Problem 8-1 (as stated in RSM Simplified

Transcription

1 Problem 8-1 (as stated in RSM Simplified) University of Minnesota food scientists (Richert et al., 1974) used a central composite design to study the effects of five factors on whey protein concentrates, a popular diet supplement for body builders. Also, whey proteins, when properly processed, are beneficial in the development of foams characteristic of frozen desserts, whipped toppings, meringues, and mousses. The factors, with ranges noted in terms of alpha (star levels), were: A. Heating temperature: 65 to 85 C/30 min. B. ph level: 4 to 8. C. Redox potential: to volts. D. Sodium oxalate (Na Ox ), 0 to 0.05 molar. E. Sodium lauryl sulfate (Na LS), 0 to 0.2 percent of solids. There were nine responses, but we will look at only three: 1. Whipping time in minutes (to produce a given amount of foam) 2. Time at first drop in minutes (a measure of foam stability) 3. Undenatured protein in percent. Denature means to render unfit, so therefore the undenatured protein must be maximized. The whipping time should be minimized and the stability maximized. The experimenters chose a rotatable CCD based on a one-half fraction for the cube portion (2 5-1 ). The data are shown below in Table 8-2. Be careful when you set up this central composite design all points must fall within the specified ranges. If supported by your software (it is in the one provided with this book), enter the limits in terms of the alpha levels, which are the axial (star) points. This is a popular design (for example, it was also used by GE scientists to study plastic processing as presented in Problem 6-2). However, as we discussed in this chapter, rotatable CCDs do a poor job of exploring the x-space, that is, the region defined by the process limits. What would have been a good alternative for these food scientists studying production of whey protein concentrates? Prob 8-1 p. 1

2 Std A: Heat (deg C) B: ph Table 8-2: Data for whey protein study C: Redox (volt) D: Na OX (Molar) E: Na LS (percent) Whip Time min. Time 1 st Drop min. Protein (percent) Develop predictive models for the three responses of interest. With this many factors (five), you may discover that many terms come out insignificant in the ANOVA. Therefore, we suggest you apply model reduction. We did this in chapter 2 for the Longley data and discussed the pros and cons of it in the sidebar titled Are You a Tosser or a Keeper? in chapter 4. Recall our advice that you be vigilant for cases where a factor and all of its dependents (for example: A, AB, AC, A 2 ) are all insignificant. Is it possible to model any or all of the whey protein responses with only a subset of the tested factors--in other words, one or more factors reduced out of the model? If so, this will simplify the response surfaces and keep things simpler all around. Prob 8-1 p. 2

3 Also, do not be shy about applying response transformations. By now you ve seen this done numerous times throughout the book to produce better predictive models. Table 8-3 provides criteria for a multiple response optimization of the process for making whey protein concentrate. Table 8-3. Multiple response optimization criteria whey protein concentrate Response Goal Lower Limit Upper Limit Importance Whip time Minimize Time 1 st drop Maximize Protein Maximize Can you find a sweet spot for the meringue and mousse lovers? Solution to Problem 8-1 If you choose to set this up from scratch using Design-Expert software, here s how to do it. Start by running the program and selecting File, New Design or click the blanksheet icon ( ) on the toolbar. Then click on the tab marked Response Surface. The default of Central Composite design is correct. Enter for Numeric Factors, 5. Figure 8-1.1: Choosing central composite design (CCD) At the bottom of the screen: 1. Select Enter factor ranges in terms of alphas 2. Click the downlist arrow [ ] and change the Type to 1/2 Fraction. Figure 8-1.2: Specifying a fractional core-factorial for the CCD Now you are ready to enter fields for the Name, Units and Alpha levels for each factor. Enter these as shown in the screen shot on Figure Prob 8-1 p. 3

4 Figure 8-1.3: Factor specification Leave the center points at their default number of 6. Continue to the screen for Responses, change the number to 3 and enter the Name and Units as shown in Figure Figure 8-1.4: Entering the responses Press Continue to get the design layout in random run order. It will be too much work to enter all the response data, and this would be prone to mistakes, so open the file named 8-1 Prob - Whey.dx7 that we posted to the RSM Simplified website. To more easily compare what you see in this file with what s listed in Table 8-2, select View from the main menu and switch to Std Order. We ve already modeled all three responses, but go ahead and look over what s been done by clicking each node under Analysis and following along the progressive toolbar of icons. For example, press the Whip time node (notice that it s labeled Analyzed ). This brings up the Transform screen. Figure 8-1.5: Viewing analysis of first response Prob 8-1 p. 4

5 Notice that we took a base-10 log of all the response data for whip time, which, according to the on-screen footnote, varied by a ratio of 4 enough to make this transformation worthwhile. Now take a look at the Fit Summary. Figure 8-1.6: Fit summary part 1 Design-Expert cannot decide in this case whether the linear or the quadratic model works best both are underlined and suggested. However, notice in the second table that linear exhibits much more significant lack of fit (p-value of ) that the quadratic (p>0.2). Scroll down to the last table on this screen to see why the software does not recommend the quadratic model alone the predicted R 2 value goes negative. Figure 8-1.6: Fit summary part 2 (R 2 predicted highlighted) On the other hand, the adjusted R 2 value for quadratic of is very good. Discrepancies like this between R 2 values crop up on occasion, more often as the number of factors and order of model increase. It frequently can be remedied via model reduction. Prob 8-1 p. 5

6 You can see that we indeed performed a model reduction by pressing the Model button. Figure 8-1.6: Model (previously analyzed) Let s reproduce the procedure we followed to select the model terms listed (ones designated M ). First change the Process Order to Quadratic the model, all terms included, that exhibited such a poor R 2 predicted. Figure 8-1.7: Full quadratic model Next, change the Selection to Backward. Then change the Alpha Out to a p-value of Figure 8-1.8: Backward selection with alpha out reduced to 0.05 Press ANOVA to see the results. Just press Yes to the following warning. Prob 8-1 p. 6

7 Figure 8-1.9: Hierarchy warning Notice how the software systematically removed terms starting with the least significant. Figure : Terms removed via backward regression (p-values highlighted) Scroll ing down from here, you can see that a few terms, B and D, come back into the model to maintain its hierarchy. These parent terms, neither of which are very significant alone, are need to support their child the significant interaction BD. Figure : The final model after hierarchy correction Prob 8-1 p. 7

8 Below the ANOVA report you will see a dramatic increase in the predicted R-squared, which now falls only a bit below the adjusted R-squared as one would normally expect. Figure : R-squared statistics Press the Diagnostics button and check out the normal plot of residuals they line up nicely! Then on the floating Diagnostics Tool, click Box Cox. Figure : Box Cox plot The minimal model residuals fall within a zone (defined by the red lines) that does not include the value of 1 for Lambda, which represents no transformation of the response (y 1 = y). As Design-Expert software indicates on the legend for this Box Cox plot, the log transformation (lambda of 0), which we already selected, will be nearer to the minimal level for residuals. To see the final response surface, press Model Graphs and select View, 3D Surface. Then change the Display Options to Responses in Original Scale. Using the Factors Tool move the C slide bar left. (Not sure why this factor might be worth changing? See View, Perturbation and note that it causes slightly more change in response other than factor A, which is already chosen for an axis on the plot.) Prob 8-1 p. 8

9 Figure : Whip Time response surface in original scale with Factor C slid to low end Take a look at the analyses for the other two responses. Notice that the second one, time at first drop, is transformed via the inverse function. Figure : Transformation of Time at first drop Recall from the Chapter 5 Appendix titled Box-Cox Plot for Transformations the quote from Box that A research worker studying athletics may measure the time t in seconds that a subject takes to run 1,000 meters, but he could equally well have considered 1,000/t, which is the athlete s speed in meters per second This transformation from t to 1000/t for runners times is an inverse. Whenever the nature of response is time or rate, consider applying this transformation. In the case of time at first drop, we found the inverse transformation beneficial in terms of model F-value, R 2 predicted and the measure for adequate precision. Feel free to check out these values by clicking on Prob 8-1 p. 9

10 the ANOVA. You may notice that, based on the lack-of-fit test, this second response cannot be modeled as well as the first, but assume that it can be predicted well enough to prove useful for purposes of multiple response optimization your next task. Under the Optimization branch of the program, select Numerical. Figure : Numerical optimization Look over the five factors. They all should be held within their factorial ranges. For example, notice that the limits for Heating (factor A) are 70 to 80 versus the range of 65 to 85 you specified for the star points (alpha levels) in this central composite design. It s best that you stay within the factorial box when searching for an optimum based on the CCD. Continue your inspection of the numerical optimization criteria by clicking on the first response Log10(Whip time). Figure : Criteria for first response Prob 8-1 p. 10

11 Notice that although this response was analyzed after transformation to the log base-10 scale, it has been changed back to original scale via the Display Options menu. Refer back to Table 8-3 for the goal, limits and level of importance of this and the other response. They all should be already set up with the proper specifications but make no assumptions: Click the Time at first drop (analyzed with the inverse transformation) and Undenatured Protein and look over what s been entered as their criteria. Press the Solutions button and on the floating Solutions Tool select Ramps. Your results may vary due to the random elements in the search algorithm. Figure : A desirable solution for making whey protein (your results may vary) The best solution comes up first. Click over to the last solution (due to the variable nature of the software s search algorithm, you may have more or less than what s shown in Figure ). Does it differ much in factor settings and the predicted response? How much less is it for desirability? Click the Graphs button and go back to Solutions 1. Then select View, 3D Surface. (If you do not see a range of colors from blue to red, right click the color bar and change the High end to 0.21 as shown in Figure ) Figure : Color bar re-scaled Prob 8-1 p. 11

12 Figure : Desirability surface for best solution Perhaps plotting something other than A versus B may be more dramatic. Select View, Perturbation. Then click the letter A on the plot so you can see its effect better. Figure : Perturbation plot of desirability A highlighted Click on each of the other four factors. Notice that factors B and C generate a broader range of desirable outcomes. Re-select View, 3D Surface. Now on the Factors Tool right-click on C:Redox pot and make it the X1 axis. Prob 8-1 p. 12

13 This looks more impressive! Figure : Axes changed on desirability surface Let s finish this case study by clicking the Graphical node under Optimization. Look over the limits for each of the responses. Figure : Graphical optimization limits for first response Remember that we want to minimize the first response, Whip time, but its Limits must not exceed an Upper level of 5. Unlike the numerical optimization where we needed to define desirability, in this graphical search for the operating window or sweet spot no lower limit should be entered. Conversely, the second response, Time at first drop, must be maximized, so its Limits must not fall below a Lower level of 10. Prob 8-1 p. 13

14 Figure : Graphical optimization limits for second response Now that you know how graphical optimization should be specified, take a look at the last response and see if its limits are entered correctly. Then press the Graphs button to see the results. Figure : The operating window or sweet spot Notice that the first solution from numerical optimization comes up by default. On the Factors Tool right-click on C:Redox pot and make it the X1 axis. You then will see a bigger sweet spot. Prob 8-1 p. 14

15 Figure : Bigger sweet spot after changing axes Click on the boundary formed by the Undenatured Protein and drag it so it increases from 80 to near 84. This will shrink the sweet spot. Figure : Shrinking sweet spot after dragging the limit for Undenatured Protein You may find this ability to drag boundaries handy when negotiating new operating specifications. In this case we ve tightened up one of them to the point that our previous optimum is no longer valid. Right-click somewhere in this new operating window and select Add Flag to see how the plotted factor levels must change. Prob 8-1 p. 15

16 Figure : Adding a flag inside the shrunken sweet spot That s it we ve messed things up enough as it is. If you want to put the limits back to where they were originally, go back a step to Criteria and change the upper limit of undenatured protein to 80. PS. Recall the question: What would have been a good alternative for these food scientists studying production of whey protein concentrates? Let s consider what was done in this case. Under the Design branch of the Design-Expert software click the Summary node. Figure : Design summary The summary reminds us that this is a central composite design (CCD), but observe that the stated lows and highs relate to the factorial core that s coded -1 and +1. These ranges fall well within the extremes originally stated by the experimenters: A. Heating temperature: 65 to 85 C/30 min. B. ph level: 4 to 8. C. Redox potential: to volts. D. Sodium oxalate (Na Ox ), 0 to 0.05 molar. E. Sodium lauryl sulfate (Na LS), 0 to 0.2 percent of solids. Prob 8-1 p. 16

17 A better choice of design would ve been the face-centered central composite (FCD). To see what this looks like, click the New Design icon on the left of the toolbar (or select File, New Design). Figure : Starting up a new design Then click No to Save Changes and Yes to use previous design information. Figure : Not saving changes, but using previous design information This time around, click the Options button and press the Face Centered button and OK. Figure : Re-designing as FCD Now, if you look closely at the specified factor levels, you will see that they are the same for 1/+1 and alpha/+alpha. Given that it explores a wider range of factor space, this FCD choice would ve made a good alternative to the chosen CCD. Cancel out of this screen to complete this exercise. Prob 8-1 p. 17