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DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07 Multifactor RSM Tutorial(Part 1 – The Basics)Response Surface Design and AnalysisThis tutorial shows the use of Design-Expert® software for response surfacemethodology (RSM). This class of designs is aimed at process optimization. A casestudy provides a real-life feel to the exercise.Due to the specific nature of the case study, a number of features that could be helpful toyou for RSM will not be exercised in this tutorial. Many of these features are used in theGeneral One Factor, RSM One Factor or Two-Level Factorial tutorials. If you have notcompleted all of these tutorials, consider doing so before starting in on this one.We will presume that you can handle the statistical aspects of RSM. For a good primeron the subject, see RSM Simplified (Anderson and Whitcomb, Productivity, Inc., NewYork). You will find overviews on RSM and how it’s done via Design-Expert in the on-line Help system. To gain a working knowledge of RSM, we recommend you attend ourResponse Surface Methods for Process Optimization workshop. Call Stat-Ease or visitour website, , for a schedule.The case study in this tutorial involves production of a chemical. The two mostimportant responses, designated by the letter “y”, are:•y1 - Conversion (% of reactants converted to product)•y2 - Activity.The experimenter chose three process factors to study. Their names and levels can beseen in the following table.Factor Units Low Level (-1) High Level (+1)A – Time minutes 40 50B – Temperature degreesC 80 90C – Catalyst percent 2 3Factors for response surface studyYou will study the chemical process with a standard RSM design called a centralcomposite design (CCD). It’s well suited for fitting a quadratic surface, which usuallyworks well for process optimization. The three-factor layout for the CCD is picturedbelow. It is composed of a core factorial that forms a cube with sides that are two codedunits in length (from -1 to +1 as noted in the table above). The stars represent axialpoints. How far out from the cube these should go is a matter for much discussionbetween statisticians? They designate this distance “alpha” – measured in terms ofcoded factor levels. As you will see Design-Expert offers a variety of options for alpha.Central Composite Design for three factorsAssume that the experiments will be conducted over a two-day period, in two blocks:1.Twelve runs: composed of eight factorial points, plus four center points.2.Eight runs: composed of six axial (star) points, plus two more center points.Design the ExperimentStart the program by finding and double clicking the Design-Expert software icon. Takethe quickest route to initiating a new design by clicking the blank-sheet icon on theleft of the toolbar. The other route is via File, New Design (or associated Alt keys).Main menu and tool barClick on the Response Surface folder tab to show the designs available for RSM.Response surface design tabThe default selection is the Central Composite design, which will be used for this casestudy. Click on the down arrow in the Numeric Factors entry box and Select 3.Ignore the option of including categoric factors in your designs (leave at default of 0).DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07To see alternative RSM designs for three factors, click on the choices for Box-Behnken (17 runs) and Miscellaneous designs, where you find the 3-Level Facto option (32 runs, includ rial ing 5 center points). Now go back and re-select the Central Composite design.oints set at1.68719 coded units from the center – a conventional choice for the CCD.Before entering the factors and ranges, click the Options at the bottom of the CCD screen. Notice that it defaults to a Rotatabledesign with the axial (star) pDefault CCD option for alpha set so design will be rotatablet of may be the “Face Centered” (alpha equal one). Press OK to accept the rotatable value. icking to each cell and entering the details given in the introduction to this case study.Many of the options are statistical in nature, but one that produces less extreme factor ranges is the “Practical” value for alpha. This is computed by taking the fourth roo the number of factors (in this case 3¼ or 1.31607). See RSM Simplified Chapter 8 “Everything You Should Know About CCDs (but dare not ask!)” for details on this practical versus other levels suggested for alpha in CCDs – the most popular of which Using the information provided in the table on page 1 of this tutorial (or on the screen capture below), type in the details for factor Name (A, B, C), Units and levels for low (-1) and high (+1), by tabbing or clCompleted factor formYou’ve now specified the cubical portion of the CCD. As you did this, Design-Exper calculated the coded distance “alpha” for placement on the star points in the central composite design. Alternatively, by clicking an option further down this scr t een, you could have entered values for alpha levels and let the software figure out the rest. This nd unless you must cut the number of runs to the bare minimum). You will need two blocks for this design, one for each day, so click on the Blocks field and select 2.would be helpful if you wanted to avoid going out of operating constraints.Now go back to the bottom of the central composite design form. Leave the Type at its default value of Full(the other option is a “small” CCD, which we do not recommeSelecting the number of blocksNotice that the software displays how this CCD will be laid out in the two blocks. Click on the Continue button to reach the second page of the “wizard” for building aresponse surface design. You now have the option of identifying Block Names . Enter Day 1 and Day 2as shown below.Block namesPress Continue to enter Responses . Select 2 from the pull down list. Then enter the response Name and Unitsfor each response as shown below.Completed response formAt any time in the design-building phase, you can return to the previous page by pressing the Back button. Then you can revise your selections. Press the Continue button to get the design layout (your run order may differ due to randomization ).DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Design-Expert offers many ways to modify the design and how it’s laid out on-screen. Preceding tutorials, especially in Part 2 for the General One Factor, delved into this in detail, so go back and look this over if you haven’t already. Click the Tips button for arefresher.Design layout (only partially shown, your run order may differ due to randomization)Save the Data to Now that you’ve invested some time into your design, it would be prudent to save your work. Click on the File menu item and select Save As .a FileSave As selectionYou can then specify the File name (we suggest tut-RSM ) to Save as type *.dx7” in the Data folder for Design-Expert (or wherever you want to Save in).File Save As dialog boxEnter the Response Data – Create Simple Scatter PlotsAssume that the experiment is now completed. Obviously at this stage the responsesmust be entered into Design-Expert. We see no benefit to making you type all thenumbers, particularly with the potential confusion due to differences in randomized runorders. Use the File, Open Design menu and select RSM.dx7 from theDesign-Expert program Data directory. Click on Open to load the data.Let’s examine the data, which came in with the file you opened (no need to type it in!).Move your cursor to the top of the Std column and perform a right-click to bring up amenu from which you should select Sort by Standard Order (this could also be donevia the View menu).Sorting by Standard (Std) OrderNext go to the Select button and do a right mouse click. Then select Point Type.DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Displaying the Point TypeNotice the new column identifying points as “Fact” for factorial and others “Center” for center point, etc. Notice how the factorial points go with the Day 1 block. Then in Day 2 the axial points are run. Center points are divided between the two blocks.Unless you change the default setting for the Select option, do not expect the Type column to appear the next time you run Design-Expert. It is only on temporarily at this stage for your information.Before doing the analysis, it might be interesting to take a look at some simple plots. Click on the Graph Columns node which branches from the design ‘root’ at the upper left of your screen. You should now see a scatter plot with factor A:Time on the X-axis and the response of Conversion on the Y-axis. It will be much more productive to see the impact of the control factors on response surface graphics to be produced later. For now it would be most useful to produce a plot showing the impact of blocks, because this will be literally blocked out in the analysis. On the floating Graph Columns tool click on the X Axis downlist symbol and select Block.Graph Columns feature for design layoutThe graph shows a slight correlation (0.152) of conversion by block. Change the Y Axis to Activity to see how it’s affected by the day-to-day blocking (not much!).Changing response (resulting graph not shown)Finally, to see how the responses correlate with each other, change the X Axis to Conversion.Plotting one response versus the other (resulting graph not shown)Feel free to make other scatter plots. Notice that you can also color the by selected factors, including run (the default). However, do not get carried away with this, because it will be much more productive to do statistical analysis first before drawing any conclusions.DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Analyze the ResultsYou will now start analyzing the responses numerically. Under the Analysis branchclick the node labeled Conversion. A new set of buttons appears at the top of yourscreen. They are arranged from left to right in the order needed to complete the analysis.What could be simpler?Begin analysis of ConversionDesign-Expert provides a full array of response transformations via the Transformoption. Click Tips for details. For now, accept the default transformation selection ofNone.Click on the Fit Summary button next. At this point Design-Expert fits linear, two-factor interaction (2FI), quadratic and cubic polynomials to the response. To movearound the display, use the side and/or bottom scroll bars. You will first see theidentification of the response, immediately followed in this case by a warning: “TheCubic Model is Aliased.” Do not be alarmed. By design, the central composite matrixprovides too few unique design points to determine all of the terms in the cubic model.It’s set up only for the quadratic model (or some subset). Next you will see severalextremely useful summary tables for model selection. Each of these tables will bediscussed briefly below.The table of “Sequential Model Sum of Squares” (technically “Type I”) shows howterms of increasing complexity contribute to the total model. The model hierarchy isdescribed below:•“Linear vs Block”: the significance of adding the linear terms to the meanand blocks,•“2FI vs Linear”: the significance of adding the two factor interaction termsto the mean, block and linear terms already in the model,•“Quadratic vs 2FI”: the significance of adding the quadratic (squared) terms to the mean, block, linear and two factor interaction terms already in themodel,•“Cubic vs Quadratic”: the significance of the cubic terms beyond all other terms.Sequential Model Sum of SquaresFor each source of terms (linear, etc.), examine the probability (“Prob > F”) to see if it falls below 0.05 (or whatever statistical significance level you choose). So far, the quadratic model looks best – these terms are significant, but adding the cubic order terms will not significantly improve the fit. (Even if they were significant, the cubic terms would be aliased, so they wouldn’t be useful for modeling purposes.) Use the handy Bookmarks or scroll down to the next table for lack of fit tests on the various model orders.Summary Table: Lack of Fit TestsThe “Lack of Fit Tests” table compares the residual error to the “Pure Error” from replicated design points. If there is significant lack of fit, as shown by a low probabilityDX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07 value (“Prob>F”), then be careful about using the model as a response predictor. In this case, the linear model definitely can be ruled out, because its Prob > F falls below 0.05. The quadratic model, identified earlier as the likely model, does not show significant lack of fit. Remember that the cubic model is aliased, so it should not be chosen.Scroll down to the last table in the Fit Summary report, which provides “Model Summary Statistics” for the ‘bottom line’ on comparing the options.Summary Table: Model Summary StatisticsThe quadratic model comes out best: It exhibits low standard deviation (“Std. Dev.”), high “R-Squared” values and a low “PRESS.”The program automatically underlines at least one “Suggested” model. Always confirm this suggestion by looking at these tables. Check Tips for more information about the procedure for choosing model(s).Design-Expert now allows you to select a model for an in-depth statistical study. Click on the Model button at the top of the screen next to see the terms in the model.Model resultsThe program defaults to the “Suggested” model from the Fit Summary screen. If you want, you can choose an alternate model from the Process Order pull-down list. (Be sure to do this in the rare cases when Design-Expert suggests more than one model.)The options for process orderAt this stage you could press the Add Term button and insert higher degree terms with integer powers, such as quartic (4th degree). However, for this case study, we’ll leave the selection at Quadratic.You could now manually reduce the model by clicking off insignificant effects. For example, you will see in a moment that several terms in this case are marginally significant at best. Design-Expert also provides several automatic reduction algorithms as alternatives to the “Manual” method: “Backward,” “Forward” and “Stepwise.” Click the down arrow on the Selection list box to use these.Click on the ANOVA button to produce the analysis of variance for the selected model. The ANOVA table is available in two views. By default it will add text providing brief explanations and guidelines to the reported statistics. To turn this off, choose View, Annotated ANOVA. Notice that this toggles off the check mark (√).Statistics for selected model: ANOVA tableThe ANOVA in this case confirms the adequacy of the quadratic model (the Model Prob>F is less than 0.05.) You can also see probability values for each individual termDX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07 in the model. You may want to consider removing terms with probability values greater than 0.10. Use process knowledge to guide your decisions.Next, see that Design-Expert presents various statistics to augment the ANOVA – most notably various R-Squared values. These look very good.Post-ANOVA statisticsScroll down (or use the Bookmark) to bring the following details on model coefficients to your screen. The mean effect shift for each block is listed here too.Coefficients for the quadratic modelAgain scroll down to bring the next section to your screen: the predictive models in terms of coded versus actual factors (shown side-by-side below). Block terms are left out. These terms can be used to re-create the results of this experiment, but they cannot be used for modeling future responses.Final equation: coded versus actualYou cannot edit any of the ANOVA outputs. However, you can copy and paste the datato your favorite Windows word processor or spreadsheet.Diagnose the Statistical Properties of the ModelThe diagnostic details provided by Design-Expert can best be digested by viewing plotsthe come with a click on the Diagnostics button. The most important diagnostic, thenormal probability plot of the residuals, comes up by default.Normal probability plot of the residualsDX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07 The data points should be approximately linear. A non-linear pattern (look for an S-shaped curve) indicates non-normality in the error term, which may be corrected by a transformation. There are no signs of any problems in our data.At the left of the screen you see the Diagnostics Tool palette. First of all, notice that residuals will be studentized unless you uncheck the first box on the floating tool palette (not advised). This counteracts varying leverages due to location of design points. For example, the center points carry little weight in the fit and thus exhibit low leverage. Each button on the palette represents a different diagnostics graph. Check out the other graphs if you like. Explanations for most of these graphs were covered in prior Tutorials. In this case, none of the graphs indicate any cause for alarm.Now click the option for Influence. Here’s where you find the find plots for externally studentized residuals (better known as “outlier t”) and other plots that may be helpful for finding problem points in the design. Also, from here you can click Report to bring up a detailed case-by-case diagnostic statistics, many of which have already been shown graphically. (In previous versions of Design-Expert, this report appeared under ANOVA.)Diagnostics reportThe note below the table (“Predicted values include block corrections.”) alerts you that any shift from block 1 to block 2 will be included for purposes of residual diagnostics. (Recall that block corrections did not appear in the predictive equations shown in the ANOVA report.) Also note that one value of DFFITS is flagged. As we discussed in the General One-Factor Tutorial (Part 2 – Advanced Features), this statistic stands for difference in fits. It measures the change in each predicted value that occurs when that response is deleted. To see what program Help says about DFFFITs, right-click the number.Accessing context-sensitive HelpGiven that only this one diagnostic is flagged, it probably is not a cause for alarm. Press X to close out the screen tip provided by the program’s Help system.Before moving ahead from the diagnostics, click the option for DFFITS, which, as noted in the screen tips “breaks down the changes in the model to each coefficient, hence the acronym standing for difference in betas.”DFBETA for term CIn this case you can evaluate all ten model terms, including the intercept, for this quadratic predictive model. (Tip: Use your mouse wheel to quickly move up and down the list!) In a similar experiment to this one, where the chemist changed catalyst, the DFBETA plot for that factor exhibited an outlier for the one run where its level went below a minimal level needed to initiate the reaction. Thus, this diagnostic proved to be very helpful in seeing where things went wrong in the experiment.DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Examine Model GraphsThe diagnosis of residuals reveals no statistical problems, so you will now generate theresponse surface plots. Click on the Model Graphs button. The 2D contour plot offactors A versus B comes up by default in graduated color shading.Response surface contour plotNote that Design-Expert will display any actual point included in the design spaceshown. In this case you see a plot of conversion as a function of time and temperature ata mid-level slice of catalyst. This slice includes six center points as indicated by the dotat the middle of the contour plot. By replicating center points, you get a very goodpower of prediction at the middle of your experimental region.The Factors Tool comes along with the default plot. Move this floating tool as neededby clicking on the top blue border and dragging it. The tool controls which factor(s) areplotted on the graph. The Gauges view is the default. Each factor listed will either havean axis label, indicating that it is currently shown on the graph, or a red slider bar, whichallows you to choose specific settings for the factors that are not currently plotted. Thered slider bars will default to the midpoint levels of the factors not currently assigned toaxes. You can change a factor level by dragging the red slider bars or by right clickingon a factor name to make it active (it becomes highlighted) and then typing the desiredlevel in the numeric space near the bottom of the tool palette. Click on the C:Catalysttoolbar to see its value. Don’t worry if the red slider bar shifts a bit – we will instructyou on how to re-set it in a moment.Factors tool with factor C highlighted and value displayedClick down on the red bar with your mouse and push it to the right.Slide bar for C pushed right to higher valueAs indicated by the color key on the left, the surface becomes ‘hot’ at higher response levels, yellow in the ’80’s and red above 90 for conversion.To enable a handy tool for reading coordinates off contour plots, go to View, Show Crosshairs Window.DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Showing crosshairs windowNow move your mouse over the contour plot and notice that Design-Expert generates the predicted response for specific values of the factors that correspond to that point. If you place the crosshair over an actual point, for example – the one at the upper left corner of the graph now on screen, you also get that observed value (in this case: 66).Prediction at coordinates of 40 and 90 where an actual run was performedNow press the Default button to put factor C back at its midpoint. Then switch to the Sheet View by clicking on the Sheet button.Factors tool – Sheet viewIn the columns labeled Axis and Value you can change the axes settings or type inspecific values for factors. Return to the default view by clicking on the Gaugesbutton.At the bottom of the Factors Tool is a pull-down list from which you can also select thefactors to plot. Only the terms that are in the model are included in this list. If youselect a single factor (such as A) the graph will change to a One Factor Plot. From thisview, if you now choose a two-factor interaction term (such as AC) the plot will becomethe interaction graph of that pair. The only way to get back to a contour graph is to usethe menu item View, Contour.Perturbation PlotWouldn’t it be handy to see all your factors on one response plot? You can do this withthe perturbation plot, which provides silhouette views of the response surface. The realbenefit from this plot is for selecting axes and constants in contour and 3D plots.Use the View, Perturbation menu item to select it.The Perturbation plot with factor A clicked to highlight itFor response surface designs the perturbation plot shows how the response changes aseach factor moves from the chosen reference point, with all other factors held constant atthe reference value. Design-Expert sets the reference point default at the middle of thedesign space (the coded zero level of each factor).Click on the curve for factor A to see it better. (The software will highlight it with adifferent color.) In this case, you can see that factor A (time) produces a relatively smalleffect as it changes from the reference point. Therefore, because you can only plotcontours for two factors at a time, it makes sense to choose B and C, and slice on A.DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Contour Plot: RevisitedLet’s look at the contour plot of factors B and C. Return to the contour plots via theView, Contour selection.Back to Contour viewIn the Factors Tool right click on the Catalyst bar. Then select X1 axis by leftclicking on it.Making factor C the x1-axisYou now see a catalyst versus temperature plot of conversion, with time held as aconstant at its midpoint. The colors are neat, but what if you must print the graphs inblack and white? That can be easily fixed by right-clicking over the graph and selectingGraph Preferences.Graph preferencesClick the Graphs 2 tab and change the Contour graph shading to Std Errorshading.Changing contour graph shadingPress OK to see the effect on your plot.Change factor A:time by dragging the slider left. See how it affects the shape of the contours. Also, notice how the plot darkens as you approach the extreme. You’re getting into areas of extrapolation. Be careful out there!Contour plot with standard error shading (shows up when A set at lowest level)Put the slider back at its center point by pressing the Default button.As you’ve no doubt observed by now, Design-Expert contour plots are highly interactive. For example, move your mouse over the center point. Notice that it turns into a crosshair (ª) and the prediction appears in the Crosshairs Window along with the X1-X2 coordinate. (If you do not see crosshairs, go to View, Show Crosshairs Window.) For more statistical detail, such as the very useful individual prediction interval (PI), can be found by pressing the Full button.DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Crosshairs Window set to Full display of statisticsWatch what happens to the numbers as you mouse the cross-hairs around the plot. We hope you are sitting down because it may make you dizzy! Perhaps it may be best if you press the Small button before passing out and falling on to the floor.Design-Expert draws five contour levels by default. They range from the minimum response to the maximum response. Click on a contour to highlight it. You can move the contours by dragging them to new values. (Place the mouse cursor on the contour and hold down the left button while moving the mouse.) Give this a try – it’s fun!Also you can add new contours via a right mouse click. Find a vacant region on the plot and check it out: Right-click and select Add contour. Then drag the contour around (it will become highlighted). You may get two contours from one click, such as those with the same response value shown below. (This pattern is indicative of a shallow valley, which will become apparent when we get to the 3D view later.)Adding contoursTo get more precise contour levels for your final report, you could right-click each one and enter the desired value.Setting a contour valueCheck this out if you like. But we recommend another approach: right click in the drawing or label area of the graph and choose Graph Preferences. Then choose Contours. Now select the Incremental option and fill in the Start at 66, Step at 3 and Levels at 8. Also, under Format choose 0.0 to display whole numbers (no decimals). Your screen should now match that shown below.Contours dialog box: Incremental optionPress OK to get a good-looking contour plot.DX71-04C-MultifactorRSM-P1.doc Rev. 1/24/07Contour plot with incremental contour valuesMove your mouse to a spot near the top of the graph, where the response hits the maximum. Click the right mouse button and Add flag.Adding a flagIt displays the value of the response at that point. Now do a right click of the mouse on the flag and select Toggle size.Flag enlarged via Toggle Size option。