Analysis of a Combined-Array Design in A Robust Parameter Experiment

This posting presents the analysis of results from the Robust Parameter Design Experiment introduced in the previous posting.  The five factors are A = CO2 pressure (bar), B = CO2 temperature oC, C = peanut moisture (% by wt), D = CO2 flow rate (liters/min), and E = peanut particle size (mm).   The factor E is the noise factor).   The purpose of the experiment is to show the effects of these factors on Solubility, S, or mg of oil removed from the peanuts.  The data for the half-factorial experiment appears below.

A B C D E S
415 25 5 40 1.28 29.2
550 25 5 40 4.05 23.0
415 95 5 40 4.05 37.0
550 95 5 40 1.28 139.7
415 25 15 40 4.05 23.3
550 25 15 40 1.28 38.3
415 95 15 40 1.28 42.6
550 95 15 40 4.05 141.36
415 25 5 60 4.05 22.4
550 25 5 60 1.28 37.2
415 95 5 60 1.28 31.3
550 95 5 60 4.05 48.6
415 25 15 60 1.28 22.9
550 25 15 60 4.05 36.2
415 95 15 60 4.05 33.6
550 95 15 60 1.28 172.6

Minitab estimated the coefficients for a regression model estimating S from a linear sum of terms including a constant, A, B, C, D, E, and all of the two factor interactions.  Minitab used a stepwise procedure where it starts with no significant terms and it adds a term at each step.   A term is introduced based on the p-value of the estimated term coefficient.   Assuming that the term true value of the coefficient is zero, a p-value is the probability of obtaining an estimate as large or larger than the one observed.   In this estimate, the p-value had to be less than or equal to .015 for the term to be added.

The only statistically significant terms were those for the constant, A (pressure), B (temperature), and AB (pressure*temperature).   The estimated regression model is:

Solubility = 80.0 – 0.144 Pressure – 3.36 Temperature + 0.00849 Pressure*Temperature

A surface plot of the regression model appears below, and it shows the effect of the two-factor interaction term.

Solubility Surface Plot

The p-values for the model coefficients are listed in the following table.

Model Coefficient p-value
Constant 0.000
Pressure 0.003
Temperature 0.002
Pressure*Temperature 0.012

This example clearly shows the advantage of a combined array design over the Taguchi crossed array design when two-factor interactions are present among the controllable variables.  As mentioned in the previous posting, the standard Taguchi crossed array design would have the same number of experiments but the estimates of two-factor interactions would be aliased.   If the experimenters were aware of the possibility of significant two factor interactions, they would have to have a larger experiment in order to use a Taguchi crossed array design.

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