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Under the Optimization branch to the left of the screen, click the Numerical node to start. We will detail POE later. The program restricts factor ranges to factorial levels plus one to minus one in coded values — the region for which this experimental design provides the most precise predictions. Response limits default to observed extremes. In this case, you should leave the settings for time, temperature, and catalyst factors alone, but you will need to make some changes to the response criteria.
Desirabilities range from zero to one for any given response. The program combines individual desirabilities into a single number and then searches for the greatest overall desirability. A value of one represents the ideal case.
A zero indicates that one or more responses fall outside desirable limits. For this tutorial case study, assume you need to increase conversion.
Click Conversion and set its Goal to maximize. As shown below, set Lower Limit to 80 the lowest acceptable value, and Upper Limit to , the theoretical high. Conversion criteria settings You must provide both these thresholds so the desirability equation works properly. By default, thresholds will be set at the observed response range, in this case 51 to Otherwise we may come up short of the potential optimum. Now click the second response, Activity.
Enter Lower Limits and Upper Limits of 60 and 66, respectively. Values outside that range are not acceptable. Activity criteria settings The above settings create the following desirability functions: 1. Close out Screen Tips by pressing X at the upper-right corner of its screen. Weights give added emphasis to upper or lower bounds or emphasize target values.
With a weight of 1, di varies from 0 to 1 in linear fashion. Weights greater than 1 maximum weight is 10 give more emphasis to goals. Weights less than 1 minimum weight is 0. Try pulling the square on the left down and the square on the right up as shown below. Before moving on from here, re- enter the Lower and Upper Weights at their default values of 1 and 1; respectively.
If you want to emphasize one over the rest, set its importance higher. By leaving all importance criteria at their defaults, no goals are favored over others. Then click Contents. From here you can open various topics and look for any details you need.
Now click the Options button to see what you can control for the numerical optimization. After doing your first search for the optimum, go back to this Option and slide it one way and the other. Observe what happens to the solutions presented by Design-Expert. If you move the Filter bar to the right, you decrease the number.
Conversely, moving the bar to the left increases the solutions. Click OK to close Optimization Options. Running the optimization Start the optimization by clicking the Solutions tab. It defaults to the Ramps view so you get a good visual on the best factor settings and the desirability of the predicted responses. Numerical Optimization Ramps view for Solutions Your results may differ The program randomly picks a set of conditions from which to start its search for desirable results — your results may differ.
Multiple cycles improve the odds of finding multiple local optimums, some of which are higher in desirability than others. Due to random starting conditions, your results are likely to be slightly different from those in the report above.
The colored dot on each ramp reflects the factor setting or response prediction for that solution. The height of the dot shows how desirable it is. Press the different solution buttons 1, 2, 3,… and watch the dots.
They may move only very slightly from one solution to the next. However, if you look closely at temperature, you should find two distinct optimums, the first few near 90 degrees; further down the solution list, others near 80 degrees.
You may see slight differences in results due to variations in approach from different random starting points. For example, click the last solution on your screen.
Does it look something like the one below? Second optimum at lower temperature, but conversion drops, so it is inferior If your search also uncovered this local optimum, note that conversion falls off, thus making it less desirable than the higher-temperature option.
The Solutions Tool provides three views of the same optimization. Drag the tool to a convenient location on the screen. Click the Solutions Tool view option Report. Desirability A:time 1 B:temperature 1 C:catalyst 1 Conversion 0. Optimization Graphs Press Graphs near the top of your screen to view a contour graph of overall desirability. On the Factors Tool palette, right-click C:Catalyst.
Make it the X2 axis. Temperature then becomes a constant factor at 90 degrees. Design-Expert software sets a flag at the optimal point. To view the responses associated with the desirability, select the desired Response from its droplist. Take a look at the Conversion plot. Then go to Surface Graphs and click Show contour grid lines.
Show contour grid lines option Grid lines help locate the optimum, but for a more precise locator right-click the flag and Toggle Size to see the coordinates plus many more predicted outcome details. To get just what you want on the flag, right-click it again and select Edit Info. Flag size toggled to see select detail By returning to Toggle size, you can change back to the smaller flag. If you like, view optimal activity response as well.
To look at the desirability surface in three dimensions, again click Response and choose Desirability. The options for process order At this stage you could make use of the Add Term feature. Also, 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. You can also see probability values for each individual term in the model. You may want to consider removing terms with probability values greater than 0.
Use process knowledge to guide your decisions. The R-Squared statistics are very good — near to 1. Post-ANOVA statistics Press forward to Coefficients to bring the following details to your screen, including the mean effect-shift for each block, that is, the difference from Day 1 to Day 1 in the response. 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. However, you can copy and paste the data to your favorite Windows word processor or spreadsheet. This might be handy for client who are phobic about statistics.
The most important diagnostic — normal probability plot of the residuals — appears by default. A non-linear pattern such as an S-shaped curve indicates non-normality in the error term, which may be corrected by a transformation.
The only sign of any problems in this data may be the point at the far right. Click this on your screen to highlight it as shown above. Find the floating Diagnostics Tool palette on your screen.
This has been discussed in prior tutorials. For example, center points carry little weight in the fit and thus exhibit low leverage.
Now go to the Diagnostics Tool and click Resid. Check out the other graphs if you like. Press Screen Tips along the way to get helpful details and suggestions on interpretation.
In this case, none of the graphs really indicates anything that invalidates the model, so press ahead. Next press the Influence side for another set of diagnostics, including a report detailed case-by-case residual statistics. Influence diagnostics Leverage is best explained by the previous tutorial on One-Factor RSM so go back to that if you did not already go through it. In a similar experiment to this one, where the chemist changed catalyst, the DFBETAS 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. Now skip ahead to the Report to bring up detailed case-by-case diagnostic statistics, many which have already been shown graphically. As we discussed in the General One- Factor Tutorial, this statistic stands for difference in fits. It measures change in each predicted value that occurs when that response is deleted. Given that only one diagnostic is flagged, there may be no real cause for alarm.
This indicates less cause for concern than red-lined outliers, that is, points outside of the plus-or-minus 2 values for DFFITS are not that unusual. Anyways, assume for purposes of this tutorial that the experiments found nothing out of the ordinary for the one run that went slightly out for DFFITS.
Click the Model Graphs tab. The 2D contour plot of factors A versus B comes up by default in graduated color shading. In this case you see a plot of conversion as a function of time and temperature at a mid-level slice of catalyst. This slice includes six center points as indicated by the dot at the middle of the contour plot.
By replicating center points, you get a very good power of prediction at the middle of your experimental region. The floating Factors Tool palette appears with the default plot. Move this floating tool as needed by clicking and dragging the top blue border. The tool controls which factor s are plotted on the graph. Each factor listed has either 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 the factors that are not currently plotted.
All red slider bars default to midpoint levels of those factors not currently assigned to axes. You can change factor levels by dragging their red slider bars or by right clicking factor names to make them active they become highlighted and then typing desired levels into the numeric space near the bottom of the tool palette. Give this a try. Click the C:Catalyst toolbar to see its value. Now move your mouse over the contour plot and notice that Design-Expert generates the predicted response for specific factor values corresponding to that point.
If you place the crosshair over an actual point, for example — the one at the far upper left corner of the graph now on screen, you also see that observed value in this case: Prediction at coordinates of 40 and 90 where an actual run was performed P.
See what happens when you press the Full option for crosshairs. Now press the Default button on the floating Factors Tool to place factor C back at its midpoint. Factors tool — Sheet view In the columns labeled Axis and Value you can change the axes settings or type in specific values for factors. Then return to the Gauges view and press the Default button. At the bottom of the Factors Tool is a pull-down list from which you can also select the factors to plot. Only the terms that are in the model are included in this list.
At this point in the tutorial this should be set at AB. If you select a single factor such as A the graph changes to a One-Factor Plot. You can do this with the perturbation plot, which provides silhouette views of the response surface. The real benefit of this plot is when selecting axes and constants in contour and 3D plots.
See it by mousing to the Graphs Tool and pressing Perturbation or pull it up via View from the main menu. The Perturbation plot with factor A clicked to highlight it For response surface designs, the perturbation plot shows how the response changes as each factor moves from the chosen reference point, with all other factors held constant at the reference value.
Design-Expert sets the reference point default at the middle of the design space the coded zero level of each factor. The software highlights it in a different color as shown above. It also highlights the legend. You can click it also — it is interactive! In this case, at the center point, you see that factor A time produces a relatively small effect as it changes from the reference point.
Therefore, because you can only plot contours for two factors at a time, it makes sense to choose B and C — and slice on A. Entering components, limits, and total Press Continue. Immediately a warning appears. Adjustment made to constraints Press OK. Design-Expert recognizes that this does not compute. Very helpful! Warning that comes up if mixture region is not a simplex You should then shift to the Optimal design choice. Now you must choose the order of the model you expect is appropriate for the system being studied.
In this case, assume that a quadratic polynomial, which includes second-order terms for curvature, will adequately model the responses. Therefore, leave the order at Quadratic. Press your Tab key to display the correct number of total runs. In this case, there are three points with highest leverage — the vertices of the triangular simplex. This makes the fourth replicate a bit awkward because it creates an imbalance in the design. Feel free to try this and see for yourself.
Press Continue to proceed to the next step in the design process. In the Responses droplist, choose 2. Then enter all response Names and Units as shown below. When you press Continue on this page, Design-Expert completes the design setup for you.
In the Design layout right click the Select column header at the upper-left corner and pick Design ID. Go back and alsso Select display the Space Point Type column. This is very helpful for insights about design geometry. Adding columns to design layout via the Select option right-click menu Next, right click the column header labeled Id, and select Sort by Design ID.
Now your screen should match that below except for the randomized run numbers. Duplicating the centroid Whenever you insert, delete, or duplicate rows, always right-click the Run column-header and chose Randomize.
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. Go ahead if you wish or simply do a File, Print Preview. Click File then Save As. The program displays a standard file dialog box.
Use it to specify the name and destination of your data file. Enter a file name in the field with default extension dxpx. We suggest tut-mix. Click Save. Analyze the Results Assume your experiments are completed. You now need to enter responses into the Design-Expert software. For tutorial purposes, we see no benefit to making you type all the numbers.
Select File, Open Design. Click the file named Mix. Press OK. You now should be displaying the response data shown below. Note the design layout returns to the default selection, which we have not changed. In some cases this improves the statistical properties of the analysis. For example, when responses vary over several orders of magnitude, the log scale usually works best. Also, leave the coding for analysis as pseudo because this re-scales the actual component levels to 0 — 1.
In the meantime, bring up Help, Contents. Select Component Scaling in Mixture Designs. After studying all the information you find here, close Help by pressing X. Next click the Fit Summary tab. Here Design-Expert fits linear, quadratic, special cubic, and full cubic polynomials to the response.
Drag it to the right. To begin your analysis, look for any warnings about aliasing. In this case, the full cubic model and beyond could not be estimated by the chosen design — an augmented simplex design.
Remember, you chose only to fit a quadratic model, so this should be no surprise. Now on the floating Bookmarks press forward to the Sum of Squares breakdown. This is the default model if none of the factors causes a significant effect on the response. The output then shows the significance of each set of additional terms. 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.
In this case, these terms are aliased. Now you must choose the order of the model you expect is appropriate for the system being studied. In this case, assume that a quadratic polynomial, which includes second-order terms for curvature, will adequately model the responses.
Therefore, leave the order at Quadratic. Press your Tab key to display the correct number of total runs. In this case, there are three points with highest leverage — the vertices of the triangular simplex. This makes the fourth replicate a bit awkward because it creates an imbalance in the design. Feel free to try this and see for yourself. Press Continue to proceed to the next step in the design process.
In the Responses droplist, choose 2. Then enter all response Names and Units as shown below. When you press Continue on this page, Design-Expert completes the design setup for you. In the Design layout right click the Select column header at the upper-left corner and pick Design ID. Go back and alsso Select display the Space Point Type column.
This is very helpful for insights about design geometry. Adding columns to design layout via the Select option right-click menu Next, right click the column header labeled Id, and select Sort by Design ID. Now your screen should match that below except for the randomized run numbers. Duplicating the centroid Whenever you insert, delete, or duplicate rows, always right-click the Run column-header and chose Randomize.
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. Go ahead if you wish or simply do a File, Print Preview.
Click File then Save As. The program displays a standard file dialog box. Use it to specify the name and destination of your data file. Enter a file name in the field with default extension dxpx. We suggest tut-mix. Click Save. Analyze the Results Assume your experiments are completed. You now need to enter responses into the Design-Expert software. For tutorial purposes, we see no benefit to making you type all the numbers.
Select File, Open Design. Click the file named Mix. Press OK. You now should be displaying the response data shown below. Note the design layout returns to the default selection, which we have not changed. In some cases this improves the statistical properties of the analysis. For example, when responses vary over several orders of magnitude, the log scale usually works best.
Also, leave the coding for analysis as pseudo because this re-scales the actual component levels to 0 — 1. In the meantime, bring up Help, Contents. Select Component Scaling in Mixture Designs. After studying all the information you find here, close Help by pressing X. Next click the Fit Summary tab. Here Design-Expert fits linear, quadratic, special cubic, and full cubic polynomials to the response.
Drag it to the right. To begin your analysis, look for any warnings about aliasing. In this case, the full cubic model and beyond could not be estimated by the chosen design — an augmented simplex design. Remember, you chose only to fit a quadratic model, so this should be no surprise. Now on the floating Bookmarks press forward to the Sum of Squares breakdown. This is the default model if none of the factors causes a significant effect on the response.
The output then shows the significance of each set of additional terms. 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.
In this case, these terms are aliased. Always confirm this suggestion by reviewing all tables under Fit Summary. On the floating Bookmarks tool click Lack of Fit to move on to the next table.
This table compares residual error with pure error from replication. If residual error significantly exceeds pure error, then deviations remain in the residuals that can be removed using a more appropriate model. Residual error from the linear model shows significant lack of fit this is bad , while quadratic, special cubic, and full cubic do not show significant lack of fit this is good. Lack of fit table At this point the quadratic model statistically looks very good indeed.
Select Multilevel Categoric for this design. If your factor is numerical, such as temperature, then you would use the One Factor option under the Response Surface tab. If any of your factors are quite hard to control, that is, not easily run at random levels, then consider using the Split- Plot Multilevel Categoric design. However, restricting randomization creates big repercussion on the power of your experiment, so do your best to allow all factors to vary run-by-run as chance dictates.
Design-Expert by default will lay out your design in a randomized run order. Enter Bowler as the name of the factor. Tab down to the Units field and enter Person. Next tab to Type. Leaving Type at its default of Nominal, tab down to the Levels field and enter 3. Now tab to L 1 level one and enter Pat. Type Mark, and Shari for the other two levels L2 and L3. Screen tips on factor Type Press Continue to specify the remaining design options.
In the Replicates field, which becomes active by default, type 6 each bowler rolls six games. Design-Expert now recalculates the number of runs for this experiment: Leave the number of Responses at the default of 1. Now click on the Name box and enter Score. Tab to the Units field and enter Pins. Response name dialog box — completed At this stage you can skip the remainder of the fields and continue on. However, it is good to gain an assessment of the power of your planned experiment.
In this case, as shown in the fields below, enter the value 20 because the bowling captain does not care if averages differ by fewer than 20 pins. Then enter the value 10 for standard deviation derived from league records as the variability of a typical bowler. Design-Expert then compute a signal- to-noise ratio of 2 10 divided by 5. Optional power calculator — necessary inputs entered Press Continue to view the happy outcome — power that exceeds 80 percent probability of seeing the desired difference.
Results of power calculation Click on Finish for Design-Expert to create the design and take you to the design layout window. Save As dialog box Click on Save. A printout provides space to write down the responses. Note: this view of the data does not To type results into the program you must switch back to the home base — the Design Layout view. You can do the same from the basic design layout if you like that format better. Enter the Response Data When performing your own experiments, you will need to go out and collect the data.
Simulate this by clicking File, Exit. Click on Yes if you are prompted to Save. You should now see your data tabulated in the randomized layout. For this example, you must enter your data in the proper order to match the correct bowlers.
Sort runs by standard std order Now enter the responses from the table on page one, or use the following screen. Except for run order, your design layout window must look like that shown below.
Standard order should only be used as a convenience for entering pre-existing design data. Bowling six games is taxing but manageable for any serious bowler. For example, right click the Select button. This allows you to control what Design-Expert displays.
For this exercise, choose Comments. If comments exceed allotted space, move the cursor to the right border of the column header until it turns into a double-headed arrow shown below. Then, just double-click for automatic column re-sizing. Adjusting column size Now, to better grasp the bowling results, order them from low-to-high as shown below by right-clicking the Response column header and selecting Sort Ascending.
It works on factors as well as responses. In this example, you quickly see that Mark bowled almost all the highest games. Under the Analysis branch of the program on the left side of your screen , click the Score node. Transform options appear in the main window of Design-Expert on a progressive tool bar. The Transform screen gives you the opportunity to select a transformation for the response. For complete details, go to the Help command on the main menu. Examine the Analysis By necessity, the tutorial now turns a bit statistical.
Design-Expert now pops up a very specialized plot that highlights factor A—the bowlers— as an emergent effect relative to the statistical error, that is, normal variation, shown by the line of green triangles.
It supports what was obvious from the raw results—who bowls does matter. Notice to the far right side of your screen that Design-Expert verifies that the results are significant. Note that the blue textual hints and explanations disappear so you can make a clean printout for statistically savvy clients. In most cases you will access helpful advice about the particular statistic.
Now click the Coefficients Bookmark button to view the output illustrated below. The intercept in this simple one-factor comparative experiment is simply the overall mean score of the three bowlers. You may wonder why only two terms, A1 and A2, are provided for a predictive model on three bowlers. It turns out that the last model term, A3, is superfluous because it can be inferred once you know the mean plus the averages of the other two bowlers. As you can see below, these are compared via pair-wise t-tests in the following part of the ANOVA report.
Back to the top Analyze Residuals Click the Diagnostics tab to bring up the normal plot of residuals. Ideally this will be a straight line, indicating no outlying abnormalities.
Does it loosely cover up all the points? If you need to re-set the line, simply double-click your left mouse button over the graph. Notice that the points are coded by color to the level of response they represent — going from cool blue for lowest values to hot red for the highest. Is this fair? More details on studentization reside in Help. Raw residuals can be displayed by choosing it off the down-list on the Diagnostics Tool shown below.
Check it out! Other ways to display residuals In any case, when runs have greater leverage another statistical term to look up in Help , only the Studentized form of residuals produces valid diagnostic graphs. Due to potential imbalances of this sort, we advise that you always leave the Studentized feature checked as done by default.
This is explored in the Two-Level Factorial Tutorial. For now, suffice it to say that the program chooses this form of residual to provide greater sensitivity to statistical outliers. On the Diagnostics Tool, select Resid. In other words, the vertical spread of the studentized residuals should be approximately the same for each bowler. In this case the plot looks OK. The spread from bottom-to-top is not out of line with his competitors, despite their protestations about the highest score still highlighted.
Bring up the next graph on the Diagnostics Tool list — Resid. Note: your graph may differ due to randomization. Residuals versus run chart Note: your graph may differ due to randomization Here you might see trends due to changing alley conditions the lane re-oiling, for example , bowler fatigue, or other time-related lurking variables.
However, even if you see a pronounced upward, downward, or shift change, it will probably not bias the outcome because the runs are completely randomized. To ensure against your experiment being sabotaged by uncontrolled variables, always randomize! These limits indicate that it is most desirable to achieve the targeted value of 63, but values in the range of are acceptable.
Values outside that range are not acceptable. Activity Criteria Settings These settings create the following desirability functions: 1. Weights give added emphasis to upper or lower bounds or emphasize a target value. With a weight of 1, the di will vary from 0 to 1 in linear fashion. Weights greater than 1 maximum weight is 10 give more emphasis to the goal. Weights less than 1 minimum weight is 0.
Leave the Weights fields at their default values of 1. Importance is a relative scale for weighting each of the resulting di in the overall desirability product. See the on-line help system for a more in-depth explanation of the construction of the desirability function, and formulas for the weights and importance. The Options button controls the number of cycles searches per optimization. If you have a very complex combination of response surfaces, increasing the number of cycles will give you more opportunities to find the optimal solution.
The Duplicate solution filter establishes the epsilon minimum difference for eliminating duplicate solutions. Leave these options at their default levels shown below.
Multiple cycles improve the odds of finding multiple local optimums, some of which will be higher in desirability than others. After grinding through 10 cycles of optimization, the results appear. The report view shows the results in tabular form. Due to the random starting conditions, your results are likely to be slightly different from those shown here.
Note that the last solution falls short of the first for conversion. There may be some duplicates in between. These passed through the filter discussed earlier. If you want to adjust the filter, go to the Options button and change the Duplicate Solutions Filter. If you move the Filter bar to the right you will decrease the number of solutions shown.
Likewise, moving the bar to the left increases the number of solutions. The Solutions tool provides three views of the same optimization. Drag the tool to a convenient location on the screen. Click on the solutions view option Ramps. Ramps Report on Numerical Optimization The ramp display combines the individual graphs for easier interpretation. The dot on each ramp reflects the factor setting or response prediction for that solution. The height of the dot shows how desirable it is. Press the different solution buttons 1, 2, 3,… and watch the dots.
They may move only very slightly from one solution to the next. However, if you look closely at temperature, you should find two distinct optimums, one near 80 degrees and the other near 90 degrees. You may see slight differences in the results due to variations in approach from the various random starting points.
Select the Histogram view. Nearly duplicate solutions, as found by the duplicate solutions filter, will be eliminated. In this example, you will find two or more somewhat different solutions. You can cycle through each of the solutions: the best is listed first. Press Graphs to view the results of the first optimization run. Go to the Factor control and right click on catalyst.
Make it the Y axis. Temperature then becomes a constant factor at 90 degrees. You can plant additional flags by doing a right mouse click at any location. Right click again on the flag and Toggle size to see the associated desirability value and the factor levels.
To view the responses associated with the desirability, select the desired Response from the drop down list. Take a look at the plot for Conversion. Conversion Contour Plot with optimum flagged If you like, look at the optimal activity response as well. To look at the desirability surface in three dimensions, click again on Response and choose Desirability.
Then select View, 3D Surface from the main menu. Drag the tools out of the way as needed to view the results. Then rotate the plot for a different perspective by using the Rotation tool. Drag the rim of the wheels with your mouse pointer to change the orientation of the 3D plot. You can change the quality of the 3D graph by going to the Edit, Preferences menu item, or by doing a right mouse click on the graph. Click on the Graph tab and then set parameters as you desire.
Then if you have a printer attached, make a hard copy by doing a File, Print. Show your colleagues what Design - Expert software will do! Then you can generate propagation of error POE plots that show how that error is transmitted to the response. Start by clicking on the Design node on the left side of the screen to get back to the design layout.
Then select View, Column Info Sheet. Enter the following info rmation into the Std. Click on the Conversion analysis node on the left to start the analysis again. Then jump past the intermediate buttons for analysis and click on the Model Graphs button. The end result is a more robust process. However, POE will only work when the response surface is non-linear, such as for the Conversion.
When the surface is linear, such as that for Activity, the error will be transmitted equally throughout the region. Call, or check our web site for a schedule. Click on Numerical optimization node. Click on the alternate solutions 2, 3,…. Watch the red dots. What changes? Select the Graphs node. Click the number 1 solution. To make the view similar to what we had before, on the Factors tool palette change catalyst to the Y axis.
❿Design expert manual pdf free download
Choose View, Contour to get a better view of the circular contours. Then do a right mouse click on the graph and select Graph preferences. Change the default X Axis value for Low to -2 and the High value to 2.
After making all these changes to the graph properties, press OK. If you see any leftover flags, right click on them and delete. These areas cannot be predicted as well as the interior region. Design - Expert provides the shading as a warning against extrapolation. It begins at one-half the standard deviation and increases linearly up to 1.
You will also see the shading on response model plots. Be wary of predictions in these nether regions! Remember that the darker shading represents higher standard error. Select Display Options, Process Factors, Actual to view the factors in their original units of measure. This feature in Design - Expert software allows you to generate predicted response s for any set of factors. To see how this works, click on the Point Prediction icon in the optimization section lower left on your screen.
Go ahead and play with them now if you like. You can either move the slider controls, or switch to the Sheet view and enter values. This tells you what to expect for an individual confirmation test.
You might be surprised at the level of variability, but it will help you manage expectations. Note: block effects are not accounted for in the prediction. You can print the results by using the File, Print command. Analyze the data for the second response, activity. Be sure you find the appropriate polynomial to fit the data, examine the residuals and plot the response surface.
Hint: The correct model is linear. Design - Expert will save your models. It's based on the data from the Response Surface Tutorial. You should go back to this section if you've not already completed it. Call or visit our web site for info rmation on content and schedules. In this section, you will work with a three-factor central composite design on a chemical reaction.
The factors are: time, temperature and catalyst. The experimenters measured two responses: yield and activity. You will optimize the process using models developed earlier in the Response Surface Tutorial. You will find the case study data, with the responses already analyzed, stored in a file named Rsm-a. The standard file open dialog box appears.
File Open Dialog Box Once you have found the proper drive, directory and file name, click on Open to load the data. To see a description of the design status, click on the Status node.
Drag the left border and open the window to see the report better. You can also re-size columns with the mouse. From the design status screen you can see that we modeled conversion with a quadratic model and activity with a linear model. We will lead you through the latter case: a multiple response optimization. Click on the Numerical node to start the process. We will get to POE later. To be safe, the program sets the factor ranges to the actual levels plus one to minus one in coded values.
The limits for the responses default to the observed extremes. In this case, you should leave the settings for time, temperature and catalyst factors alone, but you will need to make some changes to the response criteria.
The program combines the individual desirabilities into a single number and then searches for the greatest overall desirability. A value of one represents the ideal case. A zero indicates that one or more responses fall outside desirable limits. For this tutorial case study, assume that you need the conversion to be as high as possible.
Click on Conversion and set its Goal at is maximum. Set the Lower Limit to 80, the lowest acceptable value. You must enter an Upper Limit to get the desirability equation to work properly, so set it at the theoretical high of Set its Goal to is target of Enter a Lower Limit of 60 and an Upper Limit of These limits indicate that it is most desirable to achieve the targeted value of 63, but values in the range of are acceptable.
Values outside that range are not acceptable. Activity Criteria Settings These settings create the following desirability functions: 1. Weights give added emphasis to upper or lower bounds or emphasize a target value. With a weight of 1, the di will vary from 0 to 1 in linear fashion. Weights greater than 1 maximum weight is 10 give more emphasis to the goal. Weights less than 1 minimum weight is 0. Leave the Weights fields at their default values of 1.
Importance is a relative scale for weighting each of the resulting di in the overall desirability product. See the on-line help system for a more in-depth explanation of the construction of the desirability function, and formulas for the weights and importance. The Options button controls the number of cycles searches per optimization. If you have a very complex combination of response surfaces, increasing the number of cycles will give you more opportunities to find the optimal solution.
The Duplicate solution filter establishes the epsilon minimum difference for eliminating duplicate solutions. Leave these options at their default levels shown below. Multiple cycles improve the odds of finding multiple local optimums, some of which will be higher in desirability than others.
After grinding through 10 cycles of optimization, the results appear. The report view shows the results in tabular form. Due to the random starting conditions, your results are likely to be slightly different from those shown here.
Note that the last solution falls short of the first for conversion. There may be some duplicates in between. These passed through the filter discussed earlier. If you want to adjust the filter, go to the Options button and change the Duplicate Solutions Filter. If you move the Filter bar to the right you will decrease the number of solutions shown. Likewise, moving the bar to the left increases the number of solutions. The Solutions tool provides three views of the same optimization. Drag the tool to a convenient location on the screen.
Click on the solutions view option Ramps. Ramps Report on Numerical Optimization The ramp display combines the individual graphs for easier interpretation. The dot on each ramp reflects the factor setting or response prediction for that solution. The height of the dot shows how desirable it is. Press the different solution buttons 1, 2, 3,… and watch the dots. They may move only very slightly from one solution to the next.
However, if you look closely at temperature, you should find two distinct optimums, one near 80 degrees and the other near 90 degrees. You may see slight differences in the results due to variations in approach from the various random starting points.
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Design expert 9 tutorials 1. Your Design-Expert program offers rotatable 3D plots to easily view response surfaces from all angles. Use your mouse to set flags and explore the contours on interactive 2D graphs. Our numerical optimization function finds maximum desirability for dozens of responses simultaneously!
Take advantage of this information gold-mine that is literally at your fingertips. Also, do not overlook the helpful annotations provided on all reports. Anderson and Whitcomb have also written a Primer on Mixture Design. The alpha out is enforced after AHM is completed by doing a final sweep using backward selection, after which hierarchy is again corrected by the program. Simulator now provides an entry field for ratio of variance between whole and sub plots so trainers can set up split-plot exercises.
Slides available on request to stathelp statease. Largest effect not selected. These new designs can be seen at the bottom of the factorial design builder in DX9 see the box in the screenshot, below right. Randomized designs remain available, but some feature new, more descriptive names, and they have been resorted for easier access. As always in Stat-Ease software, the most commonly used designs get top priority, that is, they are listed in order of usefulness.
For a quick overview of the changes, compare the screenshots below. New Design New Designs The data, shown below, is a measure of the endplay: 61, 61, 57, 56, 60, 52, 62, 59, 62, 67, 55, 56, 52, 60, 59, 59, 60, 59, 49, 42, 55, 67, 53, 66, Then click Continue. Design- Expert then presents the needed statistics as seen here.
Also, take a look at the tool under the Post Analysis node for Point Prediction, in particular the tolerance interval, a very useful statistic for a purchaser who needs to establish incoming specifications. This concludes our feature tour of simple sample tools in DX9. Feel free to explore other tools. If you need more information at any time, press for Tips, Screen Tips off the main menu or push the light bulb icon.
If you are in a hurry, skip the boxed bits—these are sidebars for those who want to spend more time and explore things. The data for this example come from the Stat-Ease bowling league. Three bowlers Pat, Mark, and Shari are competing for the last team position. They each bowl six games in random order — ideal for proper experimentation protocol. Results are: Game Pat Mark Shari 1 2 3 4 5 6 Mean The captain needs to know if the average scores are significantly different, given the variability in individual games.
This one-factor case study provides a good introduction to the power of simple comparative design of experiments DOE. It exercises many handy features found in Design-Expert software. Many other features and outputs are detailed only in the help system, which you can access by clicking Help in the main menu, or in most places via a right click, or by pressing the F1 key context sensitive. Start the program by double clicking the Design-Expert icon.
You will then see the main menu and icon bar. Click on File in the main menu. Unavailable items are dimmed. If you prefer using your keyboard, press the Alt key and underlined letter simultaneously, in this case Alt F. File menu Select the New Design item with your mouse. To try this, press Cancel to re-activate the tool bar.
Opening a new design with the blank sheet icon Using either path, you now see four yellow tabs on the left of your screen. The Factorial tab comes up by default. Select Multilevel Categoric for this design. If your factor is numerical, such as temperature, then you would use the One Factor option under the Response Surface tab.
If any of your factors are quite hard to control, that is, not easily run at random levels, then consider using the Split- Plot Multilevel Categoric design. However, restricting randomization creates big repercussion on the power of your experiment, so do your best to allow all factors to vary run-by-run as chance dictates. Design-Expert by default will lay out your design in a randomized run order. Enter Bowler as the name of the factor.
Tab down to the Units field and enter Person. Next tab to Type. Leaving Type at its default of Nominal, tab down to the Levels field and enter 3. Now tab to L 1 level one and enter Pat. Type Mark, and Shari for the other two levels L2 and L3. Screen tips on factor Type Press Continue to specify the remaining design options. In the Replicates field, which becomes active by default, type 6 each bowler rolls six games. Design-Expert now recalculates the number of runs for this experiment: Leave the number of Responses at the default of 1.
Now click on the Name box and enter Score. Tab to the Units field and enter Pins. Response name dialog box — completed At this stage you can skip the remainder of the fields and continue on. For example, center points carry little weight in the fit and thus exhibit low leverage. Externalizing the residuals isolates each one in comparison to the others so discrepant results stand out more.
Now click the Influence option. To bring up bring up case-by-case details on many of the statistics shown graphically for diagnostic purposes: Press Report. It measures 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. Explanations for most of these graphs are addressed in earlier tutorials. Get more details via Screen Tips and Help.
Click the Model Graphs tab. The 2D contour plot comes up by default in graduated color shading. In this case you see a plot of viscosity as a function of the three mixture components. Move this floating tool as needed by clicking on the top blue border and dragging it. The tool controls which factor s are plotted on the graph.
The Gauges view is the default. Each component listed has either an axis label, indicating that it is currently appearing 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 fit on one mixture plot — a ternary diagram.
Therefore, you do not see any red slider bars. If you did, they 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 left or right. Place your mouse cursor over the contour graph. Then 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, Show Crosshairs Window. 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.
Full Crosshairs display Close the crosshairs window by clicking X. With your left mouse button held down, drag over the lower right corner of the contour graph. Corner identified for zoom Now the area you chose is magnified. You can do this with the trace plot, which provides silhouette views of the response surface.
The real benefit from this plot is for selecting axes and constants in contour and 3D plots. From the floating Graphs Tool select Trace. Trace plots show the effects of changing each component along an imaginary line from the reference blend defaulted to the overall centroid to the vertex.
For example, click on the curve for A and it changes color. Check this out by going to the Trace Graph tool and pressing Cox.
In the Cox direction, as the amount of any component increases, the amounts of all other components decrease, but their ratio to one another remains constant.
Chemists may like this because it preserves the reaction stoichiometry. However, when plotted in this direction, traces for highly constrained mixture components such as a catalyst for a chemical reaction become truncated. For this reason Piepel is the preferred plot in Design-Expert. Trace plots depend greatly on where you place the starting point by default the centroid. See for yourself by moving slide bars on the Factors Tool. When you are done, press the Default. Consider that the traces are one- dimensional only, and thus cannot provide a very useful view of a response surface.
A 3D response plot provides a better picture of the surface, and ultimately provides the basis for numerical optimization. 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 in ways that are most visually interesting. For example, if you design for four components, the experimental space is a tetrahedron. Generating a 3D View of the Response Surface Now to really get a feel for how response varies as a function of the two factors chosen for display, select View, 3D Surface.
A three-dimensional display of the response surface appears. If coordinates encompass actual design points, these emerge. Then click and hold the left mouse-button and drag. Try it! 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. Give this a try, too. Then press Default and X off the view of Rotation tool. Design-Expert offers many options for 3D graphs via its Graph Preferences, which come up with a right-click over the plot.
Response Prediction Response prediction in Design-Expert software falls under the Post Analysis branch, which will be 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 the Point Prediction node.
The Factors Tool opens along with the point prediction window. Move the floating tool as needed by clicking and dragging the top border.
You can also drag the handy red sliders on the component gauges to view other blends. Note that in a mixture you can only vary two of the three components independently. Can you find a combination that produces viscosity of 43?
Hint: push Urea up a bit. Design-Expert makes adjustments as you go — perhaps in ways you do not anticipate. Analyze the data for the second response, turbidity Y 2. Be sure you find the appropriate polynomial to fit the data, examine the residuals, and plot the response surface. A value of one represents the ideal case. A zero indicates that one or more responses fall outside desirable limits. In this case, components are allowed to range within their pre-established constraints, but be aware they can be set to desired goals.
For example, because water is cheap, you could set its goal to maximize. Options for goals on components Notice that components can be set equal to specified levels. Enter Limits as Lower of 39 and Upper of Press Tab to set your entries. Values outside that range have no zero desirability.
Now click the second response — Turbidity. Select its Goal to minimize, with Limits set at Lower of and Upper of You must provide both these thresholds to get the desirability equation to work properly. By default they are set at the observed response range, in this case to On the other hand, when turbidity exceeds , it looks as bad as it gets.
Aiming for minimum on second response of turbidity These settings create the following desirability functions: 1. Close out Tips by pressing X at the upper-right corner of its screen. Weights give added emphasis to upper or lower bounds, or emphasize a target value. With a weight of 1, di varies from 0 to 1 in linear fashion. Weights greater than 1 maximum weight is 10 give more emphasis to goals. Weights less than 1 minimum weight is 0. Try pulling the handle on the ramp down as shown below.
Weights change by grabbing handle with mouse Notice that Weight now reads Before moving on, re-enter Upper Weights to its default value of 1 and press the Tab key. This straightens the desirability ramp. If you want to emphasize one variable over the rest, set its importance higher. By leaving all importance criteria at their defaults, none of the goals is favored over any other.
When you finish viewing Help, close the screen by pressing X at the upper-right corner of its screen. Running the optimization Start the optimization by clicking the Solutions tab. Design-Expert brings up the Ramps view by default. The dot on each ramp reflects the factor setting or response prediction for that solution. The height of the dot shows how desirable it is. Press the different solution buttons 1, 2, 3,… and watch the dots.
The red ones representing the component levels move around quite a bit, but do the responses remain within their goals desirability of 1? Does your solution look something like the one below? Sub-optimum solution that ranks least desirable If your search also uncovered the above local optimum, note that viscosity falls off target and turbidity becomes excessive, thus making it less desirable than the option for higher temperature.
Go back a step up on the Solutions Tool by pressing for the Report. Then it lists solutions in order of desirability. It ends with detailing of the starting points for the search. Multiple cycles improve the odds of finding multiple local optimums, some of which will be higher in desirability than others. In this case Design-Expert grinds through 40 cycles of optimization, starting from the 10 design points plus 30 more at random. Go back now to the Solutions Tool and select the Bar Graph.
Solution to multiple-response optimization — desirability bar graph The above bar graph shows how well each variable satisfies the criteria and the overall combined desirability: Values near one are good.
This is not the best solution! Optimization Graphs Press the Graphs tab to view a contour graph of overall desirability. Design-Expert software sets a flag at the optimal point for solution 13 or whichever one is your worst. Now click back through the numbered Solutions choices atop your screen until the flag relocates to the largest sweet spot the one with the largest area at the top of the triangular mixture space.
To view the responses associated with this desirability sweet spot , press the droplist arrow for Response and select Viscosity. There are many other options on this and other Graph preferences tabs. Look them over if you like and then press OK to see how options specified by this tutorial affect your contour plot. If you like, look at the optimal turbidity response as well. For tutorial purposes, go back and press Default on all Graph Preference tabs to re-set the original layouts.
To view the desirability surface in three dimensions, again click Response and choose Desirability. Then from the floating Graphs Tool select 3D Surface. Another high point can be achieved, but it requires sharp control of the composition.
The other peak is less desirable lower. Try smoothing out the 3D desirability surface via a right-click over the graph, selecting Graph Preferences and then on the Surface Graphs tab changing the Graph resolution to Very High. Press OK for the new graph preferences. The go back and re-set things to the Default. Adding Propagation of Error POE to the Optimization If you have prior knowledge of the variation in your component amounts, this information can be fed into Design-Expert software.
Then you can generate propagation of error POE plots showing how that error transmits to the response. Start by clicking the Design node on the left side of the screen to get back to the design layout.
Enter the following information into the Std. First, click the Viscosity analysis node and press the Model Graphs tab. Next, select View, Propagation of Error, which previously was grayed out.
Also choose 3D Surface view. These minima occur at flat regions on model graphs where formulations are most robust to varying amounts of components. Rotate it so you can see the surface best. Click the Numerical optimization node. You may need to press Ramps on the Solutions Tool to get the view shown below. In this case, the number 2 solution, which you may or may not get due to the random nature of the optimization, increases the water level presumably cheaper and reduces turbidity, so it may actually be preferred by the formulators.
Viewing Trace Plots from Optimal Point Continue on to the numerical optimization Graphs to look at the desirability contour plot. For now, accept the default transformation selection of None. Now click the Fit Summary tab. At this point Design-Expert fits linear, two-factor interaction 2FI , quadratic, and cubic polynomials to the response. By design, the central composite matrix provides too few unique design points to determine all the terms in the cubic model.
Next you will see several extremely useful tables for model selection. Each table is discussed briefly via sidebars in this tutorial on RSM. So far, Design-Expert is indicating via underline the quadratic model looks best — these terms are significant, but adding the cubic order terms will not significantly improve the fit.
Use the handy Bookmarks tool to advance to the next table for Lack of Fit tests on the various model orders. 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. Always confirm this suggestion by viewing these tables.
Design-Expert allows you to select a model for in-depth statistical study. Click the Model tab at the top of the screen to see the terms in the model. Be sure to try this in the rare cases when Design-Expert suggests more than one model.
The options for process order At this stage you could make use of the Add Term feature. Also, 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. You can also see probability values for each individual term in the model. You may want to consider removing terms with probability values greater than 0. Use process knowledge to guide your decisions. The R-Squared statistics are very good — near to 1. Post-ANOVA statistics Press forward to Coefficients to bring the following details to your screen, including the mean effect-shift for each block, that is, the difference from Day 1 to Day 1 in the response.
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. However, you can copy and paste the data to your favorite Windows word processor or spreadsheet. This might be handy for client who are phobic about statistics. The most important diagnostic — normal probability plot of the residuals — appears by default. A non-linear pattern such as an S-shaped curve indicates non-normality in the error term, which may be corrected by a transformation.
The only sign of any problems in this data may be the point at the far right. Click this on your screen to highlight it as shown above. Find the floating Diagnostics Tool palette on your screen. This has been discussed in prior tutorials. For example, center points carry little weight in the fit and thus exhibit low leverage.
Now go to the Diagnostics Tool and click Resid. Check out the other graphs if you like. Press Screen Tips along the way to get helpful details and suggestions on interpretation. In this case, none of the graphs really indicates anything that invalidates the model, so press ahead. Next press the Influence side for another set of diagnostics, including a report detailed case-by-case residual statistics.
Influence diagnostics Leverage is best explained by the previous tutorial on One-Factor RSM so go back to that if you did not already go through it. In a similar experiment to this one, where the chemist changed catalyst, the DFBETAS 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. Now skip ahead to the Report to bring up detailed case-by-case diagnostic statistics, many which have already been shown graphically.
As we discussed in the General One- Factor Tutorial, this statistic stands for difference in fits. It measures change in each predicted value that occurs when that response is deleted. Given that only one diagnostic is flagged, there may be no real cause for alarm. This indicates less cause for concern than red-lined outliers, that is, points outside of the plus-or-minus 2 values for DFFITS are not that unusual. Anyways, assume for purposes of this tutorial that the experiments found nothing out of the ordinary for the one run that went slightly out for DFFITS.
Click the Model Graphs tab. The 2D contour plot of factors A versus B comes up by default in graduated color shading.
In this case you see a plot of conversion as a function of time and temperature at a mid-level slice of catalyst. This slice includes six center points as indicated by the dot at the middle of the contour plot. By replicating center points, you get a very good power of prediction at the middle of your experimental region. The floating Factors Tool palette appears with the default plot. Move this floating tool as needed by clicking and dragging the top blue border. The tool controls which factor s are plotted on the graph.
Each factor listed has either 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 the factors that are not currently plotted. All red slider bars default to midpoint levels of those factors not currently assigned to axes. You can change factor levels by dragging their red slider bars or by right clicking factor names to make them active they become highlighted and then typing desired levels into the numeric space near the bottom of the tool palette.
Give this a try. Click the C:Catalyst toolbar to see its value. Now move your mouse over the contour plot and notice that Design-Expert generates the predicted response for specific factor values corresponding to that point. If you place the crosshair over an actual point, for example — the one at the far upper left corner of the graph now on screen, you also see that observed value in this case: Prediction at coordinates of 40 and 90 where an actual run was performed P.
See what happens when you press the Full option for crosshairs. Now press the Default button on the floating Factors Tool to place factor C back at its midpoint. Factors tool — Sheet view In the columns labeled Axis and Value you can change the axes settings or type in specific values for factors. Then return to the Gauges view and press the Default button. At the bottom of the Factors Tool is a pull-down list from which you can also select the factors to plot.
Only the terms that are in the model are included in this list. At this point in the tutorial this should be set at AB. If you select a single factor such as A the graph changes to a One-Factor Plot. You can do this with the perturbation plot, which provides silhouette views of the response surface.
The real benefit of this plot is when selecting axes and constants in contour and 3D plots. See it by mousing to the Graphs Tool and pressing Perturbation or pull it up via View from the main menu. The Perturbation plot with factor A clicked to highlight it For response surface designs, the perturbation plot shows how the response changes as each factor moves from the chosen reference point, with all other factors held constant at the reference value.
Design-Expert sets the reference point default at the middle of the design space the coded zero level of each factor. The software highlights it in a different color as shown above. It also highlights the legend. You can click it also — it is interactive!
In this case, at the center point, you see that factor A time produces a relatively small effect as it changes from the reference point.
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