mixture design tutorial: the basics

8/8/2019 Mixture Design Tutorial: The basics

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Design-Expert 7.1 Users Guide Mixture Tutorial Part 1 1

Mixture Design Tutorial(Part 1 The Basics)

IntroductionIn this tutorial you will get an introduction to mixture design. It will be assumed that at

this stage youve mastered many of Design-Expert softwares basic features by

completing the preceding tutorials. At the very least you ought to first do the General

One-Factor tutorials, basic and advanced, prior to starting this one.

We will presume that you can handle the statistical aspects of mixture design. To gain a

working knowledge of this powerful tool, we recommend you attend our Mixture Design

for Optimal Formulations workshop. Call Stat-Ease or visit our website,

www.statease.com, for a schedule. If you want statistical detail on this topic, see, John

CornellsExperiments with Mixtures , 3rd edition published by John Wiley and Sons,

New York in 2002.

This tutorial provides only the essential program functions. For more details, check out

the Help system, which you can access at any time by pressing F1. Its hypertext search

capability makes it easy for you to track down the information you need.

The Case Study Formulating a Detergent

A detergent must be re-formulated to fine-tune two product attributes, which will be

measured as responses from a designed experiment:

Y1 - viscosity

Y2 - turbidity.

Three primary components will be varied as shown:

3% A (water) 8%

2% B (alcohol) 4%

2% C (urea) 4%

These components represent nine weight-percent of the total formulation, that is:

A + B + C = 9%

The other materials (held constant) make up the difference: 91 weight-percent of the

detergent. For purposes of this experiment they can be ignored.

The experimenters chose a standard mixture design called a simplex-lattice. They

augmented this design with axial check blends and the overall centroid. The vertices

and overall centroid were replicated, which increased the size of the experiment to a total

of 14 blends.


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2 Mixture Tutorial Part 1 Design-Expert 7.1 Users Guide

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Augmented simplex lattice (second degree)

This case study leads you through all the steps of design and analysis for mixtures. The

next tutorial, Part 2, will detail how you can simultaneously optimize the two responses.

Design the Experiment

Start the program by finding and double clicking the Design-Expert software icon. Take

the quickest route to initiating a new design by clicking on 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 bar New Design icon identified

Click on the Mixture folder tab. The design you want, a simplex lattice, comes up bydefault.

Choosing a mixture design

Some detail is offered on-screen, but more can found by pressing the screen tips button

(or select Help, Screen Tips).


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On-screen details, plus more brought up via Tips button

Close Tips down by pressing the X button at its upper right corner. Then explore Help,Contents.

Looking for more Help

Then double-clickMixture Designs and Mixture Design Choices and then select

Simplex-Lattice Design.

Detail on simplex-lattice design via Help contents

Close Help down by pressing X. Then change the number ofMixture Components to3. Then enter the component and their limits as shown below in the Name, Low andHigh fields, pressing the Tab key after each entry. Enter9 in the Total field and % inthe Units field.


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Entering components, limits and total

Press Continue. Immediately a warning appears.

Adjustment made to constraints

Press OK. Notice that, although you entered the high limit for water as 8%,Design-Expert adjusts it to 5% which leaves room for 2% each of the other two

ingredients within the 9% total. Otherwise, at 8% water and the low levels of alcohol

and urea the total would become 12%, which the program realizes does not compute.

Very smart and very helpful!

Press Continue and this time the software lets you move on. Now you must choose the

order of the model that you expect to be appropriate for the system being studied. In thiscase, you can assume that a quadratic polynomial, which includes second order terms for

curvature, will adequately model the responses. Therefore leave the order at

Quadratic. Accept the default check-off to Augment design, but change the Numberof runs to replicate to 3.

Simplex-lattice design form (after pressing Tab key)

By accepting Augment design you allow Design-Expert to add the overall centroid

and axial check blends to the design points. The Number of runs to replicate field,

defaulted to 4, causes the specified number of highest leverage experiments to be

duplicated. In this case, there are three points with highest leverage the vertices of the

triangular simplex.


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Press Continue to the next step in the design process: Entering Responses. For thenumber select 2. Then enter the response Names and Units as shown below.

Response entry

You can step Back through the design forms and change what you want anywhere along

the way. When you press Continue on this page, Design-Expert will complete thedesign setup for you.

Examine the Design and Modify It If You LikeThe default columns in the design layout can be modified to be more informative in this

mixture case. Right click on the Select button at the upper-left corner and pickDesignID. It then becomes a new field for uniquely identifying your mixture recipes. Thenright-clickSelect again to turn off display of the Block column superfluous for thisexperiment. Also Select off the Std Order. Finally, Select on display ofPointType. This will be very helpful for insights on the design geometry. Your selectionsshould now look like those depicted below (default shown for comparative purposes).

Changing what will be displayed on the design layout (versus the default)

Next, right click the top of the column now labeled Id, and select Sort by Design ID.

Now you get a feel for where the points are located and which ones are replicated. (Runorder is randomized and thus differs from whats shown below.)


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Initial design sorted by ID with point type shown (run order randomized)

The experimenters ran an additional centroid point, so in the box to the left of Id 0

(Type: Center) right-click and select Duplicate.

Duplicating the centroid

Whenever you insert, delete or duplicate rows, always right-click the Run column-header and chose Randomize.

Re-randomizing the run order

A box will pop up asking if youd like all blocks to be randomized. In this case there is

only one block, so simply press OK. Notice that run numbers now change.

Again, right-click the Run column-header, but this time choose Sort by Run Order.Normally youd now do a File, Print to produce a recipe sheet for doing the

experiment. Go ahead if you have a printer available.


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Save Your Work

Now that youve invested some time into your design, it would be prudent to save your

work. Click on File menu item and select Save As. The program displays a standardfile dialog box. Use it to specify the name and destination of your data file. Enter a file

name in the field with the default extension ofdx7. (We suggest tut-mix). Then click

on Save.

Analyze the Results

Assume that the experiments are now complete. You now need to enter the responses

into the Design-Expert software. For tutorial purposes, we see no benefit to making you

type all the numbers. Therefore, to save time, read the response data in from a file that

weve put on your programs Data directory. Select File, Open Design. Click on thefile called Mix.dx7. Then press OK. You now should see response data. (Note that thedesign layout returns to the default selection, which we did not bother to change.)

Theres no need for typing in this case, but normally youd have put in a lot more workat this stage, so always do a File, Save for preservation of entered response data.

Response data

Go to the Analysis branch of Design-Expert and click on the node forViscosity. Younow will work across the buttons at the top of the window.


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First step in analysis: Transformation options

First, consider doing a transformation on the response. In some cases this will improve

the statistical properties of the analysis. For example, when responses vary over several

orders of magnitude, the log scale usually works best. In this case the ratio of maximum

to minimum response is only a bit over 4, which isnt too excessive (see detail at bottom

of the screen), so leave the selection at its default, None, because no transformation willbe needed. Also, leave the coding for analysis as pseudo which re-scales the actual

component levels to 0 1. For complete details on coding mixtures see the textbook by

Cornell mentioned at the outset of the tutorial. In the meantime, bring up Help,Contents and then double-clickMixture Designs and Mixture StatisticalInformation and then select Component Scaling in Mixture Designs.

Statistical detail on coding available via Help

Close Help down by pressing X. Then click on the Fit Summarybutton. At this pointDesign-Expert fits linear, quadratic, special cubic and full cubic polynomials to the

response.


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Fit summary reports

You may need to widen your window to get the entire output showing. Just move the

cursor to the left edge until it changes to a double-ended arrow. Then drag it open. In

similar fashion, you can also adjust column widths in any table or report. This may be

necessary to uncover the entire text. To move around the display, use the side and/or

bottom scroll bars, if necessary.

First, look for any warnings about aliasing. In this case, the full cubic model could not

be estimated by the chosen design an augmented simplex design. Remember that you

chose only to fit a quadratic model, so this should be no surprise.

Next, you see the Sequential Model Sum of Squares table. The analysis proceeds

from a basis of the mean response. This is the default model if none of the factors cause

a significant effect on the response. The output then shows the significance of each set

of additional terms:

Linear vs Mean: the significance of adding the linear terms after

accounting for the mean. (Due to the constraint that the three components

must sum to a fixed total, you will see only two degrees of freedom

associated with the linear mixture model.)

Quadratic vs Linear: the significance of adding the quadratic terms to the

linear terms already in the model.

Sp Cubic vs Quadratic: the contribution of the special cubic terms beyond

the quadratic and linear terms.

Cubic vs Sp Cubic: the contribution of the full cubic terms beyond the

special cubic, quadratic, and linear terms. (In this case, these terms are

aliased.)


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For each set of terms, the probability (Prob > F) should be examined to see if it falls

below 0.05 (or whatever statistical significance level you choose). Adding terms up to

quadratic will significantly improve this particular model, but when you get to the

special cubic level, theres no further improvement. The program automatically

underlines at least one Suggested model. Always confirm this suggestion by

reviewing all the tables under Fit Summary.

Moving on to next table via the Bookmarks, click the Lack of Fit, which comparesthe residual error to the pure error from replication. If the residual error significantly

exceeds the pure error, then something remains in the residuals that can be removed by a

more appropriate model. The residual error from the linear model shows significant lack

of fit (bad), while the quadratic, special cubic and full cubic do not show significant lack

of fit (good). At this point the quadratic model looks very good.

Lack of fit table

Now, take a look at the last table: Model Summary Statistics. Here you see several

comparative measures for model selection.

Summary statistics

Ignoring the aliased cubic model, the quadratic model comes out best: low standard

deviation (Std Dev), high R-Squared statistics and low PRESS.

Before moving on, you may want to print the Fit Summary tables by doing a File, Print.

These tables, or any selected subset, can be also cut and pasted into a word processor,

spreadsheet or any other Windows application. Youre now ready to take an in-depth

look at the quadratic model.


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Model Selection and Analysis of Variance (ANOVA)

Click the Modelbutton to see the model suggested by Design-Expert software. Pressthe screen tips button to see some very helpful information about what you can do at

this stage.

Choosing the model (Tips up on screen)

You may select alternate models to the default, in this case quadratic, from the pull down

list if you want. (Be sure to do this in the rare cases when Design-Expert suggests more

than one model.) On this screen you are allowed to manually reduce the model byclicking off terms that are not statistically significant. For example, in this case, you will

see in a moment that the AB term is not statistically significant.

As noted under the heading Model Reduction in Tips, Design-Expert provides several

automatic reduction algorithms as alternatives to the manual method: Backward,

Forward and Stepwise. Click the down arrow on the list box if youd like to try one.

We recommend that you not reduce mixture models unless youre sure from subject

matter knowledge that this makes sense.

Close Tips by clicking X. The press the ANOVAbutton for the details on the quadraticmodel. There are two views available for the ANOVA report: plain or annotated. For

tutorial purposes its best you keep it in View, Annotated ANOVA to help with theinterpretation. The program will default to whichever view was last chosen.


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ANOVA report with annotations toggled on

The statistics look very good. The model has a high F value, low probability values

(Prob > F). The probability values show the significance of each term. Because the

mixture model does not contain an intercept term, the main effect coefficients (linearterms) incorporate the overall average response and are tested together. Use the handy

Bookmarks for jumping to the next report with R-Squared statistics.

R-squared and other statistics after the ANOVA

These statistics, many of these youve seen already in the Model Summary Statistics

table, all look good. Note the more than adequate precision value of 27.9.

Next, take a look at the coefficients and associated confidence intervals for the quadratic

model.


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Coefficients for the quadratic model

Continue on to see several alternative models that vary by how the components are

coded. This may be of most interest to students of mixture design rather than actual

formulators, so its time to move on: Press the next button Diagnostics.

Diagnose the Statistical Properties of the Model

The most important diagnostic, the normal probability plot of the residuals, comes up by

default.

Normal Probability Plot of the Residuals

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 thatresiduals will be studentized unless you uncheck the first box on the floating tool palette


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(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 forInfluence. Heres where you find the find plots for externallystudentized 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 bring up case-by-case

details on many of the statistics shown graphically for diagnostic purposes: Press

Report. (In previous versions of Design-Expert, this report appeared under ANOVA.)

Diagnostics report

Notice that one value is flagged for exceeding suggested limits: DFFITs for standardorder 11. 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. Given that only this one

diagnostic is flagged, it probably is not a cause for alarm. However, observe that its

one of the highest observed viscosity responses (130.00 Actual Value), so the

experimenter might want to double-check the accuracy of this response.

Examine Model Graphs

The diagnosis of residuals reveals no statistical problems, so you will now generate the

response surface plots. Click on the Model Graphsbutton. The 2D contour plot of

comes up by default in graduated color shading.


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Response surface contour plot

Note that Design-Expert will display any actual point included in the design space

shown. In this case you see a plot of viscosity as a function of the three mixture

components. This slice includes two centroids as indicated by the dot at the middle of

the contour plot.

The Factors Toolcomes 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) are

plotted on the graph. The Gaugesview is the default. Each component listed will eitherhave an axis label, indicating that it is currently shown on the graph, or a red slider bar,

which allows you to choose specific settings for those not currently plotted. This case

study involves only three components, all of which can fit on one mixture plot a

ternary diagram. Therefore, you will not see a slider bar. If you did, it would default to

the midpoint levels of the components not currently assigned to axes. You could then

change a level by dragging the red slider bars. If youd like to see a demonstration of

this feature, do the Multifactor RSM Tutorial (Part 1 The Basics).

Place your mouse cursor over the contour graph. Notice in the lower left corner of the

screen that Design-Expert displays the predicted response and coordinates.

Coordinates display at lower, left corner of screen

To enable a handier tool for reading coordinates off contour plots, go to View, ShowCrosshairs Window.


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Showing crosshairs window

Now 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 check-blend above the

centroid, you also get that observed value (in this case: 35.1).

Prediction at upper check blend where an actual run was performed (Small vs Full)

Press the Full button to get confidence and prediction intervals in addition to thecoordinates and predicted response. Then close out the crosshairs window by clicking

X.

Zooming In and Out

Lets say you were particularly interested in highest values for viscosity. With your left

mouse button pressed, drag over the lower right corner of the contour graph.


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Corner identified for zoom

Now you see just the area you chose magnified.

Zoomed-in area on contour plot

To revert back to the full triangle, right-click over the plot and select Zoom out.

Zoom out

Trace Plot

Wouldnt it be handy to see all your factors on one response plot? You can do this with

the trace plot, which provides silhouette views of the response surface. The real benefit


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If you experiment on more than three mixture components, use the trace plot to find

those components that most affect the response. Choose these influential components

for the axes on the contour plots. Set as constants those components that create

relatively small effects. Your 2D contour and 3D plots will then be sliced up in way

thats most interesting visually.

Contour Plot: Revisited

Return to the contour plots via the View, Contour selection. The colors are neat, butwhat if you must print the graphs in black and white? That can be easily fixed by right-

clicking over the graph and selecting Graph Preferences. Then click the Graphs 2tab and change the Contour graph shading to Std Error shading.

Graph preferences: Changing contour graph shading

Press OK to see the effect on your plot.

Contour plot with standard error shading

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 its fun!


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We recommend another approach to laying out contours. 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 35, Step at 10 and Levelsat 20. Also, underFormat choose 0 to display whole numbers (no decimals). Yourscreen should now match that shown below.

Contours dialog box: Incremental option

Press OK to get a good-looking contour plot suitable for printing. This one wasproduced via Edit, Copy from Design-Expert and Edit, Paste into Microsoft Word. (The

triangle may not come in looking equilateral, but if this is bothersome, it can be easily

remedied by dragging the right border and widening the picture.)

Plot with incremental contours


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You can add new contours via a right mouse click. Find a vacant region on the plot, for

example between the 35 and 45 contours, and check it out: Right-click and select Addcontour. Then drag the contour around (it will become highlighted) until it becomes40.

Adding contours

To get a more precise contour level and locate it accurately, you can right-click it and

enter the desired value. On the new contour select Set contour value and enter40.

Setting a contour value

That fills a gap nicely!

Move your mouse to a spot near the top of the graph, where the response hits the

minimum. Click the right mouse button and Add flag. To get more digits displayed,right click forGraph Preferences and underContours change Format to 0.00.Then press OK.

Adding a flag (format re-set via Graph Preferences)

It displays the value of the response at that point. Now do a right click of the mouse on

the flag and select Toggle size.


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Flag enlarged via Toggle Size option

The expanded flag displays a 95 percent confidence interval (CI) on the mean prediction.

It also provides the prediction interval (PI). Use this to manage expectations about

individual confirmation runs near this point. The PI conveys how natural variability in

the process/sampling/testing plus imprecision in the estimate of the mean (SE mean)

causes actual outcomes to differ from the prediction. The larger flag also lists the point

coordinates. If you have a printer, you can print the contour plot by using the File, Printmenu item.

Generating a 3D View of the Response Surface

Now to really get a feel for how the response varies as a function of the two factors

chosen for display, select View, 3D Surface. You then will see three-dimensionaldisplay of the response surface. If the coordinates encompass actual design points, these

will be displayed.

3D response surface plot

The Rotation tool allows you to view the 3D surface plot from any angle.


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Control for rotating 3D plot

Move your cursor over the tool. The pointer changes to a hand. Now use the hand to

rotate the vertical or horizontal wheel. Watch the 3D surface change. Its fun! Whats

really neat is how it becomes transparent so you can see hidden points falling below the

surface.

Transparent view as surface is rotated, which allows points to be seen better

Also, be aware that Design-Expert offers many options for 3D graphs via its preferences,

which come up via a right-click over the plot. For example, if you dont like the

graduated colors, go to the Graphs 2 tab and change the 3D graph to the hidden wire

view (a transparent look).

Press the Default button when youre done playing. The graph then re-sets to its original

position. Notice that you can also specify the horizontal (h) and vertical (v)

coordinates.

Response Prediction

This feature in Design-Expert software falls under the Optimization branch, which willbe explored more fully in the next tutorial in this series. It allows you to generate

predicted response(s) for any set of factors. To see how this works, click on the PointPrediction node (lower left on your screen).


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