher (1925) and their importance ers (Bingham and Sitter (1999, 2001, 2003), Bingham in industrial experimentation has been long recog- et al. (2004), Loeppky and Sitter (2002)), the devel- nized (Yates (1936)). It is also well known that many opment of equivalent-estimation split-plot designs for industrial experiments are fielded as split-plot exper- response surface and mixture designs (Kowalski et iments and yet erroneously analyzed as if they were al. (2002), Vining et al. (2005), Vining and Kowal- completely randomized designs. This is frequently ski (2008)), and the development of optimal split- the case when hard-to-change factors exist and eco- plot designs by Goos, Vandebroek, and Jones and nomic constraints preclude the use of complete ran- their coworkers (Goos (2002), Goos and Vandebroek domization. Recent work, most notably by Lucas and (2001, 2003, 2004), Jones and Goos (2007 and 2009)). his coworkers (Anbari and Lucas (1994), Ganju and Lucas (1997, 1999, 2005), Ju and Lucas (2002), Webb Our goal here is to review these recent develop- et al. (2004)) has demonstrated that many experi- ments and to provide guidelines for the practicing statistician on their use. Dr. Jones is Director of Research and Development for What Is a Split-Plot Design? the JMP Division of SAS Institute, Inc. His email address is [email protected]. In simple terms, a split-plot experiment is a Dr. Nachtsheim is the Curtis L. Carlson Professor and blocked experiment, where the blocks themselves Chair of the Operations and Management Science Department serve as experimental units for a subset of the factors. at the Carlson School of Management of the University of Min- Thus, there are two levels of experimental units. The nesota. His email address is [email protected]. blocks are referred to as whole plots, while the experi- Journal of Quality Technology 340 Vol. 41, No. 4, October 2009 SPLIT-PLOT DESIGNS: WHAT, WHY, AND HOW 341 TABLE 1. Split-Plot Design and Data for Studying the Corrosion Resistance of Steel Bars (Box et al. (2005)) Temperature Coating Whole-plot (◦C) (randomized order) 1. 360 C2 C3 C1 C4 73 83 67 89 2. 370 C1 C3 C4 C2 65 87 86 91 FIGURE 1. Split Plot Agricultural Layout. (Factor A is the whole-plot factor and factor B is the split-plot factor.) 3. 380 C3 C1 C2 C4 147 155 127 212 mental units within blocks are called split plots, split units, or subplots. Corresponding to the two levels of 4. 380 C4 C3 C2 C1 experimental units are two levels of randomization. 153 90 100 108 One randomization is conducted to determine the assignment of block-level treatments to whole plots. 5. 370 C4 C1 C3 C2 Then, as always in a blocked experiment, a random- 150 140 121 142 ization of treatments to split-plot experimental units occurs within each block or whole plot. 6. 360 C1 C4 C2 C3 33 54 8 46 Split-plot designs were originally developed by Fisher (1925) for use in agricultural experiments. As a simple illustration, consider a study of the effects of two irrigation methods (factor A) and two fertiliz- to-change factor. The experiment was designed to ers (factor B) on yield of a crop, using four available study the corrosion resistance of steel bars treated with four coatings, C1, C2, C3,andC4, at three fur- fields as experimental units. In this investigation, it ◦ ◦ ◦ is not possible to apply different irrigation methods nace temperatures, 360 C, 370 C, and 380 C. Fur- (factor A) in areas smaller than a field, although dif- nace temperature is the hard-to-change factor be- ferent fertilizer types (factor B) could be applied in cause of the time it takes to reset the furnace and relatively small areas. For example, if we subdivide reach a new equilibrium temperature. Once the equi- each whole plot (field) into two split plots, each of the librium temperature is reached, four steel bars with two fertilizer types can be applied once within each randomly assigned coatings C1, C2, C3,andC4 are whole plot, as shown in Figure 1. In this split-plot randomly positioned in the furnace and heated. The design, a first randomization assigns the two irriga- layout of the experiment as performed is given in Ta- tion types to the four fields (whole plots); then within ble 1. Notice that each whole-plot treatment (tem- each field, a separate randomization is conducted to perature) is replicated twice and that there is just assign the two fertilizer types to the two split plots one complete replicate of the split-plot treatments within each field. (coatings) within each whole plot. Thus, we have six whole plots and four subplots within each whole plot. In industrial experiments, factors are often differ- Other illustrative examples of industrial exper- entiated with respect to the ease with which they can iments having both easy-to-change and hard-to- be changed from experimental run to experimental change factors include run. This may be due to the fact that a particular treatment is expensive or time-consuming to change, 1. Box and Jones (1992) describe an experiment or it may be due to the fact that the experiment is in which a packaged-foods manufacturer wished to be run in large batches and the batches can be to develop an optimal formulation of a cake subdivided later for additional treatments. Box et mix. Because cake mixes are made in large al. (2005) describe a prototypical split-plot experi- batches, the ingredient factors (flour, shorten- ment with one easy-to-change factor and one hard- ing, and egg powder) are hard to change. Many Vol. 41, No. 4, October 2009 www.asq.org 342 BRADLEY JONES AND CHRISTOPHER J. NACHTSHEIM packages of cake mix are produced from each the ANOVA sums of squares are orthogonal, a stan- batch, and the individual packages of cake mix dard, mixed-model, ANOVA-based approach to the can be baked using different baking times and analysis is possible. This is the approach that is taken baking temperatures. Here, the easy-to-change, by most DOE textbooks that treat split-plot de- split-plot factors are time and temperature. signs (see, e.g., Kutner et al. (2005), Montgomery 2. Another example involving large batches and (2008), Box et al. (2005)). With this approach, all the subsequent processing of subbatches was experimental factor effects are assumed to be fixed. discussed by Bingham and Sitter (2001). In this The standard ANOVA model for the balanced two- instance, a company wished to study the ef- factor split-plot design, where there are a levels of fect of various factors on the swelling of a wood the whole-plot factor A and b levels of the split-plot product after it has been saturated with water treatment B, where each level of whole-plot factor A and allowed to dry. A batch is characterized is applied to c whole plots (so that the whole plot ef- by the type and size of wood being tested, the fect is nested within factor A), and where the number amounts of two additives used, and the level of runs is n = abc,isgivenby of moisture saturation. Batches can later be Yijk = μ... + αi + βj +(αβ)ij + γk(i) + εijk, (1) subdivided for processing, which is character- ized by three easy-to-change factors, i.e., pro- where: μ... is a constant; αi,thea whole-plot treat- cess time, pressure, and material density. ment effects, are constants subject to αi =0;βj, the b split-plot treatment effects, are constants sub- 3. Bisgaard (2000) discusses the use of split-plot ject to βj =0;(αβ)ij,theab interaction effects, arrangements in parameter design experiments. are constants subject to (αβ)ij =0forallj and In one example, he describes an experiment in- i (αβ)ij =0foralli; γk i ,thenw = ac whole-plot volving four prototype combustion engine de- j ( ) 2 signs and two grades of gasoline. The engineers errors, are independent N(0,σw); εijk are indepen- 2 preferred to make up the four motor designs dent N(0, σ ); i =1,...,a, j =1,...,b, k =1,...,c. and test both of the gasoline types while a par- The ANOVA model (1) is appropriate for balanced ticular motor was on a test stand.
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