Alias patterns and effects that can be estimated in the example design The two-factor alias patterns for X4 are: Original experiment: X4 = X1X2 = X3X7 = X5X6; "Reverse X4" foldover experiment: X4 = -X1X2 = -X3X7 = -X5X6. The following effects can be estimated by combining the original with the "Reverse X4" foldover fraction: X1 + X3X5 + X6X7 X2 + X3X6 + X5X7 X3 + X1X5 + X2X6 X4 X5 + X1X3 + X2X7 X6 + X2X3 + X1X7 X7 + X2X5 + X1X6 X1X4 X2X4 X3X4 X4X5 X4X6 X4X7 X1X2 + X3X7 + X5X6 Note: The 16 runs allow estimating the above 14 effects, with one degree of freedom left over for a possible block effect. Advantage and disadvantage of this example design The advantage of this follow-up design is that it permits estimation of the X4 effect and each of the six two-factor interaction terms involving X4. The disadvantage is that the combined fractions still yield a resolution III design, with all main effects other than X4 aliased with two-factor interactions. Case when purpose is simply to estimate all two-factor interactions of a single factor Reversing a single factor column to obtain de-aliased two-factor interactions for that one factor works for any resolution III or IV design. When used to follow-up a resolution IV design, there are relatively few new effects to be estimated (as compared to designs). When the original resolution IV fraction provides sufficient precision, and the purpose of the follow-up runs is simply to estimate all two-factor interactions for one factor, the semifolding option should be considered. Semifolding 5.3.3.8.2. Alternative foldover designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3382.htm (2 of 3) [5/1/2006 10:30:44 AM] Number of runs can be reduced for resolution IV designs For resolution IV fractions, it is possible to economize on the number of runs that are needed to break the alias chains for all two-factor interactions of a single factor. In the above case we needed 8 additional runs, which is the same number of runs that were used in the original experiment. This can be improved upon. Additional information on John's 3/4 designs We can repeat only the points that were set at the high levels of the factor of choice and then run them at their low settings in the next experiment. For the given example, this means an additional 4 runs instead 8. We mention this technique only in passing, more details may be found in the references (or see John's 3/4 designs). 5.3.3.8.2. Alternative foldover designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3382.htm (3 of 3) [5/1/2006 10:30:44 AM] A notation such as "20" means that factor A is at its high level (2) and factor B is at its low level (0). The 3 3 design The model and treatment runs for a 3 factor, 3-level design This is a design that consists of three factors, each at three levels. It can be expressed as a 3 x 3 x 3 = 3 3 design. The model for such an experiment is where each factor is included as a nominal factor rather than as a continuous variable. In such cases, main effects have 2 degrees of freedom, two-factor interactions have 2 2 = 4 degrees of freedom and k-factor interactions have 2 k degrees of freedom. The model contains 2 + 2 + 2 + 4 + 4 + 4 + 8 = 26 degrees of freedom. Note that if there is no replication, the fit is exact and there is no error term (the epsilon term) in the model. In this no replication case, if one assumes that there are no three-factor interactions, then one can use these 8 degrees of freedom for error estimation. In this model we see that i = 1, 2, 3, and similarly for j and k, making 27 5.3.3.9. Three-level full factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri339.htm (2 of 4) [5/1/2006 10:30:44 AM] treatments. Table of treatments for the 3 3 design These treatments may be displayed as follows: TABLE 3.37 The 3 3 Design Factor A Factor B Factor C 0 1 2 0 0 000 100 200 0 1 001 101 201 0 2 002 102 202 1 0 010 110 210 1 1 011 111 211 1 2 012 112 212 2 0 020 120 220 2 1 021 121 221 2 2 022 122 222 Pictorial representation of the 3 3 design The design can be represented pictorially by FIGURE 3.24 A 3 3 Design Schematic 5.3.3.9. Three-level full factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri339.htm (3 of 4) [5/1/2006 10:30:44 AM] Two types of 3 k designs Two types of fractions of 3 k designs are employed: Box-Behnken designs whose purpose is to estimate a second-order model for quantitative factors (discussed earlier in section 5.3.3.6.2) ● 3 k-p orthogonal arrays.● 5.3.3.9. Three-level full factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri339.htm (4 of 4) [5/1/2006 10:30:44 AM] Table illustrating the generation of a design with one factor at 2 levels and another at 3 levels from a 2 3 design A X L X L AX L AX L X Q AX Q TRT MNT Run A B C AB AC BC ABC A X 1 -1 -1 -1 +1 +1 +1 -1 Low Low 2 +1 -1 -1 -1 -1 +1 +1 High Low 3 -1 +1 -1 -1 +1 -1 +1 Low Medium 4 +1 +1 -1 +1 -1 -1 -1 High Medium 5 -1 -1 +1 +1 -1 -1 +1 Low Medium 6 +1 -1 +1 -1 +1 -1 -1 High Medium 7 -1 +1 +1 -1 -1 +1 -1 Low High If quadratic effect negligble, we may include a second two-level factor If we believe that the quadratic effect is negligible, we may include a second two-level factor, D, with D = ABC. In fact, we can convert the design to exclusively a main effect (resolution III) situation consisting of four two-level factors and one three-level factor. This is accomplished by equating the second two-level factor to AB, the third to AC and the fourth to ABC. Column BC cannot be used in this manner because it contains the quadratic effect of the three-level factor X. More than one three-level factor 3-Level factors from 2 4 and 2 5 designs We have seen that in order to create one three-level factor, the starting design can be a 2 3 factorial. Without proof we state that a 2 4 can split off 1, 2 or 3 three-level factors; a 2 5 is able to generate 3 three-level factors and still maintain a full factorial structure. For more on this, see Montgomery (1991). Generating a Two- and Four-Level Mixed Design Constructing a design with one 4-level factor and two 2-level factors We may use the same principles as for the three-level factor example in creating a four-level factor. We will assume that the goal is to construct a design with one four-level and two two-level factors. Initially we wish to estimate all main effects and interactions. It has been shown (see Montgomery, 1991) that this can be accomplished via a 2 4 (16 runs) design, with columns A and B used to create the four level factor X. Table showing design with 4-level, two 2-level factors in 16 runs TABLE 3.39 A Single Four-level Factor and Two Two-level Factors in 16 runs Run (A B) = X C D 1 -1 -1 x 1 -1 -1 2 +1 -1 x 2 -1 -1 3 -1 +1 x 3 -1 -1 4 +1 +1 x 4 -1 -1 5 -1 -1 x 1 +1 -1 6 +1 -1 x 2 +1 -1 7 -1 +1 x 3 +1 -1 8 +1 +1 x 4 +1 -1 5.3.3.10. Three-level, mixed-level and fractional factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri33a.htm (2 of 5) [5/1/2006 10:30:45 AM] 9 -1 -1 x 1 -1 +1 10 +1 -1 x 2 -1 +1 11 -1 +1 x 3 -1 +1 12 +1 +1 x 4 -1 +1 13 -1 -1 x 1 +1 +1 14 +1 -1 x 2 +1 +1 15 -1 +1 x 3 +1 +1 16 +1 +1 x 4 +1 +1 Some Useful (Taguchi) Orthogonal "L" Array Designs L 9 design L 9 - A 3 4-2 Fractional Factorial Design 4 Factors at Three Levels (9 runs) Run X1 X2 X3 X4 1 1 1 1 1 2 1 2 2 2 3 1 3 3 3 4 2 1 2 3 5 2 2 3 1 6 2 3 1 2 7 3 1 3 2 8 3 2 1 3 9 3 3 2 1 L 18 design L 18 - A 2 x 3 7-5 Fractional Factorial (Mixed-Level) Design 1 Factor at Two Levels and Seven Factors at 3 Levels (18 Runs) Run X1 X2 X3 X4 X5 X6 X7 X8 1 1 1 1 1 1 1 1 1 2 1 1 2 2 2 2 2 2 3 1 1 3 3 3 3 3 3 4 1 2 1 1 2 2 3 3 5 1 2 2 2 3 3 1 1 6 1 2 3 3 1 1 2 2 7 1 3 1 2 1 3 2 3 8 1 3 2 3 2 1 3 1 9 1 3 3 1 3 2 1 2 10 2 1 1 3 3 2 2 1 11 2 1 2 1 1 3 3 2 12 2 1 3 2 2 1 1 3 13 2 2 1 2 3 1 3 2 14 2 2 2 3 1 2 1 3 15 2 2 3 1 2 3 2 1 16 2 3 1 3 2 3 1 2 17 2 3 2 1 3 1 2 3 18 2 3 3 2 1 2 3 1 5.3.3.10. Three-level, mixed-level and fractional factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri33a.htm (3 of 5) [5/1/2006 10:30:45 AM] L 27 design L 27 - A 3 13-10 Fractional Factorial Design Thirteen Factors at Three Levels (27 Runs) Run X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 2 2 2 2 2 2 2 2 2 3 1 1 1 1 3 3 3 3 3 3 3 3 3 4 1 2 2 2 1 1 1 2 2 2 3 3 3 5 1 2 2 2 2 2 2 3 3 3 1 1 1 6 1 2 2 2 3 3 3 1 1 1 2 2 2 7 1 3 3 3 1 1 1 3 3 3 2 2 2 8 1 3 3 3 2 2 2 1 1 1 3 3 3 9 1 3 3 3 3 3 3 2 2 2 1 1 1 10 2 1 2 3 1 2 3 1 2 3 1 2 3 11 2 1 2 3 2 3 1 2 3 1 2 3 1 12 2 1 2 3 3 1 2 3 1 2 3 1 2 13 2 2 3 1 1 2 3 2 3 1 3 1 2 14 2 2 3 1 2 3 1 3 1 2 1 2 3 15 2 2 3 1 3 1 2 1 2 3 2 3 1 16 2 3 1 2 1 2 3 3 1 2 2 3 1 17 2 3 1 2 2 3 1 1 2 3 3 1 2 18 2 3 1 2 3 1 2 2 3 1 1 2 3 19 3 1 3 2 1 3 2 1 3 2 1 3 2 20 3 1 3 2 2 1 3 2 1 3 2 1 3 21 3 1 3 2 3 2 1 3 2 1 3 2 1 22 3 2 1 3 1 3 2 2 1 3 3 2 1 23 3 2 1 3 2 1 3 3 2 1 1 3 2 24 3 2 1 3 3 2 1 1 3 2 2 1 3 25 3 3 2 1 1 3 2 3 2 1 2 1 3 26 3 3 2 1 2 1 3 1 3 2 3 2 1 27 3 3 2 1 3 2 1 2 1 3 1 3 2 L 36 design L36 - A Fractional Factorial (Mixed-Level) Design Eleven Factors at Two Levels and Twelve Factors at 3 Levels (36 Runs) Run 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 2 2 2 2 2 2 2 2 2 2 2 2 3 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 4 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3 5 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 1 1 1 1 6 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 7 1 1 2 2 2 1 1 1 2 2 2 1 1 2 3 1 2 3 3 1 2 2 3 8 1 1 2 2 2 1 1 1 2 2 2 2 2 3 1 2 3 1 1 2 3 3 1 9 1 1 2 2 2 1 1 1 2 2 2 3 3 1 2 3 1 2 2 3 1 1 2 10 1 2 1 2 2 1 2 2 1 1 2 1 1 3 2 1 3 2 3 2 1 3 2 11 1 2 1 2 2 1 2 2 1 1 2 2 2 1 3 2 1 3 1 3 2 1 3 12 1 2 1 2 2 1 2 2 1 1 2 3 3 2 1 3 2 1 2 1 3 2 1 5.3.3.10. Three-level, mixed-level and fractional factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri33a.htm (4 of 5) [5/1/2006 10:30:45 AM] 13 1 2 2 1 2 2 1 2 1 2 1 1 2 3 1 3 2 1 3 3 2 1 2 14 1 2 2 1 2 2 1 2 1 2 1 2 3 1 2 1 3 2 1 1 3 2 3 15 1 2 2 1 2 2 1 2 1 2 1 3 1 2 3 2 1 3 2 2 1 3 1 16 1 2 2 2 1 2 2 1 2 1 1 1 2 3 2 1 1 3 2 3 3 2 1 17 1 2 2 2 1 2 2 1 2 1 1 2 3 1 3 2 2 1 3 1 1 3 2 18 1 2 2 2 1 2 2 1 2 1 1 3 1 2 1 3 3 2 1 2 2 1 3 19 2 1 2 2 1 1 2 2 1 2 1 1 2 1 3 3 3 1 2 2 1 2 3 20 2 1 2 2 1 1 2 2 1 2 1 2 3 2 1 1 1 2 3 3 2 3 1 21 2 1 2 2 1 1 2 2 1 2 1 3 1 3 2 2 2 3 1 1 3 1 2 22 2 1 2 1 2 2 2 1 1 1 2 1 2 2 3 3 1 2 1 1 3 3 2 23 2 1 2 1 2 2 2 1 1 1 2 2 3 3 1 1 2 3 2 2 1 1 3 24 2 1 2 1 2 2 2 1 1 1 2 3 1 1 2 2 3 1 3 3 2 2 1 25 2 1 1 2 2 2 1 2 2 1 1 1 3 2 1 2 3 3 1 3 1 2 2 26 2 1 1 2 2 2 1 2 2 1 1 2 1 3 2 3 1 1 2 1 2 3 3 27 2 1 1 2 2 2 1 2 2 1 1 3 2 1 3 1 2 2 3 2 3 1 1 28 2 2 2 1 1 1 1 2 2 1 2 1 3 2 2 2 1 1 3 2 3 1 3 29 2 2 2 1 1 1 1 2 2 1 2 2 1 3 3 3 2 2 1 3 1 2 1 30 2 2 2 1 1 1 1 2 2 1 2 3 2 1 1 1 3 3 2 1 2 3 2 31 2 2 1 2 1 2 1 1 1 2 2 1 3 3 3 2 3 2 2 1 2 1 1 32 2 2 1 2 1 2 1 1 1 2 2 2 1 1 1 3 1 3 3 2 3 2 2 33 2 2 1 2 1 2 1 1 1 2 2 3 2 2 1 2 1 1 3 1 1 3 3 34 2 2 1 1 2 1 2 1 2 2 1 1 3 1 2 3 2 3 1 2 2 3 1 35 2 2 1 1 2 1 2 1 2 2 1 2 1 2 3 1 3 1 2 3 3 1 2 36 2 2 1 1 2 1 2 1 2 2 1 3 2 3 1 2 1 2 3 1 1 2 3 Advantages and Disadvantages of Three-Level and Mixed-Level "L" Designs Advantages and disadvantages of three-level mixed-level designs The good features of these designs are: They are orthogonal arrays. Some analysts believe this simplifies the analysis and interpretation of results while other analysts believe it does not. ● They obtain a lot of information about the main effects in a relatively few number of runs. ● You can test whether non-linear terms are needed in the model, at least as far as the three-level factors are concerned. ● On the other hand, there are several undesirable features of these designs to consider: They provide limited information about interactions. ● They require more runs than a comparable 2 k-p design, and a two-level design will often suffice when the factors are continuous and monotonic (many three-level designs are used when two-level designs would have been adequate). ● 5.3.3.10. Three-level, mixed-level and fractional factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri33a.htm (5 of 5) [5/1/2006 10:30:45 AM] [...]... addition, the role of engineering judgment should not be underestimated http://www.itl.nist.gov/div898 /handbook/ pri/section4/pri41.htm (2 of 2) [5/1 /20 06 10:30:46 AM] 5.4 .2 How to "look" at DOE data Plots for viewing main effects and 2- factor interactions, explanation of normal or half-normal plots to detect possible important effects Subsequent Plots: Main Effects, Comparisons and 2- Way Interactions... prediction q Normal probability plot of residuals Model Predictions q Contour plots http://www.itl.nist.gov/div898 /handbook/ pri/section4/pri 42. htm (2 of 3) [5/1 /20 06 10:30:46 AM] 5.4 .2 How to "look" at DOE data http://www.itl.nist.gov/div898 /handbook/ pri/section4/pri 42. htm (3 of 3) [5/1 /20 06 10:30:46 AM] 5.4.3 How to model DOE data Response surface designs Response surface initial models include quadratic... validation Of course, as in all cases of model fitting, residual analysis and other tests of model fit are used to confirm or adjust models, as needed http://www.itl.nist.gov/div898 /handbook/ pri/section4/pri43.htm (2 of 2) [5/1 /20 06 10:30:46 AM] ... DOE plots (which assume standard models for effects and errors) s Main effects mean plots s Block plots s Normal or half-normal plots of the effects http://www.itl.nist.gov/div898 /handbook/ pri/section4/pri41.htm (1 of 2) [5/1 /20 06 10:30:46 AM] 5.4.1 What are the steps in a DOE analysis? s Interaction plots Sometimes the right graphs and plots of the data lead to obvious answers for your experimental objective... help? If a transformation is used, return to step 3 with a new model Use the results to answer the questions in your experimental objectives finding important factors, finding optimum settings, etc r 2 3 4 5 Flowchart is a guideline, not a hard-and -fast rule Note: The above flowchart and sequence of steps should not be regarded as a "hard-and-fast rule" for analyzing all DOE's Different analysts may . 3 2 2 1 25 2 1 1 2 2 2 1 2 2 1 1 1 3 2 1 2 3 3 1 3 1 2 2 26 2 1 1 2 2 2 1 2 2 1 1 2 1 3 2 3 1 1 2 1 2 3 3 27 2 1 1 2 2 2 1 2 2 1 1 3 2 1 3 1 2 2 3 2 3 1 1 28 2 2 2 1 1 1 1 2 2 1 2 1 3 2 2 2 1. 1 21 2 1 2 2 1 1 2 2 1 2 1 3 1 3 2 2 2 3 1 1 3 1 2 22 2 1 2 1 2 2 2 1 1 1 2 1 2 2 3 3 1 2 1 1 3 3 2 23 2 1 2 1 2 2 2 1 1 1 2 2 3 3 1 1 2 3 2 2 1 1 3 24 2 1 2 1 2 2 2 1 1 1 2 3 1 1 2 2 3 1 3 3 2. 2 2 2 2 2 2 1 1 1 1 2 2 2 2 3 3 3 3 5 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 1 1 1 1 6 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 1 1 1 1 2 2 2 2 7 1 1 2 2 2 1 1 1 2 2 2 1 1 2 3 1 2 3 3 1 2 2 3 8 1 1 2 2 2