Engineering Statistics Handbook Episode 6 Part 13 docx

5. Process Improvement 5.3. Choosing an experimental design 5.3.3. How do you select an experimental design? 5.3.3.2. Randomized block designs 5.3.3.2.3.Hyper-Graeco-Latin square designs These designs handle 4 nuisance factors Hyper-Graeco-Latin squares, as described earlier, are efficient designs to study the effect of one treatment factor in the presence of 4 nuisance factors. They are restricted, however, to the case in which all the factors have the same number of levels. Randomize as much as design allows Designs for 4- and 5-level factors are given on this page. These designs show what the treatment combinations should be for each run. When using any of these designs, be sure to randomize the treatment units and trial order, as much as the design allows. For example, one recommendation is that a hyper-Graeco-Latin square design be randomly selected from those available, then randomize the run order. Hyper-Graeco-Latin Square Designs for 4- and 5-Level Factors Designs for 4-level factors (there are no 3-level factor Hyper-Graeco Latin square designs) 4-Level Factors X1 X2 X3 X4 X5 row blocking factor column blocking factor blocking factor blocking factor treatment factor 1 1 1 1 1 1 2 2 2 2 1 3 3 3 3 1 4 4 4 4 2 1 4 2 3 2 2 3 1 4 2 3 2 4 1 2 4 1 3 2 3 1 2 3 4 5.3.3.2.3. Hyper-Graeco-Latin square designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3323.htm (1 of 3) [5/1/2006 10:30:25 AM] 3 2 1 4 3 3 3 4 1 2 3 4 3 2 1 4 1 3 4 2 4 2 4 3 1 4 3 1 2 4 4 4 2 1 3 with k = 5 factors (4 blocking factors and 1 primary factor) L 1 = 4 levels of factor X1 (block) L 2 = 4 levels of factor X2 (block) L 3 = 4 levels of factor X3 (primary) L 4 = 4 levels of factor X4 (primary) L 5 = 4 levels of factor X5 (primary) N = L1 * L2 = 16 runs This can alternatively be represented as (A, B, C, and D represent the treatment factor and 1, 2, 3, and 4 represent the blocking factors): A11 B22 C33 D44 C42 D31 A24 B13 D23 C14 B41 A32 B34 A43 D12 C21 Designs for 5-level factors 5-Level Factors X1 X2 X3 X4 X5 row blocking factor column blocking factor blocking factor blocking factor treatment factor 1 1 1 1 1 1 2 2 2 2 1 3 3 3 3 1 4 4 4 4 1 5 5 5 5 2 1 2 3 4 2 2 3 4 5 2 3 4 5 1 2 4 5 1 2 2 5 1 2 3 3 1 3 5 2 3 2 4 1 3 3 3 5 2 4 5.3.3.2.3. Hyper-Graeco-Latin square designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3323.htm (2 of 3) [5/1/2006 10:30:25 AM] 3 4 1 3 5 3 5 2 4 1 4 1 4 2 5 4 2 5 3 1 4 3 1 4 2 4 4 2 5 3 4 5 3 1 4 5 1 5 4 3 5 2 1 5 4 5 3 2 1 5 5 4 3 2 1 5 5 4 3 2 with k = 5 factors (4 blocking factors and 1 primary factor) L 1 = 5 levels of factor X1 (block) L 2 = 5 levels of factor X2 (block) L 3 = 5 levels of factor X3 (primary) L 4 = 5 levels of factor X4 (primary) L 5 = 5 levels of factor X5 (primary) N = L1 * L2 = 25 runs This can alternatively be represented as (A, B, C, D, and E represent the treatment factor and 1, 2, 3, 4, and 5 represent the blocking factors): A11 B22 C33 D44 E55 D23 E34 A45 B51 C12 B35 C41 D52 E31 A24 E42 A53 B14 C25 D31 C54 D15 E21 A32 B43 Further information More designs are given in Box, Hunter, and Hunter (1978). 5.3.3.2.3. Hyper-Graeco-Latin square designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3323.htm (3 of 3) [5/1/2006 10:30:25 AM] 5. Process Improvement 5.3. Choosing an experimental design 5.3.3. How do you select an experimental design? 5.3.3.3. Full factorial designs 5.3.3.3.1.Two-level full factorial designs Description Graphical representation of a two-level design with 3 factors Consider the two-level, full factorial design for three factors, namely the 2 3 design. This implies eight runs (not counting replications or center point runs). Graphically, we can represent the 2 3 design by the cube shown in Figure 3.1. The arrows show the direction of increase of the factors. The numbers `1' through `8' at the corners of the design box reference the `Standard Order' of runs (see Figure 3.1). FIGURE 3.1 A 2 3 two-level, full factorial design; factors X1, X2, X3 5.3.3.3.1. Two-level full factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3331.htm (1 of 3) [5/1/2006 10:30:26 AM] The design matrix In tabular form, this design is given by: TABLE 3.3 A 2 3 two-level, full factorial design table showing runs in `Standard Order' run X1 X2 X3 1 -1 -1 -1 2 1 -1 -1 3 -1 1 -1 4 1 1 -1 5 -1 -1 1 6 1 -1 1 7 -1 1 1 8 1 1 1 The left-most column of Table 3.3, numbers 1 through 8, specifies a (non-randomized) run order called the `Standard Order.' These numbers are also shown in Figure 3.1. For example, run 1 is made at the `low' setting of all three factors. Standard Order for a 2 k Level Factorial Design Rule for writing a 2 k full factorial in "standard order" We can readily generalize the 2 3 standard order matrix to a 2-level full factorial with k factors. The first (X1) column starts with -1 and alternates in sign for all 2 k runs. The second (X2) column starts with -1 repeated twice, then alternates with 2 in a row of the opposite sign until all 2 k places are filled. The third (X3) column starts with -1 repeated 4 times, then 4 repeats of +1's and so on. In general, the i-th column (X i ) starts with 2 i-1 repeats of -1 folowed by 2 i-1 repeats of +1. Example of a 2 3 Experiment 5.3.3.3.1. Two-level full factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3331.htm (2 of 3) [5/1/2006 10:30:26 AM] Analysis matrix for the 3-factor complete factorial An engineering experiment called for running three factors; namely, Pressure (factor X1), Table speed (factor X2) and Down force (factor X3), each at a `high' and `low' setting, on a production tool to determine which had the greatest effect on product uniformity. Two replications were run at each setting. A (full factorial) 2 3 design with 2 replications calls for 8*2=16 runs. TABLE 3.4 Model or Analysis Matrix for a 2 3 Experiment Model Matrix Response Variables I X1 X2 X1*X2 X3 X1*X3 X2*X3 X1*X2*X3 Rep 1 Rep 2 +1 -1 -1 +1 -1 +1 +1 -1 -3 -1 +1 +1 -1 -1 -1 -1 +1 +1 0 -1 +1 -1 +1 -1 -1 +1 -1 +1 -1 0 +1 +1 +1 +1 -1 -1 -1 -1 +2 +3 +1 -1 -1 +1 +1 -1 -1 +1 -1 0 +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 +6 +5 The block with the 1's and -1's is called the Model Matrix or the Analysis Matrix. The table formed by the columns X1, X2 and X3 is called the Design Table or Design Matrix. Orthogonality Properties of Analysis Matrices for 2-Factor Experiments Eliminate correlation between estimates of main effects and interactions When all factors have been coded so that the high value is "1" and the low value is "-1", the design matrix for any full (or suitably chosen fractional) factorial experiment has columns that are all pairwise orthogonal and all the columns (except the "I" column) sum to 0. The orthogonality property is important because it eliminates correlation between the estimates of the main effects and interactions. 5.3.3.3.1. Two-level full factorial designs http://www.itl.nist.gov/div898/handbook/pri/section3/pri3331.htm (3 of 3) [5/1/2006 10:30:26 AM] Graphical representation of the factor level settings We want to try various combinations of these settings so as to establish the best way to run the polisher. There are eight different ways of combining high and low settings of Speed, Feed, and Depth. These eight are shown at the corners of the following diagram. FIGURE 3.2 A 2 3 Two-level, Full Factorial Design; Factors X1, X2, X3. (The arrows show the direction of increase of the factors.) 2 3 implies 8 runs Note that if we have k factors, each run at two levels, there will be 2 k different combinations of the levels. In the present case, k = 3 and 2 3 = 8. Full Model Running the full complement of all possible factor combinations means that we can estimate all the main and interaction effects. There are three main effects, three two-factor interactions, and a three-factor interaction, all of which appear in the full model as follows: A full factorial design allows us to estimate all eight `beta' coefficients . Standard order 5.3.3.3.2. Full factorial example http://www.itl.nist.gov/div898/handbook/pri/section3/pri3332.htm (2 of 6) [5/1/2006 10:30:27 AM] Coded variables in standard order The numbering of the corners of the box in the last figure refers to a standard way of writing down the settings of an experiment called `standard order'. We see standard order displayed in the following tabular representation of the eight-cornered box. Note that the factor settings have been coded, replacing the low setting by -1 and the high setting by 1. Factor settings in tabular form TABLE 3.6 A 2 3 Two-level, Full Factorial Design Table Showing Runs in `Standard Order' X1 X2 X3 1 -1 -1 -1 2 +1 -1 -1 3 -1 +1 -1 4 +1 +1 -1 5 -1 -1 +1 6 +1 -1 +1 7 -1 +1 +1 8 +1 +1 +1 Replication Replication provides information on variability Running the entire design more than once makes for easier data analysis because, for each run (i.e., `corner of the design box') we obtain an average value of the response as well as some idea about the dispersion (variability, consistency) of the response at that setting. Homogeneity of variance One of the usual analysis assumptions is that the response dispersion is uniform across the experimental space. The technical term is `homogeneity of variance'. Replication allows us to check this assumption and possibly find the setting combinations that give inconsistent yields, allowing us to avoid that area of the factor space. 5.3.3.3.2. Full factorial example http://www.itl.nist.gov/div898/handbook/pri/section3/pri3332.htm (3 of 6) [5/1/2006 10:30:27 AM] Factor settings in standard order with replication We now have constructed a design table for a two-level full factorial in three factors, replicated twice. TABLE 3.7 The 2 3 Full Factorial Replicated Twice and Presented in Standard Order Speed, X1 Feed, X2 Depth, X3 1 16, -1 .001, -1 .01, -1 2 24, +1 .001, -1 .01, -1 3 16, -1 .005, +1 .01, -1 4 24, +1 .005, +1 .01, -1 5 16, -1 .001, -1 .02, +1 6 24, +1 .001, -1 .02, +1 7 16, -1 .005, +1 .02, +1 8 24, +1 .005, +1 .02, +1 9 16, -1 .001, -1 .01, -1 10 24, +1 .001, -1 .01, -1 11 16, -1 .005, +1 .01, -1 12 24, +1 .005, +1 .01, -1 13 16, -1 .001, -1 .02, +1 14 24, +1 .001, -1 .02, +1 15 16, -1 .005, +1 .02, +1 16 24, +1 .005, +1 .02, +1 Randomization No randomization and no center points If we now ran the design as is, in the order shown, we would have two deficiencies, namely: no randomization, and1. no center points.2. 5.3.3.3.2. Full factorial example http://www.itl.nist.gov/div898/handbook/pri/section3/pri3332.htm (4 of 6) [5/1/2006 10:30:27 AM] [...]... Order 1 0 0 0 2 5 -1 -1 +1 3 15 -1 +1 +1 4 9 -1 -1 -1 5 7 -1 +1 +1 6 3 -1 +1 -1 7 12 +1 +1 -1 8 6 +1 -1 +1 9 0 0 0 10 4 +1 +1 -1 11 2 +1 -1 -1 12 13 -1 -1 +1 13 8 +1 +1 +1 14 16 +1 +1 +1 15 1 -1 -1 -1 16 14 +1 -1 +1 17 11 -1 +1 -1 18 10 +1 -1 -1 19 0 0 0 http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3332.htm (6 of 6) [5/1/20 06 10:30:27 AM] 5.3.3.3.3 Blocking of full factorial designs Graphical... X3 Order Order 1 5 -1 -1 +1 2 15 -1 +1 +1 3 9 -1 -1 -1 4 7 -1 +1 +1 5 3 -1 +1 -1 6 12 +1 +1 -1 7 6 +1 -1 +1 8 4 +1 +1 -1 9 2 +1 -1 -1 10 13 -1 -1 +1 11 8 +1 +1 +1 12 16 +1 +1 +1 13 1 -1 -1 -1 14 14 +1 -1 +1 15 11 -1 +1 -1 16 10 +1 -1 -1 http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3332.htm (5 of 6) [5/1/20 06 10:30:27 AM] 5.3.3.3.2 Full factorial example Table showing design matrix with randomization... +1 II http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3333.htm (3 of 4) [5/1/20 06 10:30:27 AM] 5.3.3.3.3 Blocking of full factorial designs http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3333.htm (4 of 4) [5/1/20 06 10:30:27 AM] 5.3.3.4 Fractional factorial designs http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri334.htm (2 of 2) [5/1/20 06 10:30:35 AM] ... http://www.itl.nist.gov/div898 /handbook/ pri/section3/pri3333.htm (2 of 4) [5/1/20 06 10:30:27 AM] 5.3.3.3.3 Blocking of full factorial designs Table showing blocking scheme Formally, consider the 23 design table with the three-factor interaction column added Block by assigning the "Block effect" to a high-order interaction Rows that have a `-1' in the three-factor interaction column are assigned to `Block I' (rows 1, 4, 6, 7),... software will do blocking for you The general rule for blocking is: use one or a combination of high-order interaction columns to construct blocks This gives us a formal way of blocking complex designs Apart from simple cases in which you can design your own blocks, your statistical/DOE software will do the blocking if asked, but you do need to understand the principle behind it Block effects are confounded . +1 9 16, -1 .001, -1 .01, -1 10 24, +1 .001, -1 .01, -1 11 16, -1 .005, +1 .01, -1 12 24, +1 .005, +1 .01, -1 13 16, -1 .001, -1 .02, +1 14 24, +1 .001, -1 .02, +1 15 16, -1 .005, +1 .02, +1 16 24,. 7 -1 +1 +1 5 3 -1 +1 -1 6 12 +1 +1 -1 7 6 +1 -1 +1 8 4 +1 +1 -1 9 2 +1 -1 -1 10 13 -1 -1 +1 11 8 +1 +1 +1 12 16 +1 +1 +1 13 1 -1 -1 -1 14 14 +1 -1 +1 15 11 -1 +1 -1 16 10 +1 -1 -1 5.3.3.3.2 +1 4 9 -1 -1 -1 5 7 -1 +1 +1 6 3 -1 +1 -1 7 12 +1 +1 -1 8 6 +1 -1 +1 9 0 0 0 10 4 +1 +1 -1 11 2 +1 -1 -1 12 13 -1 -1 +1 13 8 +1 +1 +1 14 16 +1 +1 +1 15 1 -1 -1 -1 16 14 +1 -1 +1 17 11 -1 +1 -1 18