Engineering Statistics Handbook Episode 6 Part 15 pps

5. Process Improvement 5.3. Choosing an experimental design 5.3.3. How do you select an experimental design? 5.3.3.4. Fractional factorial designs 5.3.3.4.7.Summary tables of useful fractional factorial designs Useful fractional factorial designs for up to 10 factors are summarized here There are very useful summaries of two-level fractional factorial designs for up to 11 factors, originally published in the book Statistics for Experimenters by G.E.P. Box, W.G. Hunter, and J.S. Hunter (New York, John Wiley & Sons, 1978). and also given in the book Design and Analysis of Experiments, 5th edition by Douglas C. Montgomery (New York, John Wiley & Sons, 2000). Generator column notation can use either numbers or letters for the factor columns They differ in the notation for the design generators. Box, Hunter, and Hunter use numbers (as we did in our earlier discussion) and Montgomery uses capital letters according to the following scheme: Notice the absence of the letter I. This is usually reserved for the intercept column that is identically 1. As an example of the letter notation, note that the design generator "6 = 12345" is equivalent to "F = ABCDE". 5.3.3.4.7. Summary tables of useful fractional factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3347.htm (1 of 3) [5/1/2006 10:30:37 AM] Details of the design generators, the defining relation, the confounding structure, and the design matrix TABLE 3.17 catalogs these useful fractional factorial designs using the notation previously described in FIGURE 3.7. Clicking on the specification for a given design provides details (courtesy of Dataplot files) of the design generators, the defining relation, the confounding structure (as far as main effects and two-level interactions are concerned), and the design matrix. The notation used follows our previous labeling of factors with numbers, not letters. Click on the design specification in the table below and a text file with details about the design can be viewed or saved TABLE 3.17 Summary of Useful Fractional Factorial Designs Number of Factors, k Design Specification Number of Runs N 3 2 III 3-1 4 4 2 IV 4-1 8 5 2 V 5-1 16 5 2 III 5-2 8 6 2 VI 6-1 32 6 2 IV 6-2 16 6 2 III 6-3 8 7 2 VII 7-1 64 7 2 IV 7-2 32 7 2 IV 7-3 16 7 2 III 7-4 8 8 2 VIII 8-1 128 8 2 V 8-2 64 8 2 IV 8-3 32 8 2 IV 8-4 16 9 2 VI 9-2 128 9 2 IV 9-3 64 9 2 IV 9-4 32 5.3.3.4.7. Summary tables of useful fractional factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3347.htm (2 of 3) [5/1/2006 10:30:37 AM] 9 2 III 9-5 16 10 2 V 10-3 128 10 2 IV 10-4 64 10 2 IV 10-5 32 10 2 III 10-6 16 11 2 V 11-4 128 11 2 IV 11-5 64 11 2 IV 11-6 32 11 2 III 11-7 16 15 2 III 15-11 16 31 2 III 31-26 32 5.3.3.4.7. Summary tables of useful fractional factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3347.htm (3 of 3) [5/1/2006 10:30:37 AM] Saturated Main Effect designs PB designs also exist for 20-run, 24-run, and 28-run (and higher) designs. With a 20-run design you can run a screening experiment for up to 19 factors, up to 23 factors in a 24-run design, and up to 27 factors in a 28-run design. These Resolution III designs are known as Saturated Main Effect designs because all degrees of freedom are utilized to estimate main effects. The designs for 20 and 24 runs are shown below. 20-Run Plackett- Burnam design TABLE 3.19 A 20-Run Plackett-Burman Design X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 2 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 3 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 4 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 5 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 6 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 7 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 8 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 9 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 10 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 11 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 12 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 13 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 14 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 15 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 16 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 17 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 18 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 19 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 20 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 24-Run Plackett- Burnam design TABLE 3.20 A 24-Run Plackett-Burman Design X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 3 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 4 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 5 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 6 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 7 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 8 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 9 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 10 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 11 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 12 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 13 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 5.3.3.5. Plackett-Burman designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri335.htm (2 of 3) [5/1/2006 10:30:38 AM] 14 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 15 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 16 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 17 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 18 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 19 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 20 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 21 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 22 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 23 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 24 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 No defining relation These designs do not have a defining relation since interactions are not identically equal to main effects. With the designs, a main effect column X i is either orthogonal to X i X j or identical to plus or minus X i X j . For Plackett-Burman designs, the two-factor interaction column X i X j is correlated with every X k (for k not equal to i or j). Economical for detecting large main effects However, these designs are very useful for economically detecting large main effects, assuming all interactions are negligible when compared with the few important main effects. 5.3.3.5. Plackett-Burman designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri335.htm (3 of 3) [5/1/2006 10:30:38 AM] Quadratic models almost always sufficient for industrial applications If the experimenter has defined factor limits appropriately and/or taken advantage of all the tools available in multiple regression analysis (transformations of responses and factors, for example), then finding an industrial process that requires a third-order model is highly unusual. Therefore, we will only focus on designs that are useful for fitting quadratic models. As we will see, these designs often provide lack of fit detection that will help determine when a higher-order model is needed. General quadratic surface types Figures 3.9 to 3.12 identify the general quadratic surface types that an investigator might encounter FIGURE 3.9 A Response Surface "Peak" FIGURE 3.10 A Response Surface "Hillside" FIGURE 3.11 A Response Surface "Rising Ridge" FIGURE 3.12 A Response Surface "Saddle" Factor Levels for Higher-Order Designs 5.3.3.6. Response surface designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri336.htm (2 of 6) [5/1/2006 10:30:39 AM] Possible behaviors of responses as functions of factor settings Figures 3.13 through 3.15 illustrate possible behaviors of responses as functions of factor settings. In each case, assume the value of the response increases from the bottom of the figure to the top and that the factor settings increase from left to right. FIGURE 3.13 Linear Function FIGURE 3.14 Quadratic Function FIGURE 3.15 Cubic Function A two-level experiment with center points can detect, but not fit, quadratic effects If a response behaves as in Figure 3.13, the design matrix to quantify that behavior need only contain factors with two levels low and high. This model is a basic assumption of simple two-level factorial and fractional factorial designs. If a response behaves as in Figure 3.14, the minimum number of levels required for a factor to quantify that behavior is three. One might logically assume that adding center points to a two-level design would satisfy that requirement, but the arrangement of the treatments in such a matrix confounds all quadratic effects with each other. While a two-level design with center points cannot estimate individual pure quadratic effects, it can detect them effectively. Three-level factorial design A solution to creating a design matrix that permits the estimation of simple curvature as shown in Figure 3.14 would be to use a three-level factorial design. Table 3.21 explores that possibility. Four-level factorial design Finally, in more complex cases such as illustrated in Figure 3.15, the design matrix must contain at least four levels of each factor to characterize the behavior of the response adequately. 5.3.3.6. Response surface designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri336.htm (3 of 6) [5/1/2006 10:30:39 AM] 3-level factorial designs can fit quadratic models but they require many runs when there are more than 4 factors TABLE 3.21 Three-level Factorial Designs Number of Factors Treatment Combinations 3 k Factorial Number of Coefficients Quadratic Empirical Model 2 9 6 3 27 10 4 81 15 5 243 21 6 729 28 Fractional factorial designs created to avoid such a large number of runs Two-level factorial designs quickly become too large for practical application as the number of factors investigated increases. This problem was the motivation for creating `fractional factorial' designs. Table 3.21 shows that the number of runs required for a 3 k factorial becomes unacceptable even more quickly than for 2 k designs. The last column in Table 3.21 shows the number of terms present in a quadratic model for each case. Number of runs large even for modest number of factors With only a modest number of factors, the number of runs is very large, even an order of magnitude greater than the number of parameters to be estimated when k isn't small. For example, the absolute minimum number of runs required to estimate all the terms present in a four-factor quadratic model is 15: the intercept term, 4 main effects, 6 two-factor interactions, and 4 quadratic terms. The corresponding 3 k design for k = 4 requires 81 runs. Complex alias structure and lack of rotatability for 3-level fractional factorial designs Considering a fractional factorial at three levels is a logical step, given the success of fractional designs when applied to two-level designs. Unfortunately, the alias structure for the three-level fractional factorial designs is considerably more complex and harder to define than in the two-level case. Additionally, the three-level factorial designs suffer a major flaw in their lack of `rotatability.' Rotatability of Designs 5.3.3.6. Response surface designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri336.htm (4 of 6) [5/1/2006 10:30:39 AM] "Rotatability" is a desirable property not present in 3-level factorial designs In a rotatable design, the variance of the predicted values of y is a function of the distance of a point from the center of the design and is not a function of the direction the point lies from the center. Before a study begins, little or no knowledge may exist about the region that contains the optimum response. Therefore, the experimental design matrix should not bias an investigation in any direction. Contours of variance of predicted values are concentric circles In a rotatable design, the contours associated with the variance of the predicted values are concentric circles. Figures 3.16 and 3.17 (adapted from Box and Draper, `Empirical Model Building and Response Surfaces,' page 485) illustrate a three-dimensional plot and contour plot, respectively, of the `information function' associated with a 3 2 design. Information function The information function is: with V denoting the variance (of the predicted value ). Each figure clearly shows that the information content of the design is not only a function of the distance from the center of the design space, but also a function of direction. Graphs of the information function for a rotatable quadratic design Figures 3.18 and 3.19 are the corresponding graphs of the information function for a rotatable quadratic design. In each of these figures, the value of the information function depends only on the distance of a point from the center of the space. 5.3.3.6. Response surface designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri336.htm (5 of 6) [5/1/2006 10:30:39 AM] [...]... factorial part of the design as a function of the TABLE 3.23 Determining for Rotatability Number of Factorial Scaled Value for Factors Portion Relative to ±1 2 3 4 5 5 6 6 22 23 24 25-1 25 26- 1 26 22/4 = 1.414 23/4 = 1 .68 2 24/4 = 2.000 24/4 = 2.000 25/4 = 2.378 25/4 = 2.378 26/ 4 = 2.828 http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3 361 .htm (4 of 5) [5/1/20 06 10:30:40 AM] 5.3.3 .6. 1 Central... categories: Box-Wilson central composite designs and Box-Behnken designs The next sections describe these design classes and their properties http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3 36. htm (6 of 6) [5/1/20 06 10:30:39 AM] 5.3.3 .6. 1 Central Composite Designs (CCD) A CCD design with k factors has 2k star points A central composite design always contains twice as many star points as there are... where the star points are placed for the 3 types of CCD designs FIGURE 3.21 Comparison of the Three Types of Central Composite Designs http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3 361 .htm (3 of 5) [5/1/20 06 10:30:40 AM] 5.3.3 .6. 1 Central Composite Designs (CCD) Comparison of the 3 central composite designs The diagrams in Figure 3.21 illustrate the three types of central composite designs... design with each factor level of the CCC design divided by to generate the CCI design) This design also requires 5 levels of each factor http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3 361 .htm (2 of 5) [5/1/20 06 10:30:40 AM] 5.3.3 .6. 1 Central Composite Designs (CCD) Face Centered CCF In this design the star points are at the center of each face of the factorial space, so = ± 1 This variety requires... Box-Behnken designs are described BLOCK X1 X2 1 1 1 1 1 1 2 2 2 2 2 2 -1 1 -1 1 0 0 -1.414 1.414 0 0 0 0 -1 -1 1 1 0 0 0 0 -1.414 1.414 0 0 http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3 361 .htm (5 of 5) [5/1/20 06 10:30:40 AM] 5.3.3 .6. 2 Box-Behnken designs Geometry of the design The geometry of this design suggests a sphere within the process space such that the surface of the sphere protrudes through... face with the surface of the sphere tangential to the midpoint of each edge of the space Examples of Box-Behnken designs are given on the next page http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3 362 .htm (2 of 2) [5/1/20 06 10:30:40 AM] ...5.3.3 .6 Response surface designs FIGURE 3. 16 FIGURE 3.17 Three-Dimensional Contour Map of the Information Function Illustration for the Information Function of a for a 32 Design 32 Design FIGURE 3.18 Three-Dimensional Illustration . 3 2 III 3-1 4 4 2 IV 4-1 8 5 2 V 5-1 16 5 2 III 5-2 8 6 2 VI 6- 1 32 6 2 IV 6- 2 16 6 2 III 6- 3 8 7 2 VII 7-1 64 7 2 IV 7-2 32 7 2 IV 7-3 16 7 2 III 7-4 8 8 2 VIII 8-1 128 8 2 V 8-2 64 8 2 IV 8-3 32 8 2 IV 8-4 16 9 2 VI 9-2 128 9 2 IV 9-3 64 9 2 IV 9-4 32 5.3.3.4.7 AM] 9 2 III 9-5 16 10 2 V 10-3 128 10 2 IV 10-4 64 10 2 IV 10-5 32 10 2 III 10 -6 16 11 2 V 11-4 128 11 2 IV 11-5 64 11 2 IV 11 -6 32 11 2 III 11-7 16 15 2 III 15- 11 16 31 2 III 31- 26 32 5.3.3.4.7 2.000 5 2 5 2 5/4 = 2.378 6 2 6- 1 2 5/4 = 2.378 6 2 6 2 6/ 4 = 2.828 5.3.3 .6. 1. Central Composite Designs (CCD) http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3 361 .htm (4 of 5) [5/1/20 06 10:30:40