Engineering Statistics Handbook Episode 8 Part 2 pps

5.6.1.11. Conclusions and Next Step http://www.itl.nist.gov/div898/handbook/pri/section6/pri61b.htm (2 of 2) [5/1/2006 10:31:50 AM] 3. Plots for interaction effects 1. Generate a dex interaction effects matrix plot. 1. The dex interaction effects matrix plot does not show any major interaction effects. 4. Block plots for main and interaction effects 1. Generate block plots. 1. The block plots show that the factor 1 and factor 2 effects are consistent over all combinations of the other factors. 5. Estimate main and interaction effects 1. Perform a Yates fit to estimate the main effects and interaction effects. 1. The Yates analysis shows that the factor 1 and factor 2 main effects are significant, and the interaction for factors 2 and 3 is at the boundary of statistical significance. 6. Model selection 1. Generate half-normal probability plots of the effects. 2. Generate a Youden plot of the effects. 1. The probability plot indicates that the model should include main effects for factors 1 and 2. 2. The Youden plot indicates that the model should include main effects for factors 1 and 2. 7. Model validation 1. Compute residuals and predicted values from the partial model suggested by the Yates analysis. 2. Generate residual plots to validate the model. 1. Check the link for the values of the residual and predicted values. 2. The residual plots do not indicate any major problems with the model using main effects for factors 1 and 2. 5.6.1.12. Work This Example Yourself http://www.itl.nist.gov/div898/handbook/pri/section6/pri61c.htm (2 of 3) [5/1/2006 10:31:51 AM] 8. Dex contour plot 1. Generate a dex contour plot using factors 1 and 2. 1. The dex contour plot shows X1 = -1 and X2 = +1 to be the best settings. 5.6.1.12. Work This Example Yourself http://www.itl.nist.gov/div898/handbook/pri/section6/pri61c.htm (3 of 3) [5/1/2006 10:31:51 AM] 5. Process Improvement 5.6. Case Studies 5.6.2. Sonoluminescent Light Intensity Case Study 5.6.2.1.Background and Data Background and Motivation Sonoluminescence is the process of turning sound energy into light. An ultrasonic horn is used to resonate a bubble of air in a medium, usually water. The bubble is ultrasonically compressed and then collapses to light-emitting plasma. In the general physics community, sonoluminescence studies are being carried out to characterize it, to understand it, and to uncover its practical uses. An unanswered question in the community is whether sonoluminescence may be used for cold fusion. NIST's motive for sonoluminescent investigations is to assess its suitability for the dissolution of physical samples, which is needed in the production of homogeneous Standard Reference Materials (SRMs). It is believed that maximal dissolution coincides with maximal energy and maximal light intensity. The ultimate motivation for striving for maximal dissolution is that this allows improved determination of alpha-and beta-emitting radionuclides in such samples. The objectives of the NIST experiment were to determine the important factors that affect sonoluminescent light intensity and to ascertain optimal settings of such factors that will predictably achieve high intensities. An original list of 49 factors was reduced, based on physics reasons, to the following seven factors: molarity (amount of solute), solute type, pH, gas type in the water, water depth, horn depth, and flask clamping. Time restrictions caused the experiment to be about one month, which in turn translated into an upper limit of roughly 20 runs. A 7-factor, 2-level fractional factorial design (Resolution IV) was constructed and run. The factor level settings are given below. Eva Wilcox and Ken Inn of the NIST Physics Laboratory conducted this experiment during 1999. Jim Filliben of the NIST Statistical Engineering Division performed the analysis of the experimental data. 5.6.2.1. Background and Data http://www.itl.nist.gov/div898/handbook/pri/section6/pri621.htm (1 of 3) [5/1/2006 10:31:51 AM] Response Variable, Factor Variables, and Factor- Level Settings This experiment utilizes the following response and factor variables. Response Variable (Y) = The sonoluminescent light intensity.1. Factor 1 (X1) = Molarity (amount of Solute). The coding is -1 for 0.10 mol and +1 for 0.33 mol. 2. Factor 2 (X2) = Solute type. The coding is -1 for sugar and +1 for glycerol. 3. Factor 3 (X3) = pH. The coding is -1 for 3 and +1 for 11.4. Factor 4 (X4) = Gas type in water. The coding is -1 for helium and +1 for air. 5. Factor 5 (X5) = Water depth. The coding is -1 for half and +1 for full. 6. Factor 6 (X6) = Horn depth. The coding is -1 for 5 mm and +1 for 10 mm. 7. Factor 7 (X7) = Flask clamping. The coding is -1 for unclamped and +1 for clamped. 8. This data set has 16 observations. It is a 2 7-3 design with no center points. Goal of the Experiment This case study demonstrates the analysis of a 2 7-3 fractional factorial experimental design. The goals of this case study are: Determine the important factors that affect the sonoluminescent light intensity. Specifically, we are trying to maximize this intensity. 1. Determine the best settings of the seven factors so as to maximize the sonoluminescent light intensity. 2. Data Used in the Analysis The following are the data used for this analysis. This data set is given in Yates order. Y X1 X2 X3 X4 X5 X6 X7 Light Solute Gas Water Horn Flask Intensity Molarity type pH Type Depth Depth Clamping 80.6 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 -1.0 66.1 1.0 -1.0 -1.0 -1.0 -1.0 1.0 1.0 59.1 -1.0 1.0 -1.0 -1.0 1.0 -1.0 1.0 68.9 1.0 1.0 -1.0 -1.0 1.0 1.0 -1.0 5.6.2.1. Background and Data http://www.itl.nist.gov/div898/handbook/pri/section6/pri621.htm (2 of 3) [5/1/2006 10:31:51 AM] 75.1 -1.0 -1.0 1.0 -1.0 1.0 1.0 1.0 373.8 1.0 -1.0 1.0 -1.0 1.0 -1.0 -1.0 66.8 -1.0 1.0 1.0 -1.0 -1.0 1.0 -1.0 79.6 1.0 1.0 1.0 -1.0 -1.0 -1.0 1.0 114.3 -1.0 -1.0 -1.0 1.0 1.0 1.0 -1.0 84.1 1.0 -1.0 -1.0 1.0 1.0 -1.0 1.0 68.4 -1.0 1.0 -1.0 1.0 -1.0 1.0 1.0 88.1 1.0 1.0 -1.0 1.0 -1.0 -1.0 -1.0 78.1 -1.0 -1.0 1.0 1.0 -1.0 -1.0 1.0 327.2 1.0 -1.0 1.0 1.0 -1.0 1.0 -1.0 77.6 -1.0 1.0 1.0 1.0 1.0 -1.0 -1.0 61.9 1.0 1.0 1.0 1.0 1.0 1.0 1.0 Reading Data into Dataplot These data can be read into Dataplot with the following commands SKIP 25 READ INN.DAT Y X1 TO X7 5.6.2.1. Background and Data http://www.itl.nist.gov/div898/handbook/pri/section6/pri621.htm (3 of 3) [5/1/2006 10:31:51 AM] Plot the Data: Dex Scatter Plot The next step in the analysis is to generate a dex scatter plot. Conclusions from the DEX Scatter Plot We can make the following conclusions based on the dex scatter plot. Important Factors: Again, two points dominate the plot. For X1, X2, X3, and X7, these two points emanate from the same setting, (+, -, +, -), while for X4, X5, and X6 they emanate from different settings. We conclude that X1, X2, X3, and X7 are potentially important, while X4, X5, and X6 are probably not important. 1. Best Settings: Our first pass at best settings yields (X1 = +, X2 = -, X3 = +, X4 = either, X5 = either, X6 = either, X7 = -). 2. Check for Main Effects: Dex Mean Plot The dex mean plot is generated to more clearly show the main effects: 5.6.2.2. Initial Plots/Main Effects http://www.itl.nist.gov/div898/handbook/pri/section6/pri622.htm (2 of 4) [5/1/2006 10:31:52 AM] Conclusions from the DEX Mean Plot We can make the following conclusions from the dex mean plot. Important Factors: X2 (effect = large: about -80) X7 (effect = large: about -80) X1 (effect = large: about 70) X3 (effect = large: about 65) X6 (effect = small: about -10) X5 (effect = small: between 5 and 10) X4 (effect = small: less than 5) 1. Best Settings: Here we step through each factor, one by one, and choose the setting that yields the highest average for the sonoluminescent light intensity: (X1,X2,X3,X4,X5,X6,X7) = (+,-,+,+,+,-,-) 2. 5.6.2.2. Initial Plots/Main Effects http://www.itl.nist.gov/div898/handbook/pri/section6/pri622.htm (3 of 4) [5/1/2006 10:31:52 AM] Comparison of Plots All of the above three plots are used primarily to determine the most important factors. Because it plots a summary statistic rather than the raw data, the dex mean plot shows the ordering of the main effects most clearly. However, it is still recommended to generate either the ordered data plot or the dex scatter plot (or both). Since these plot the raw data, they can sometimes reveal features of the data that might be masked by the dex mean plot. In this case, the ordered data plot and the dex scatter plot clearly show two dominant points. This feature would not be obvious if we had generated only the dex mean plot. Interpretation-wise, the most important factor X2 (solute) will, on the average, change the light intensity by about 80 units regardless of the settings of the other factors. The other factors are interpreted similarly. In terms of the best settings, note that the ordered data plot, based on the maximum response value, yielded +, -, +, -, +, -, - Note that a consensus best value, with "." indicating a setting for which the three plots disagree, would be +, -, +, ., +, -, - Note that the factor for which the settings disagree, X4, invariably defines itself as an "unimportant" factor. 5.6.2.2. Initial Plots/Main Effects http://www.itl.nist.gov/div898/handbook/pri/section6/pri622.htm (4 of 4) [5/1/2006 10:31:52 AM] [...]... 2- factor interactions (cross-products): X1: +, X2: - with X1*X2: X1: +, X3: + with X1*X3: + X1: +, X7: - with X1*X7: X2: -, X3: + with X2*X3: X2: -, X7: - with X2*X7: + X3: +, X7: - with X3*X7: - http://www.itl.nist.gov/div8 98 /handbook/ pri/section6/pri 623 .htm (2 of 2) [5/1 /20 06 10:31: 52 AM] 5.6 .2. 4 Main and Interaction Effects: Block Plots Conclusions from the Block Plots We can make the following conclusions... Factors: Because of the expanded vertical axis, due to the two "outliers", the block plot is not particularly revealing Block plots based on alternatively scaled data (e.g., LOG(Y)) would be more informative http://www.itl.nist.gov/div8 98 /handbook/ pri/section6/pri 624 .htm (2 of 2) [5/1 /20 06 10:31:53 AM] 5.6 .2. 5 Important Factors: Youden Plot Conclusions from the Youden plot We can make the following conclusions... three (X2*X3, X4*X5, X1*X7, and X1*X2, X5*X6, X3*X7) 2 Best Settings: Reading down the diagonal plots, we select, as before, the best settings "on the average": (X1,X2,X3,X4,X5,X6,X7) = (+,-,+,+,+,-,-) For the more important factors (X1, X2, X3, X7), we note that the best settings (+, -, +, -) are consistent with the best settings for the 2- factor interactions (cross-products): X1: +, X2: - with X1*X2:... structure is given (e.g., 13: 13 +27 +46), which suggests that the information on X1*X3 (on the plot) must be tempered with the fact that X1*X3 is confounded with X2*X7 and X4*X6 http://www.itl.nist.gov/div8 98 /handbook/ pri/section6/pri 625 .htm (2 of 2) [5/1 /20 06 10:31:53 AM] ... important factors are: X2, X7, X1, and X3 These four factors have |effect| > 60 The remaining three factors have |effect| < 10 r The off-diagonal plots are the 2- factor interaction effects Of the 21 2- factor interactions, 9 are nominally important, but they fall into three groups of three: s X1*X3, X4*X6, X2*X7 (effect = 70) s X2*X3, X4*X5, X1*X7 (effect approximately 63.5) s X1*X2, X5*X6, X3*X7 (effect... X1 and X3 2 In the lower right corner are the main effects X2 and X7 and the interaction terms X2*X3 and X1*X2 3 The remaining terms are clustered in the center, which indicates that such effects have averages that are similar (and hence the effects are near zero), and so such effects are relatively unimportant 4 On the far right of the plot, the confounding structure is given (e.g., 13: 13 +27 +46), which... remaining 2- factor interactions are small having an |effect| < 20 A virtue of the interaction effects matrix plot is that the confounding structure of this Resolution IV design can be read off the plot In this case, the fact that X1*X3, X4*X6, and X2*X7 all have effect estimates identical to 70 is not a mathematical coincidence It is a reflection of the fact that for this design, the three 2- factor...5.6 .2. 3 Interaction Effects Conclusions from the DEX Interaction Effects Plot We make the following conclusions from the dex interaction effects plot 1 Important Factors: Looking for the plots that have the . X1*X7: - X2: -, X3: + with X2*X3: - X2: -, X7: - with X2*X7: + X3: +, X7: - with X3*X7: - 2. 5.6 .2. 3. Interaction Effects http://www.itl.nist.gov/div8 98 /handbook/ pri/section6/pri 623 .htm (2 of 2) [5/1 /20 06. intensity: (X1,X2,X3,X4,X5,X6,X7) = (+,-,+,+,+,-,-) 2. 5.6 .2. 2. Initial Plots/Main Effects http://www.itl.nist.gov/div8 98 /handbook/ pri/section6/pri 622 .htm (3 of 4) [5/1 /20 06 10:31: 52 AM] Comparison of. an "unimportant" factor. 5.6 .2. 2. Initial Plots/Main Effects http://www.itl.nist.gov/div8 98 /handbook/ pri/section6/pri 622 .htm (4 of 4) [5/1 /20 06 10:31: 52 AM] Conclusions from the DEX Interaction Effects