I'm trying to learn about industrial designed experiments, i.e., full factorial and Taguchi designs. I understand the ANOVA calculations for determining the significance of effects for these sorts of designs where each factor is studied at only 2 levels. However, I'm unable to find equations that would let me do ANOVA (i.e., calculations and comparisons of mean square between treatment error, MSB, and mean square error, MSE, using the F-distribution) for systems in which factors are studied at 3 or more levels. Can anyone provide these equations or a reference to a readable book with some worked examples using them?
I'm also interested in learning about the method used to derive a model equation that predicts system response as functions of the factors studied for systems analyzed by the Taguchi method. The model equation derived using full factorial designs, at least for systems where each factor is examined at only 2 levels, is straightforward.
I know that I could just buy a computer program that would do all of these things automatically. However, I want to understand why and how the program makes the calculations (to avoid potential GIGO results that can sometimes occur when blindly relying on programs). Thanks.
When you run an analysis on your data, you should not use equations and try to derive coefficients by your self. Use software to do it. If you use something like Design Expert, JMP, SAS, Minitab or Matlab, you can be assured their programmers got it right.
The basic idea behind analysis is to use a pseudo-inverse of the 'X' matrix. The coding method you use will determine the coefficients. When you look at your 'X' matrix, you will have columns for B0, X1, X2,...Xn, X12, X13,...Xmn, (X1)^2, (X2)^2,... Suppose you have 3 categorical levels for X2, X2a, X2b and X2c. One way to code these categorical levels is to create 2 columns in your 'X' matrix, say X2a and X2b. When you have X2c, X2a and X2b have zeros. When you have X2a, you set X2b to zero. When you have X2b, X2a is zero. There are other methods for coding X2a, X2b and X2c. You'll have to look those up.
I would look for methods other than Taguchi and Full Factorial designs for your experiments. If you have several factors to test, you can use Plackett Burman or Definitive Screening Designs to test, then reduce, the number of factors you need to look at and test. Usually, you use a screening design to test all the factors you might think are significant. Then you use the results of this test to perform a further design that is a bit more in depth.
I think, for instance, Montgomery, D.C. 'Design and Analysis of Experiments' includes factorial designs with three factors.
Yes, Montgomery's 'Design and Analysis of Experiments' provides sufficient clues you need. R-package (free and open source) can help you in doing the analysis. R provides many packages regarding to the DOE.
Yes, Montgomery's 'Design and Analysis of Experiments' provides sufficient clues you need. R-package (free and open source) can help you in doing the analysis. R provides many packages regarding to the DOE.
Thank you all. I will get a copy of Montgomery's book and take a close look at it. After I'm sure I understand the math and workings for multilevel designs I'll try using a program to analyze any experimental results. No sense in reinventing the wheel.
Previously, if you are not a frequent user of these techniques you should understand the differences between classical and Taguchi methods to design an experiment and a good reference could be the next work: Jiju Antony, (2006) "Taguchi or classical design of experiments: a perspective from a practitioner", Sensor Review, Vol. 26 Iss: 3, pp.227 - 230. I think i have some calculations made by hand for factors with three and four levels using the Taguchi signal to noise equations and If you send me your email i could scan and send them.
Thanks Prof. Limon. You can send your calculations to me at [email protected] and I will get and read the reference you recommended.
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