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.
"All models are wrong, some are useful" G.E.P. Box
