Multifactor RSM Tutorial (D-optimal design) - Statease.info

DX7-04F-MultifactorRSM-Dopt.doc Rev. 4/12/06Multifactor RSM Tutorial(D-optimal design)Dealing with multifactor constraintsAfter a series of screening and in-depth factorial designs, the search for a processoptimum has been narrowed to two factors, ranging as follows:A. 110 to 180.B. 17 to 23.However, it’s been discovered that due to equipment limitations the combination offactors A and B must not fall below the constraint depicted below.23B1917110 A 180Constraint at lower levels of factorsNotice that, at the lowest level of A, factor B must be at least 19. At the lowest level ofB factor A must be at least 180. To complicate matters further, the experimenters suspectthat the response surface may be wavy, in other words, the standard quadratic modelused for RSM may fall short for providing accurate predictions. Therefore, a cubicmodel is recommended for the design.A problem like this can be easily handled by Design-Expert ® software via its d-optimaldesign tools. However, it requires that multifactor constraints like that pictured abovemust be entered as an equation that takes the form of:β L ≤ β 1 A+ β 1 B…≤ β Uwhere β L and β U are lower and upper limits, respectively.Anderson and Whitcomb provide guidelines for developing constraint equations inAppendix 7A of their book RSM Simplified. If, as in this case for both A and B, youwant factors to exceed their constraint points (CP), this equation describes the boundaryfor the experimental region:Design-Expert 7 User’s Guide D-Optimal RSM Tutorial • 1

A −LLAB −LLB1≤ +CP − LL CP − LLA A B Bwhere LL is the lower level. In this case, the lower levels are 110 for A and 17 for B.These factors’ CPs are 180 for A and 19 for B (the endpoints on the diagonal constraintline shown in the figure above). Plugging in these values produces this equation:1 A −110 B −17≤ +180 − 110 19 − 17However, the software requires formatting as a linear equation. This can be dealt withvia some simple arithmetic:A −110 B −171≤ +70 2( 70)( 2)( 1) ≤2( A − 110) + 70( B −17)140 ≤2A − 220 + 70B −11901550 ≤ 2A + 70BDesign the ExperimentStart Design-Expert and initiate the design process by clicking the blank-sheet icon onthe left of the toolbar or select File, New Design.Main menu and tool barClick on the Response Surface folder tab, select D-optimal for the design and enterthe lower and upper limits as shown belowEntering factor levels for d-optimal design2 • D-Optimal RSM Tutorial Design-Expert 7 User’s Guide

DX7-04F-MultifactorRSM-Dopt.doc Rev. 4/12/06Press the Edit Constraints button at the bottom of the screen and, as shown below,enter the multifactor equation derived at the outset of this tutorial: 1550 ≤ 2A + 70B .Notice from the screen shot that lower case letters are acceptable for the factors.Selecting the number of blocksPress OK to accept this constraint. Then press Continue to specify the d-optimaldesign via Point Exchange – the default choice (the optional Coordinate Exchange willbe explored later). Next, press the Edit Model button and upgrade the default model ofquadratic to Cubic.Cubic model specifiedThen press Create candidate points.Candidate points enumeratedDesign-Expert 7 User’s Guide D-Optimal RSM Tutorial • 3

The program now identifies 29 candidate points at various locations in the experimentalregion, from which it will select d-optimally the 10 points needed at a minimum for thecubic model. It will then fill remaining spaces with 5 points for lack of fit testing.Finally, it chooses 5 of these 13 points for replication. For details on how the softwaredoes all this, select Help, Contents and work down the Contents tree to the topic onRSM D-optimal Design.Help on how Design-Expert does d-optimal designScroll down to the bottom of this topic to see how Design-Expert picks points for its d-optimal design. At the end of this Help topic, you can link to more detail on thealgorithm employed for picking model points (“D-optimal Algorithm (Advanced).”After absorbing as much as this information as you would like, press the X button andclose out the Help screen.As noted in the Help referenced above, Design-Expert creates a number of d-optimaldesigns via random bootstraps. Click the Options button to see the design parametersthat can be adjusted by users.Options for d-optimal design buildBecause of the random element in the build, designs may vary somewhat, but all will beessentially as good by d-optimal criterion. See what you get by now pressing Continue4 • D-Optimal RSM Tutorial Design-Expert 7 User’s Guide

DX7-04F-MultifactorRSM-Dopt.doc Rev. 4/12/06to move on to the response entry (leave defaults) and another Continue again to bringup the design template. Then to see the selection of points, right-click the header for theStd column and select Display Design ID.Display design IDThen right-click the header for the Id column (formerly “Std”), but this time select Sortby Design ID. Finally, right-click the header for the Block column and selectDisplay Point Type. You should now see a list of design points that are identified andsorted. Due to the random bootstraps, it may vary from that shown below.Design layout – point type shown (your design may vary)It’s hard to assess whether these combinations of factors meet the equipment limitationsnoted at the start of this tutorial and if they provide sufficient information to fit a cubicmodel. This can be determined by clicking the Evaluation node.Design-Expert 7 User’s Guide D-Optimal RSM Tutorial • 5

Design evaluation for cubic modelBased on the d-optimal specification, the program automatically chooses the cubic orderfor evaluation. Press Results to see how well it designed the experiment. (We went toView and turned off the Annotated Evaluation option before making the screen shotbelow, but you may want to keep this informative feature enabled.)Results of design evaluation (your’s may vary)No aliases are found: That’s a good start! The degrees of freedom appear adequate.However, you may be surprised at the low power statistics and how they vary by term.This is fairly typical for d-optimal designs, which are set up to keep runs to a minimum.6 • D-Optimal RSM Tutorial Design-Expert 7 User’s Guide

DX7-04F-MultifactorRSM-Dopt.doc Rev. 4/12/06Feel free to scroll down and look over the leverage results. If you would also like to seematrix measures, go back to the Model screen and change the Options.Press the Graphs button to see where your points are located. How do they differ fromwhat our random bootstraps generated in the screen shot below? (Probably not much.)Standard error plotOur points appear to be well-spaced and the choices for replication (points labeled “2”)seem reasonable. Select View, 3D Surface to see the surface for standard error. Tilt itdown a bit by clicking on the Rotation wheel.Response surface for standard errorDesign-Expert 7 User’s Guide D-Optimal RSM Tutorial • 7

Rebuilding the D-optimal Design with Coordinate ExchangeClick the blank-sheet icon on the left of the toolbar (or select File, New Design) torebuild this design using an optional algorithm for d-optimal point selection called“coordinate exchange”. (For details on this “candidate-free approach for exact optimaldesigns”, see the similarly-titled sidebar at the end of Chapter 7 in RSM Simplified.)When asked to save changes, click No.No to save changesBut click Yes to use previous design ‘info’.Yes to use previous design informationYou should now see the design set at the D-Optimal design.Re-building the D-optimal designYour design selection and factor entries should remain preserved as before. If not, recreatewhat you did earlier in this tutorial.Press Continue and select the option for Coordinate Exchange.8 • D-Optimal RSM Tutorial Design-Expert 7 User’s Guide

DX7-04F-MultifactorRSM-Dopt.doc Rev. 4/12/06Selecting coordinate exchangeThis is a candidate-free approach so these points now disappear (total zero). PressContinue to the response entry screen and Continue again to the design.The coordinate exchange algorithm in Design-Expert includes random selectionelements that cause each design to differ somewhat, even though they will all beessentially the same according to d-optimal criterion. Therefore your design may varyfrom what we show. Take a look by going to the Evaluation and clicking the Graphsbutton.Standard error plot for d-optimal design done via coordinate exchange (may vary)This looks like a reasonable alternative to the design created via point exchange.However, what if you have other ideas on where points should be located and whichones should be replicated? This can be easily remedied with Design-Expert software.Design-Expert 7 User’s Guide D-Optimal RSM Tutorial • 9

Modifying the designClick the point at 110 A and 19 B. (Recall that this is the equipment limitation.) Noticethat the program identifies the run by standard order.Identifying a point (in this design – replicated)If you see a “2” by this point, click it again to identify its replicate.Click back to the Design, right-click the header for the Std column and select Sort byStandard Order.Sorting design by standard orderNext, right-click the header for the Block column and select Display Point Type.Displaying point typeNow you see standard order 2 identified as a vertex. Since this design was created bycoordinate exchange, non-vertex points are located more freely than by the candidaterestrictedpoint exchange used for the initial d-optimal design, so they cannot beidentified (shown as Type “??”).10 • D-Optimal RSM Tutorial Design-Expert 7 User’s Guide

DX7-04F-MultifactorRSM-Dopt.doc Rev. 4/12/06Now that you know how to identify points, feel free to delete or duplicate them to seewhat this does for your design properties.Let’s assume that the standard operating condition is (110, 20) for (A,B); respectively.This setup must be included in the experiment as a control point. Right-click the box tothe left of the first row and select Insert Row.Inserting a new rowEnter 110 for A and 20 for B as shown below.New point added to experimental designRe-Randomizing Run OrderWhenever you add runs like this, you should re-randomize the run order beforebeginning experimentation. This can be done via a right-click on the Run column andselecting “Randomize”.Randomizing run orderThe program pops up options on what it will randomize.Design-Expert 7 User’s Guide D-Optimal RSM Tutorial • 11

Randomization optionsIn this case there’s only a single block, so simply pressing OK will get the job done.Re-Evaluating the Modified DesignYou are now on your own: Go back and evaluate the design. Check the 3D view ofstandard error and notice how much lower it becomes at the point you just inserted.12 • D-Optimal RSM Tutorial Design-Expert 7 User’s Guide