Note

Screenshots may differ slightly depending on software version.

One-Factor RSM (pt 2)

(Part 2 – Advanced topics)

If you don’t have it open from the previous tutorial, load the Drive Time data using the Help, Tutorial Data menu and selecting Drive Time from the list.

The wavy curve we saw in the last tutorial on the one-factor plot is characteristic of a third-order (cubic) polynomial model. Could an even higher-order model be applied to the data from this case? If so, would it improve the fit? Under the Information header on the left, click the Evaluation node.

Change the Order to Quartic or double-click the term A4 to put it in the model.

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Model changed to quartic (4th order)

Click Results to see the evaluation of this higher-order model. Check the Alias Matrix tab to see that there are no aliases for this design. Then click the Leverage tab.

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Leverage

Note the design point with the unusually high leverage of 0.9743. This is the departure time near 50 minutes due to Mark oversleeping, causing a ‘botched’ factor setting. It should not be surprising to see this stand out so poorly for leverage.

Move ahead to the Graphs to see the plot of FDS – fraction of design space. Click the curve of standard error at a fraction near 0.8 (80 percent) to generate cross-reference lines like those shown in the screen shot below.

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Fraction of Design Space

Note

The FDS plot is a line graph showing the relationship between the “volume” of the design space (area of interest) and amount of prediction error. The curve indicates what fraction (percentage) of the design space has a given prediction error or lower. In general, a lower and flatter FDS curve is better. The FDS graph provides very helpful information on scaled prediction variance (SPV) for comparing alternative test matrices – simple enough that even non-statisticians can see differences at a glance, and versatile for any type of experiment – mixture, process, or combined. For example, one could rerun the FDS graph for the cubic model and compare results and/or try some other experiment designs.

Let’s not belabor the evaluation. In the navigation pane, click Analysis [+] to create a new analysis for the Drive time response. In the Analysis name field, change the name to “Drive time - Quartic” to differentiate it from the previous cubic analysis. Then click Start Analysis.

Continue to the Model tab, as the Fit Summary has not changed. Select Quartic from the Process Order: dropdown. Now click on the ANOVA tab. Notice that not only does the A4 term come out insignificant (p-value of 0.91), but the Predicted R2 goes negative – not a good sign!

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ANOVA for quartic model

Propagation of Error (POE)

Seeing such a rapid increase in drive time predicted for late departures makes Mark more aware of how much the response depends on what time he leaves home. He realizes that a 5-minute deviation one way or the other would not be an unreasonable expectation. How will this cause the drive time to vary? Perhaps by aiming for a specific departure time, Mark might reduce drive-time variation caused by day-to-day differences when he leaves for work. Via its capability to calculate and plot propagation of error (POE), Design-Expert can provide enlightenment on these issues.

Click the Design node to bring up the run-sheet (recipe procedure) for the experiment. Then right-click the column-header for Factor 1 (A:Departure) and select Edit Info…. In the Std dev field, enter 5.

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Entering standard deviation for the factor

Press OK and click on the Drive time node under Analysis in the navigation pane. This time, go directly to the Model Graphs tab.

Then from the View menu select Propagation of Error.

../../_images/poe-plot.png

Plot for POE

Notice that POE is minimized at two times for departure, which correspond with flats on the wavy response plot you looked at earlier.

Multiple Response Optimization

Ideally, Mark would like to leave as late as possible (to get more sleep every morning!) while minimizing his drive time – but making it the least variable. These goals can be established in Design-Expert software so it can look for the most desirable outcomes.

Under Optimization in the navigation pane, choose the Numerical node. For Departure, which comes up by default, click Goal and select maximize.

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Setting goal for Departure

The program pictures this goal as an upward ramp to indicate that the higher this variable goes the more desirable it becomes.

Next, click the response for Drive time. For this response we have two different analyses we could set a goal for. Since we determined that the quartic model was not a good fit, choose the first option (“Drive time”). For its goal select minimize.

../../_images/drive-time-goal.png

Drive time minimized

Notice the ramp now goes downward to show that for this variable, lesser is better, that is, more desirable.

Lastly, to reduce variation in drive time caused by deviation in departure, click POE (Drive time) and set its goal to minimize.

../../_images/poe-goal.png

Minimizing POE

Press the Solutions tab to see in “ramps” view what Design-Expert recommends for the most desirable departure. There is a degree of randomness to the results due to the optimization algorithm used, but in this case, roughly the same point is found every time – a departure around 33 minutes beyond the earliest start acceptable by Mark for his morning commute.

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Ramps view of the most desirable solution

Now Mark knows when it’s best to leave for work while simultaneously maximizing the departure (and gaining more ‘shut-eye’), minimizing his drive time, and minimizing propagation of error. The only thing that could possibly go wrong would be if all the other commuters learn how to use RSM and make use of Design-Expert. Mark hopes that none of you who are reading this tutorial live in his neighborhood and work downtown.