Stat-Ease Blog

Quite often, when providing statistical help for Stat-Ease software users, our consulting team sees an over-selection of effects from two-level factorial experiments. Generally, the line gets crossed when picking three-factor interactions (3FI), as I documented in the lead article for the June 2007 Stat-Teaser. In this case, the experimenter picked all the estimable effects when only one main effect (factor B) really stood out on the Pareto plot. Check it out!

In my experience, the true 3FIs emerge only when one of the variables is categorical with a very strong contrast. For example, early in my career as an R&D chemical engineer with General Mills, I developed a continuous process for hydrogenating a vegetable oil. By cranking up the pressure and temperature and using an expensive, noble-metal catalyst (palladium on a fixed bed of carbon), this new approach increased the throughput tremendously over the old batch process, which deployed powered nickel to facilitate the reaction. When setting up my factorial experiment, our engineering team knew better than to make the type of reactor one of the inputs, because being so different, this would generate many complications of time-temperature interactions differing from on process to the other. In cases like this, you are far better off doing separate optimizations and then seeing which process wins out in the end. (Unfortunately for me, I lost this battle due to the color bodies in the oil poisoning my costly catalyst.)

A response must really behave radically to require a 3FI for modeling as illustrated hypothetically in Figures 1 versus 2 for two factors—catalyst level (B) and temperature (D)—as a function of a third variable (E)—the atmosphere in the reactor.

Figures 1 & 2: 3FI (BDE) surface with atmosphere of nitrogen vs air (Factor E at low & high levels)

These surfaces ‘flip-flop’ completely like a bird in flight. Although factor E being categorical does lead to a strong possibility of complex behavior from this experiment, the dramatic shift caused by it changing from one level to the other would be highly unusual by my reckoning.

It turns out that there is a middle ground with factorial models that obviates the need for third-order terms: Multiple two-factor interactions (2FIs) that share common factors. The actual predictive model, derived from a case study we present in our Modern DOE for Process Optimization workshop, is:

Yield = 63.38 + 9.88*B + 5.25*D − 3.00*E + 6.75*BD − 5.38*DE

Notice that this equation features two 2FIs, BD and DE, that share a common factor (D). This causes the dynamic behavior shown in Figures 3 and 4 without the need for 3FI terms.

Figure 3 & 4: 2FI surface (BD) for atmosphere of nitrogen vs air (Factor E at low & high levels)

This simpler model sufficed to see that it would be best to blanket the batch reactor with nitrogen, that is, do not leave the hatch open to the air—a happy ending.

Conclusion

If it seems from graphical or other methods of effect selection that 3FI(s) should be included in your factorial model, be on guard for:

• Over-selection of effects (my first case)
• The need for a transformation (such as log): Be sure to check the Box-Cox plot (always!).
• Outlier(s) in your response (look over the diagnostic plots, especially the residual versus run).
• A combination of these and other issues—ask stathelp@statease.com for guidance if you use Stat-Ease software (send in the file, please).

I never say “never”, so if you really do find a 3FI, get back to me directly.

-Mark (mark@statease.com)

As a Stat-Ease statistical consultant, I am often asked, “What are the green triangles (Christmas trees) on my half-normal plot of effects?”

Factorial design analysis utilizes a half-normal probability plot to identify the largest effects to model, leaving the remaining small effects to provide an error estimate. Green triangles appear when you have included replicates in the design, often at the center point. Unlike the orange and blue squares, which are factor effect estimates, the green triangles are noise effect estimates, or “pure error”. The green triangles represent the amount of variation in the replicates, with the number of triangles corresponding to the degrees of freedom (df) from the replicates. For example, five center points would have four df, hence four triangles appear. The triangles are positioned within the factor effects to reflect the relative size of the noise effect. Ideally, the green triangles will land in the lower left corner, near zero. (See Figure 1). In this position, they are combined with the smallest (insignificant) effects and help position the red line. Factor effects that jump off that line to the right are most likely significant. Consider the triangles as an extra piece of information that increases your ability to find significant effects.

Once in a while we encounter an effects plot that looks like Figure 2. “What does it mean when the green triangles are out of place - on the upper right side instead of the lower left?”

This indicates that the variation between the replicates is greater than the largest factor effects! Since this error is part of the normal process variation, you cannot say that any of the factor effects are statistically significant. At this point you should first check the replicate data to make sure it was both measured and recorded correctly. Then, carefully consider the sources of process variation to determine how the variation could be reduced. For a situation like this, either reduce the noise or increase the factor ranges. This generates larger signals that allow you to discover the significant effects.

- Shari Kraber

For statistical details, read “Use of Replication in Almost Unreplicated Factorials” by Larntz and Whitcomb.

For more frequently asked questions, sign up for Mark’s bi-monthly e-mail, The DOE FAQ Alert.

Keep your experiment planned, but random

When I wrote my introduction to factorial design (Greg’s DOE Adventure - Factorial Design, Part 1), there were a couple of points that I left out. I’ll amend that post here to talk about making sure your experiment is planned out yet random.

Wait. What?

You’ll see. Let me explain.

Getting organized

During the initial phase of an experiment, you should make sure that it is well planned out. First, think about the factors that affect the outcome of your experiment. You want to create a list that’s as all-encompassing as possible. Anything that may change the outcome, put on your list. Then pare it down into the ones that you know are going to be the biggest contributors.

Once you have done that, you can set the levels at which to run each factor. You want the low and high levels to be as far apart as possible. Not too low that you won’t see an effect (if your experiment is cooking something, don’t set the temperature so low that nothing happens). Not too high that it’s dangerous (as in cooking, you don’t want to burn your product).

Finally, you want to make sure your experiment is balanced when it comes to the factors in your experiment. Taking the cooking example above a little further, suppose you have three factors you are testing: time, temperature, and ingredient quality. Let’s also say that you are testing at two different levels: low and high (symbolized by minus and plus signs, respectively). We can write this out in a table:

This table contains all the possible combinations of the three factors. It’s called an ‘orthogonal array’ because it’s balanced. Each column has the same number of pluses and minuses (4 in this case). This balance in the array allows all factors to be uncorrelated and independent from each other.

With these steps, you have ensured that your experiment is well planned out and balanced when looking at your factors.

Always randomize

At the start of this post, I said that an experiment should be planned out, yet random. Well we have the planned-out part, now let’s get into the random part.

In any experimentation, influence from external sources (variables you are not studying) should be kept to a minimum. One way to do this is randomizing your runs.

As an example, let’s look at the table above with the cooking example. Let’s say that it represents the order of how the experiment was run. So, all the low temperature runs were made together and then all the high ones together. This makes sense, right? Perform all the runs at one temperature before adjusting up to the next setting.

The problem is, what if there is an issue with your oven that causes the temperature to fluctuate more, early in the experiment and less later. This time-related issue introduces variation (bias) into your results that you didn’t know about.

To reduce the influence of this variable, randomize your run order. It may take more time adjusting your oven for every run, but it will remove that unwanted variation.

Temperature is a popular example to illustrate randomization. But this can be said of any factor that may have time-related problems. It could be warm-up time on a machine or the physical tiring of an operator. Randomization is used to guard against bias as much as you can when running an experiment.

Conclusions

Hopefully, you see now why I said to keep your experiments planned but random. It sounds like an oxymoron, but it’s not. Not in the way I’m talking about it here!

So, I’ve gotten thru some of the basics (Greg's DOE Adventure: Important Statistical Concepts behind DOE and Greg’s DOE Adventure - Simple Comparisons). These are the ‘building blocks’ of design of experiments (DOE). However, I haven’t explored actual DOE. I start today with factorial design.

In this case, the horizontal line (x-axis) is time and vertical line (y-axis) is temperature. The area in the box formed is called the Experimental Space. Each corner of this box is labeled as follows:

1 – low time, low temperature (resulting in crunchy, matchstick-like pasta), which can be coded minus-minus (-,-)

2 – high time, low temperature (+,-)

3 – low time, high temperature (-,+)

4 – high time, high temperature (making a mushy mass of nasty) (+,+)

One takeaway at this point is that when a test is run at each point above, we have 2 results for each level of each factor (i.e. 2 tests at low time, 2 tests at high time). In factorial design, the estimates of the effects (that the factors have on the results) is based on the average of these two points; increasing the statistical power of the experiment.

If we look at the same experiment from the perspective of altering just one factor at a time (OFAT), things change a bit. In OFAT, we start at point #1 (low time, low temp) just like in the Factorial model we just outlined (illustrated below).

Here, we go from point #1 to #2 by lengthening the time in the water. Then we would go from #1 to #3 by changing the temperature. See how this limits the number of data points we have? To get the same power as the Factorial design, the experimenter will have to make 6 different tests (2 runs at each point) in order to get the same power in the experiment.

After seeing these results of Factorial Design vs OFAT, you may be wondering why OFAT is still used. First of all, OFAT is what we are taught from a young age in most science classes. It’s easy for us, as humans, to comprehend. When multiple factors are changed at the same time, we don’t process that information too well. The advantage these days is that we live in a computerized world. A computer running software like Design-Expert®, can break it all down by doing the math for us and helping us visualize the results.

Additionally, with the factorial design, because we have results from all 4 corners of the design space, we have a good idea what is happening in the upper right-hand area of the map. This allows us to look for interactions between factors.

That is my introduction to Factorial Design. I will be looking at more of the statistical end of this method in the next post or two. I’ll try to dive in a little deeper to get a better understanding of the method.

As I learn about design of experiments, it’s natural to start with simple concepts; such as an experiment where one of the inputs is changed to see if the output changes. That seems simple enough.

For example, let’s say you know from historical data that if 100 children brush their teeth with a certain toothpaste for six months, 10 will have cavities. What happens when you change the toothpaste? Does the number with cavities go up, down, or stay the same? That is a simple comparative experiment.

“Well then,” I say, “if you change the toothpaste and 6 months later 9 children have cavities, then that’s an improvement.”

Not so fast, I’m told. I’ve already forgotten about that thing called variability that I defined in my last post. Great.

In that first example, where 10 kids got cavities. That result comes from that particular sample of 100 kids. A different sample of 100 kids may produce an outcome of 9, other times it’s 11. There is some variability in there. It’s not 10 every time.

[Note: You can and should remove as much variability as you can. Make sure the children brush their teeth twice a day. Make sure it’s for exactly 2 minutes each time. But there is still going to be some variation in the experiment. Some kids are just more prone to cavities than others.]

How do you know when your observed differences are due to the changes to the inputs, and not from the variation?

It’s called the F-Test.

I’ve seen it written as:

Where:

s = standard deviation

s2 = variance

n = sample size

y = response

ӯ (“y bar”) = average response

In essence, this is the amount of variance for individual observations in the new experiment (multiplied by the number of observations) divided by the total variation in the experiment.

Now that, by itself, does not mean much to me (see disclaimer above!). But I was told to think of it as the ratio of signal to noise. The top part of that equation is the amount of signal you are getting from the new condition; it’s the amount of change you are seeing from the established mean with the change you made (new toothpaste). The bottom part is the total variation you see in all your data. So, the way I’m interpreting this F-Test is (again, see disclaimer above): measuring the amount of change you see versus the amount of change that is naturally there.

If that ratio is equal to 1, more than likely there is no difference between the two. In our example, changing the toothpaste probably makes no difference in the number of cavities.

As the F-value goes up, then we start to see differences that can likely be credited to the new toothpaste. The higher the value of the F-test, the less likely it is that we are seeing that difference by chance and the more likely it is due to the change in the input (toothpaste).

Trivia

Question: Why is this thing called an F-Test?

Answer: It is named after Sir Ronald Fisher. He was a geneticist who developed this test while working on some agricultural experiments. “Gee Greg, what kind of experiments?”. He was looking at how different kinds of manure effected the growth of potatoes. Yup. How “Peculiar Poop Promotes Potato Plants”. At least that would have been my title for the research.