Response Surface Design DoE for Min & Maximum Settings
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The way you would determine the minimum and maximum settings shown on page 28 would be to take the regression models for collapse and burst pressure and put them in an optimizing program and run them. I don’t know Minitab but I do know it has such an option.
The problem is the author has not provided the models he generated using the data. Since he did provide the raw data for Case 2 I’m assuming he expects you to use that data to generate models for collapse and burst pressure (assuming you wish to check the validity of his slide)….and thereby hangs a tale….
Since you have asked and since you have indicated in your post you have been trying to confirm everything in his presentation, I went ahead and made up an Excel file of the data on the last slide and had a go at model building.
A test of the data matrix using VIF’s (Variance Inflation Factors) and condition indices indicates the design will support a model consisting of all main effects, all curvilinear effects, and all two way interactions.
Given the differences in the magnitudes of the three variables of interest it is best if, before you start running regressions, you take the time to scale pressure, weld time, and amplitude to the ranges of -1 to 1. The reason you want to do this is because significant parameter selection can be influenced by parameter magnitude and all you really want is for parameter selection to be influenced only by the degree of correlation.***
To this end you would compute the following for each of the X variables
A = (Max X + Min X)/2
B = (Max X – Min X)/2
Scaled X = (X – A)/B
In order to identify the reduced model (the regression model containing only significant terms – in this case terms with P < .05) you will want to run both backward elimination and stepwise (forward selection with replacement) regression on the design/response matrix. The reason you want to do this is because you want to make sure both methods converge to the same reduced model.
Once you have the reduced model in terms of the scaled X’s you will take these terms and re-run the reduced model using the raw X’s so your final model will have coefficients which are associated with actual X values. If you do all of this what you get are the following models:
Predicted Collapse = pcollapse = -0.01985 + 0.00011785*Amplitude + 0.00138*Pressure + 0.05207*Weld_T -0.00001279*Pressure*Pressure
Predicted Burst = pburstp = -1715.33689 + 10.20952*Amplitude + 69.08662*Pressure + 3832.92683*Weld_T -0.96918*Pressure*Pressure
The problem is, when you plug in the optimal values for Amplitude, Pressure, and Weld Time, as indicated in the presentation, you do not get the indicated predicted values for Collapse and Burst Pressure.
As check I built the full model just for burst pressure
fmburstp = 1718.56889 -20.45806*Amplitude + 21.56008*Pressure -10670*Weld_T + 0.18066*amplitude*amplitude
-0.92215*Pressure*Pressure + 18780*weld_t*weld_t + 0.06500*amplitude*pressure + 15.00000*amplitude*weld_t
+ 162.50000*pressure*weld_t
The predicted results for the full model for burst pressure are quite close to the values the author reports but they are not exact.
(pressure, weld time, amplitude) = (18, 0.26, 72) which gives (pcollapse, pburstp, fmburstp) = (0.022869, 945.85, 892.30)
(pressure, weld time, amplitude) = (22, 0.29, 78) which gives (pcollapse, pburstp, fmburstp) = (0.028612, 1243.38, 1222.66)
My guess is the author did one or both of the following:
1. He built the full model and did not run the proper regression analysis to identify the reduced models.
2. He rounded off the values for the optimum pressure, weld time, and amplitude when he wrote the report.
If he did use the full model then his analysis is wrong. The whole point of a design is to identify the significant parameters and to use the resulting model (once it is verified) for purposes of prediction and control.
If he rounded off the predicted optimum values then the best predictions you can generate with those values will not give you the Y values shown in his presentation.
*** If, in this instance, we run backward elimination and stepwise regression on the raw X values the two methods do not converge to the same model.
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