Stat-Ease Blog

Hi folks! It was wonderful to meet with so many new prospects and long-standing clients at the Advanced Manufacturing Minneapolis expo last week. One highlight of the show was running our own design of experiments (DOE) in-booth: a test to pinpoint the height and distance of a foam cat launched from our Cat-A-Pult toy. Visitors got to choose a cat and launch it based on our randomized run sheet. We got lots of takers coming in to watch the cats fly, and we even got a visit from local mascot Goldy Gopher!

But I’m getting a bit ahead of myself: this experiment primarily shows off the ease of use and powerful analytical capabilities of Design-Expert® and Stat-Ease® 360 software. I’m no statistician – the last math class I took was in high school, over a decade ago – but even a marketer like me was able to design, run, and analyze a DOE with just a little advice. Here’s how it worked.

Let’s start at the beginning, with the design. My first task was to decide what factors I wanted to test. There were lots of options! The two most obvious were the built-in experimental parts of the toy: the green and orange knobs on either side of the Cat-A-Pult, with spring tension settings from 1 to 5.

However, there were plenty of other places where there could be variation in my ‘pulting system:

• The toy comes with 5 Cat-A-Pults: are there variations between each 'pult?
• Does it matter what kind of surface the Cat-A-Pult is on, e.g., wood, concrete, or carpet?
• The toy came with 5 colors of foam cat; would the pigment change the cat’s weight enough to matter?
• What about where on the plate we apply launch pressure, or how much pressure is applied?

Some of these questions can be answered with subject matter knowledge – in the case of launch pressure, by reading the instruction manual.

For our experiment, the surface question was moot: we had no way to test it, as the convention floor was covered in carpet. We also had no way to test beforehand if there were differences in mass between colors of cat, since we lacked a tool with sufficient precision. I settled on just testing the experimental knobs, but decided to account for some of this variation in other ways. We divided the experiment into blocks based on which specific Cat-A-Pult we were using, and numbered them from 1 to 5. And, while I decided to let people choose their cat color to enhance the fun aspect, we still tracked which color of cat was launched for each run - just in case.

Since my chosen two categoric factors had five levels each, I decided to use the Multilevel Categoric design tool to set up my DOE. One thing I learned from Mark is that these are an “ordinal” type of categoric factor: there is an order to the levels, as opposed to a factor like the color of the cat or the type of flooring (a “nominal” factor). We decided to just test 3 of the 5 Cat-A-Pults, trying to be reasonable about how many folks would want to play with the cats, so we set the design to have 3 replicates separated out into 3 blocks. This would help us identify if there were any differences between the specific Cat-A-Pults.

For my responses, I chose the Cat-A-Pult’s recommended ones: height and distance. My Stat-Ease software then gave me the full, 5x5, 25-run factorial design for this, with a total of 75 runs for the 3 replicates blocked by 'pult, meaning we would test every combination of green knob level and orange knob level on each Cat-A-Pult. More runs means more accurate and precise modeling of our system, and we expected to be able to get 75 folks to stop by and launch a cat.

And so, armed with my run sheet, I set up our booth experiment! I brought two measuring tapes for the launch zone: one laid along the side of it to measure distance, and one hanging from the booth wall to measure height. My measurement process was, shall we say, less than precise: for distance, the tester and I eyeballed the point at which the cat first landed after launch, then drew a line over to our measuring tape. For height, I took a video of the launch, then scrolled back to the frame at the apex of the cat’s arc and once again eyeballed the height measurement next to it. In addition to blocking the specific Cat-A-Pult used, we tracked which color of cat was selected in case that became relevant. (We also had to append A and B to the orange cat after the first orange cat was mistaken for swag!)

Over the course of the conference, we completed 50 runs, getting through the full range of settings for ‘pults 1 and 2. While that’s less than we had hoped, it’s still plenty for a good analysis. I ran the analysis for height, following the steps I learned in our Finding the Vital Settings via Factorial Analysis eLearning module.

The green knob was the only significant effect on Height, but the relatively low Predicted R² model-fit-statistic value of 0.36 tells us that there’s a lot of noise that the model doesn’t explain. Mark directed me to check the coefficients, where we discovered that there was a 5-inch variation in height between the two Cat-A-Pult! That’s a huge difference, considering that our Height response peaked at 27 inches.

With that caveat in mind, we looked at the diagnostic plots and the one-factor plot for Height. The diagnostics all looked fine, but the Least Significant Difference bars showed us something interesting: there didn’t seem to be significant differences between setting the green knob at 1-3, or between settings 4-5, but there was a difference between those two groups.

With this analysis under my belt, I moved on to Distance. This one was a bit trickier, because while both knobs were clearly significant to the model, I wasn’t sure whether or not to include the interaction. I decided to include it because that’s what multifactor DOE is for, as opposed to one-factor-at-a-time experimentation: we’re trying to look for interactions between factors. So once again, I turned to the diagnostics.

Here's where I ran into a complication: our primary diagnostic tools told me there was something off with our data. There’s a clear S-shaped pattern in the Normal Plot of Residuals and the Residuals vs. Predicted graph shows a slight megaphone shape. No transform was recommended according to the Box-Cox plot, but Mark suggested I try a square-root transform anyways to see if we could get more of the data to fit the model. So I did!

Unfortunately, that didn’t fix the issues I saw in the diagnostics. In fact, it revealed that there’s a chance two of our runs were outliers: runs #10 and #26. Mark and I reviewed the process notes for those runs and found that run #10 might have suffered from operator error: he was the one helping our experimenter at the booth while I ran off for lunch, and he reported that he didn’t think he accurately captured the results the way I’d been doing it. With that in mind, I decided to ignore that run when analyzing the data. This didn’t result in a change in the analysis for Height, but it made a large difference when analyzing Distance. The Box-Cox plot recommended a log transform for analyzing Distance, so I applied one. This tightened the p-value for the interaction down to 0.03 and brought the diagnostics more into line with what we expected.

While this interaction plot is a bit trickier to read than the one-factor plot for Height, we can still clearly see that there’s a significant difference between certain sets of setting combinations. It’s obvious that setting the orange knob to 1 keeps the distance significantly lower than other settings, regardless of the green knob’s setting. The orange knob’s setting also seems to matter more as the green knob’s setting increases.

Normally, this is when I’d move on to optimization, and figuring out which setting combinations will let me accurately hit a “sweet spot” every time. However, this is where I stopped. Given the huge amount of variation in height between the two Cat-A-Pults, I’m not confident that any height optimization I do will be accurate. If we’d gotten those last 25 runs with ‘pult #3, I might have had enough data to make a more educated decision; I could set a Cat-A-Pult on the floor and know for certain that the cat would clear the edge of the litterbox when launched! I’ll have to go back to the “lab” and collect more data the next time we’re out at a trade show.

One final note before I bring this story to a close: the instruction manual for the Cat-A-Pult actually tells us what the orange and green knobs are supposed to do. The orange knob controls the release point of the Cat-A-Pult, affecting the trajectory of the cat, and the green knob controls the spring tension, affecting the force with which the cat is launched.

I mentioned this to Mark, and it surprised us both! The intuitive assumption would be that the trajectory knob would primarily affect height, but the results showed that the orange knob’s settings didn’t significantly affect the height of the launch at all. “That,” Mark told me, “is why it’s good to run empirical studies and not assume anything!”

We hope to see you the next time we’re out and about. Our next planned conference is our 8th European DOE User Meeting in Amsterdam, the Netherlands on June 18-20, 2025. Learn more here, and happy experimenting!

As a chemical engineer with roots as an R&D process developer, the appeal of design of experiments (DOE) is its ability to handle multiple factors simultaneously. Traditional scientific methods restrict experimenters to one factor at a time (OFAT), which is inefficient and does not reveal interactions. However, a simple-comparative OFAT often suffices for a process improvement. If this is all that’s needed, you may as well do it right statistically. As industrial-statistical guru George Box reportedly said “DOE is a wonderful comparison machine.”

A fellow named William Sealy Gosset developed the statistical tools for simple-comparative experiments (SCE) in the early 1900s. As Head Experimental Brewer for Guiness in Dublin, he evaluated hops from various regions soft resin content—a critical ingredient for optimizing the bitterness on preserving their beer.1 To compare the results from one source versus another with statistical rigor, Gosset invented the t-test—a great tool for DOE even today (and far easier to do with modern software!).

The t-test simply compares two means relative to the standard deviation of the difference. The result can be easily interpreted with a modicum of knowledge about normal distributions: As t increases beyond 2 standard deviations, the difference becomes more and more significant. Gosset’s breakthrough came by his adjustment of the distribution for small sample sizes, which make the tails on the bell shape curve slightly fatter and the head somewhat lower as shown in Figure 1. The correction, in this case for a test comparing a sample of 4 results for one level versus 4 at the other, is minor but very important to get the statistics right.

To illustrate a simple comparative DOE, consider a case study on the filling of 16-ounce plastic bottles with two production machines—line 1 and line 2.2 The packaging engineers must assess whether they differ. To make this determination, they set up an experiment to randomly select 10 bottles from each machine. Stat-Ease software makes this easy via its Factorial, Randomized, Multilevel Categorical design option as shown by the screen shot in Figure 2.

The resulting volumes in ounces are shown below (mean outcome shown in parentheses).

1. 16.03, 16.04, 16.05, 16.05, 16.02, 16.01, 15.96, 15.98, 16.02, 15.99 (16.02)
2. 16.02, 15.97, 15.96, 16.01, 15.99, 16.03, 16.04, 16.02, 16.01, 16.00 (16.01)

Stat-Ease software translates the mean difference between the two machines (0.01 ounce) a t value of 0.7989, that is, less than one standard deviation apart, which produces a p-value of 0.4347—far above the generally acceptable standard of p<0.05 for significance. Its Model Graph in Figure 3 displays all the raw data, the means of each level and their least significant difference (LSD) bars based on a t test at p of 0.05—notice how they overlap from left to right—clearly the difference is not significant.

Thus, from the stats and at first glance of the effect graph it seems that the packaging engineers need not worry about any differences between the two machine lines. But hold on before jumping to a final conclusion: What if a difference of 0.01 ounce adds up to a big expense over a long period of time? The managers overseeing the annual profit and loss for the filling operation would then be greatly concerned. Before doing any designed experiment, it pays to do a power calculation to work out how many runs are needed to see a minimal difference (signal ‘delta’) of importance relative to the variation (noise ‘sigma). In this case, the power for sample size 10 for a delta of 0.01 ounce with a sigma (standard deviation) of 0.028 ounces (provided by Stat-Ease software) generates a power of only 11.8%—far short of the generally acceptable level of 80%. Further calculations reveal that if this small of a difference really needed to be detected, they should fill 125 or more bottles on each line.

In conclusion, it turns out that simple comparative DOEs are not all that simple to do correctly from a statistical perspective. Some keys to getting these two level OFAT experiments done right are:

• Randomizing the run order (a DOE fundamental for washing out the impact of time-related lurking factors such as steadily increasing temperature or humidity).
• Performing at least 4 runs at each level—more if needed to achieve adequate power (always calculate this before pressing ahead!).
• Blocking out know sources of variation via a paired t-test,3 e.g., when assessing two runners, rather than them each running a number of time trials one after the other, race them together side-by-side, thus eliminating the impact of changing wind and other environmental conditions.
• Always deploying a non-directional two-tailed t-test4 (a fun alliteration!)—as done by default in Stat-Ease software; the option for a one-tailed t-test requires an assumption that one level of the tested factor will certainly be superior to the other (i.e., directional), which may produce false-positive significance; before going this route consult with our StatHelp consulting team.

Footnotes

1. For more background on Gosset and his work for Guiness, see my 8/9/24 StatsMadeEasy blog on The secret sauce in Guinness beer?
2. From Chapter 2, “Simple Comparative Experiments”, problem 2.24, Design and Analysis of Experiments, 8th Edition, Douglas C. Montgomery, John Wiley and Sons, New York, NY, 2013.
3. “Letter to a Young Statistician: On ‘Student’ and the Lanarkshire Milk Experiment”, Chance Magazine: Volume 37, No. 1, Stephen T. Ziliak.
4. Wikipedia, One- and two-tailed tests.

Other resources made freely available by Stat-Ease

• One Factor tutorials in Program Help.
• Stat-Ease Academy eLearning PreDOE course, which includes a t-statistic software tutorial.

In a previous Stat-Ease blog, my colleague Shari Kraber provided insights into Improving Your Predictive Model via a Response Transformation. She highlighted the most commonly used transformation: the log. As a follow up to this article, let’s delve into another transformation: the square root, which deals nicely with count data such as imperfections. Counts follow the Poisson distribution, where the standard deviation is a function of the mean. This is not normal, which can invalidate ordinary-least-square (OLS) regression analysis. An alternative modeling tool, called Poisson regression (PR) provides a more precise way to deal with count data. However, to keep it simple statistically (KISS), I prefer the better-known methods of OLS with application of the square root transformation as a work-around.

When Stat-Ease software first introduced PR, I gave it a go via a design of experiment (DOE) on making microwave popcorn. In prior DOEs on this tasty treat I worked at reducing the weight of off-putting unpopped kernels (UPKs). However, I became a victim of my own success by reducing UPKs to a point where my kitchen scale could not provide adequate precision.

With the tools of PR in hand, I shifted my focus to a count of the UPKs to test out a new cell-phone app called Popcorn Expert. It listens to the “pops” and via the “latest machine learning achievements” signals users to turn off their microwave at the ideal moment that maximizes yield before they burn their snack. I set up a DOE to compare this app against two optional popcorn settings on my General Electric Spacemaker™ microwave: standard (“GE”) and extended (“GE++”). As an additional factor, I looked at preheating the microwave with a glass of water for 1 minute—widely publicized on the internet to be the secret to success.

Table 1 lays out my results from a replicated full factorial of the six combinations done in random order (shown in parentheses). Due to a few mistakes following the software’s plan (oops!), I added a few more runs along the way, increasing the number from 12 to 14. All of the popcorn produced tasted great, but as you can see, the yield varied severalfold.

I then analyzed the results via OLS with and without a square root transformation, and then advanced to the more sophisticated Poisson regression. In this case, PR prevailed: It revealed an interaction, displayed in Figure 1, that did not emerge from the OLS models.

Going to the extended popcorn timing (GE++) on my Spacemaker makes time-wasting preheating unnecessary—actually producing a significant reduction in UPKs. Good to know!

By the way, the app worked very well, but my results showed that I do not need my cell phone to maximize the yield of tasty popcorn.

To succeed in experiments on counts, they must be:

• discrete whole numbers with no upper bound
• kept with within over a fixed area of opportunity
• not be zero very often—avoid this by setting your area of opportunity (sample size) large enough to gather 20 counts or more per run on average.

For more details on the various approaches I’ve outlined above, view my presentation on Making the Most from Measuring Counts at the Stat-Ease YouTube Channel.

My colleague Richard Williams just completed a very thorough three-part series of blogs detailing experiment designs aimed at building robustness against external noise factors, internal process variation, and combinations of both. In this follow up, I present another, more simplistic, approach to achieve on target results with minimal variation: Model not only the mean outcome but also the standard deviation. Experimenters making multiple measurements for every run in their design often overlook this opportunity.

For example, consider the paper helicopter experiment done by students of my annual DOE class at South Dakota Mines. The performance of these flying machines depends on paper weight, wing and body dimensions and other easily controlled factors such as putting on a paper clip to stabilize rotation. To dampen down variability in launching and air currents, students are strongly encouraged to drop each of their ‘copters three times and model the means of the flight time and distance from target. I also urge them to analyze the standard deviations of these two measures. Those who do discover that ‘copters without paper clips exhibit significantly higher variability in on-target landings. This can be seen in the interaction plot pictured, which came from a split plot factorial on paper helicopters done by me and colleagues at Stat-Ease (see this detailed here).

Putting on a paper clip dramatically decreased the standard deviation of distance from target for wide bodied ‘copters, but not for narrow bodied ones. Good to know!

When optimizing manufacturing processes via response surface methods, measuring variability as well as the mean response can provide valuable insights. For example, see this paper by me and Pat Whitcomb on Response Surface Methods (RSM) for Peak Process Performance at the Most Robust Operating Conditions for more details. The variability within the sample collection should represent the long-term variability of the process. As few as three per experimental run may be needed with the proper spacing.

By simply capturing the standard deviation, experimenters become enabled to deal with unknown external sources of variation. If the design is an RSM, this does not preempt them from also applying propagation of error (POE) to minimize internal variation transmitted to responses from poorly controlled process factors. However, to provide the greatest assurance for a robust operating system, take one of the more proactive approaches suggested by Richard.

The goal of robustness studies is to demonstrate that our processes will be successful upon implementation in the field when they are exposed to anticipated noise factors. There are several assumptions and underlying concepts that need to be understood when setting out to conduct a robustness study. Carefully considering these principles, three distinct types of designs emerge that address robustness:

I. Having settled on process settings, we desire to demonstrate the system is insensitive to external noise-factor variation, i.e., robust against Z factor influence.

II. Given we may have variation between our selected process settings and the actual factor conditions that may be seen in the field, we wish to find settings that are insensitive to this variation. In other words, we may set our controlled X factors, but these factors wander from their set points and cause variation in our results. Our goal is to achieve our desired Y values while minimizing the variation stemming from imperfect process factor control. The impact of external noise factors (Z’s) is not explored in this type of study.

III. Given a system having controllable factors (X’s) and non-controllable noise factors (Z’s) impacting one or more desirable properties (Y’s), we wish to find the ideal settings for the controllable factors that simultaneously maximize the properties of interest, while minimizing the impact of variation from both types of noise factors.

Read part 1 here.

Read part 2 here.

Design-Type III: A combination of the first two types

The idea here is to simultaneously involve the process factors (X’s) and the noise factors (Z’s) in the same DOE so as to identify the right process (controllable) factor settings to deliver the intended responses (Y’s) with the minimal variation when both the process and noise factors vary.

One of the first to consider this holistic approach was Taguchi in the 1980’s. Taguchi envisioned a factorial space whereby the controllable factors are changed in the usual way, and the noise factors are studied at each corner of the factorial space as a secondary factorial design. In his vocabulary, there was an inner array (of controllable factors) and an outer array (of noise factors). The design principle was as shown below, with 16 data points collected as indicated in blue (for 2 process factors, and two noise factors).

The principle of Taguchi’s approach is sound. There were, however, challenges made regarding the analytical approach and that led to further efforts by others to advance the science. Several concepts have been proposed. A solid candidate would be a dual response surface approach, where process factors and noise factors are combined in the same study, and two responses are measured: the process mean (predicted Y values), and also the variance for the predicted Y values at any given point within the design space. Armed with this knowledge the experimenter can seek regions within the design space where the desired Y values are achieved but are also relatively insensitive to the variation of both process and noise factors.

How are these dual response surface studies done? Essentially the same as described before under Type II Robustness studies, with one key exception. The external noise factors are included as factors within the study, and their influence evaluated as though they were controllable factors (which they are, of course, during the DOE itself). And the propagated error from all factors upon the responses are evaluated as per Type II.

The difference comes during the numeric optimization. Since Z factors cannot be controlled in the field, these factors are set to their nominal value (usually the center of the range studied) in numeric optimization. The standard deviation for these factors will still influence the POE assessment for responses.

Then, the experimenter can use numeric optimization to achieve the desirable Y value criteria while simultaneously minimizing the POE response, resulting in the identification of a robust region that is relative insensitive to variation in X and Z factor variation.

For additional information on using a combined array, and an introduction to using POE to address both process factor and noise factor variation, read the 2002 white paper by Mark Anderson and Shari Kraber on Cost-Effective and Information-Efficient Robust Design For Optimizing Processes And Accomplishing Six Sigma Objectives.

Read a follow-up by Mark Anderson here.