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DOE with Design and Noise Factors Montgomery 12.2 walk through

awelsh

Does anyone know how I could do something similar to what Montgomery describes in 12.2 using JMP? (see attachment)

Experiment Objective: Determine design robustness to noise factors. The desired output would be factor level setting recommendations to design teams that aren't versed in statistics. Essentially looking for plateaus in the response surface. But not design factor plateaus. Areas where the output is consistent while noise factors are changing, so a response surface generated from the noise factor model and design factor models combined.

Experimental Design: Montgomery proposes a customized fractional CCD removing the axial points for the noise factors. That's fine. But for me any type of design where I've got design factors and noise factors. The specific design isn't critical to the discussion. It could be a CCD, it could be a sequence of DOEs from a steepest ascent, Taguchi inner/outer, etc. Just something where I've got combinations of design factors and noise factors manipulated in a 2 or 3 level factorial DOE.

Analysis: Here is where my question is. Two models are proposed in the example. A mean model and a variance model. The mean model is trivial, I can replicate that easily. What he describes as the variance model I have no idea how to do in JMP though. Well I have no idea how he's calculating that at all to be honest.

I'm working in some spaces now where this type of question seems to be coming up and I'd like to add this approach to my knowledge base. DOE with design and noise factors. And the analysis provides insight on areas of design space where it's robust or at least less susceptible to those noises. So the analysis needs to model the effects of the design and the noise and then combine those to indicate areas in the response surface with less variance in the response (areas where noise effects and noiseXdesign interactions are minimized)

Visualizing that space on contour to cube plots is essential. The recipients of the output are not statistically trained. Montgomery creates a POE curve as well, calling it the square root of the variance, I'm assuming that's just to make it relevant to the mean model so they're both in the same units to put on a plot together.

Thanks for any guidance. If there is a different approach you'd recommend over this towards the initial goal, I'd love to hear about that too.

Montgomery Ex 12.2.pdf

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Victor_G

Hi @awelsh,

I quickly read your question, and I think you may find answers in the following JMP Help sections:

For the creation of robust designs involving noise factors : https://www.jmp.com/support/help/en/18.2/#page/jmp/experiments-for-robust-process-and-product-design.shtml
For the analysis of mean and variance response: https://www.jmp.com/support/help/en/18.2/#page/jmp/loglinear-variance-models.shtml#
For the use of Prediction Profiler with a noise factor: https://www.jmp.com/support/help/en/18.2/#page/jmp/example-of-a-noise-factor-in-the-prediction-profiler.shtml

The last link may be the most interesting for you regarding what you're describing: finding factors settings where the response is not affected by small changes of noise factors (or said otherwise, find the optimal settings where the derivative of the response relative to noise factors is close to 0).
I can work on your example only next week, so I hope these ressources will help you get started.

Best,

statman

First I think you need to recognize, this is Doug's "response" to Taguchi's inner/outer array (aka cross-over or coss-product I believe first discussed by Cox, 1958) methodology which received a certain amount of attention in the 80's and 90's. Instead of using a SN ratio as the response, he suggests modeling with the mean and the variance as separate response variables and then overlaying their respective contour plots. I think there are multiple options to handling this situation.

What I like about the methodology, is recognition of the importance of including noise in your experiments. Experimenting on noise using a factorial type design matrix is a great idea. Cross product arrays (originally discussed in Cox should be fundamental experiments for design engineers. The analysis is certainly a personal choice. There is nothing wrong with taking the data from such an experiment and analyzing multiple ways. This is one of the reasons JMP is so useful.

Given the situation of multiple noise factors, you have the following options for experimentation (I'm not discussing optimality criteria, I'm focussing on noise strategies not design factor strategies):

1. Full resolution design including all design and noise factors. This is complicated, expensive and decreases the precision of the design. And you likely don't need this resolution.

2. Fractional factorial with all design and noise factors. Less expensive, but still does not improve the precision.

3. Take all of the noise factors and confound them into 2 blocks. That is set levels for the noise factors and run all of them at level -1 in the first block and the 1 level in the second block. Seems more efficient, but you don't get specificity in noise by factor interactions. Would require subsequent experiments to identify the specific interactions. Better precision.

4. Do #3 in BIB. Less resources, but you confound noise by factor interactions with design factor interactions.

5. Use split-plots (my bias). Where either the noise is in the WP (making the design easier to run), or the noise is in the subplot. With this type of design, you get potentially the most efficient (has the greatest precision in both the WP and SP) and exposes specific noise bye factor interactions.

For more information on this technique, see:

Box, G.E.P., Stephen Jones (1992), “Split-plot designs for robust product experimentation”, Journal of Applied Statistics, Vol. 19, No. 1

6. Fractional split-plots see

Bisgaard, Søren, Murat Kulahei, (2001), “Robust Product Design: Saving Trials with Split-Plot Confounding”, Quality Engineering, 13(3), 525-530

Analyze it in the following ways (in no particular order):

1. The noise matrix is a subplot of a split-plot design and the design factors make up the whole plot. You can also switch the WP and SP roles.

2. Create summary statistics across the noise matrix. These include mean, variance, SN ratio, CV... and analyze each of these summary statistics and ratios as response variables in the design factor matrix.

My critique of the methodology:

1. This method “hides” the ability to assign effects to each noise variable and noise by factor interactions. This is because he summarizes the data across the changing noise matrix (the subplot of a split-plot design). In other words, by summarizing, he literally throws out information. This is also a critique I have of the Taguchi SN ratio. While the experimenter (design engineer) does not really care about the size of the noise factor effects (or their significance), he should care about whether the design factors he does control are consistent over the changing noise variable. If they are, this is the definition of robust (i.e., the absence of noise by factor interactions). If they are, then he should celebrate as he has identified conditions in which the effects of his deign factors change given certain noise conditions. Celebrate because he found them before the customer did. Celebrate because he has way more options to remedy the situation BEFORE the design is released.

2. This method does not simplify the models at all (remove insignificant terms) before creating the prediction formula he uses for creating the contour plots.

3. This method doesn’t do any diagnostics with the data prior to summarizing it. (e.g.., check for outliers)

4. The analysis assumes the Vz (variance of the noise factors) is 1. "where we have substituted parameter estimates from the ﬁtted response model into the equations for the mean and variance models and, as in the previous example, assumed that Sz^2 = 1.

awelsh

I"m going to make a workflow concept map on this topic and report back after reviewing all the references and organizing my thoughts.

I hope to include some example datasets in that as well. I'll need a couple weeks. I think it would be helpful for me as well as students who are asking for more detail on this general idea.

Victor_G

As an inspiration, it's worth mentioning the blog serie of Stat-Ease concerning robust design methodologies :

Design-Type I: Robustness against external noise factors: https://statease.com/blog/achieving-robust-processes-via-three-experiment-design-options-part-1/
Design-Type II: Robustness against variation in our set points for process factors: https://statease.com/blog/achieving-robust-processes-via-three-experiment-design-options-part-2/
Design-Type III: A combination of the first two types: https://statease.com/blog/achieving-robust-processes-via-three-experiment-design-options-part-3/

These ressources may help you map the different design and analysis strategies,

Rily_Maya

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