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Response Surface Design DoE for Min & Maximum Settings

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  • #249786

    howardp
    Participant

    Hi all,

    I am new to response surface DoE  and would like to seek guidance in utilizing the response optimizer.

    Using a case study I found online (pls refer to slide 23 – 32, 36 of attached pdf), I managed to replicate slides 23 – 27 but was unable to do so for slide 28.

    Question: How do I determine the minimum & maximum settings shown in slide 28?

    Pls find attached the case study, my Minitab files and excel data for your reference.

    Attachments:
    1. processcharacterization-goodrich-160601193124.pdf
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    2. casestudy2.xlsx
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    3. CaseStudy2.mpj
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    #249838

    Robert Butler
    Participant

    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.

    • This reply was modified 2 months, 3 weeks ago by Robert Butler. Reason: typo
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    #249852

    Robert Butler
    Participant

    ….an additional thought.  I was going over the case study and looking at the data set for Case #2 and it occurred to me that, if you want to, you can use that data set as a starting point for what would amount to an excellent self teaching exercise.

    In the attachment I’ve rearranged the data to make the composite nature of the design more apparent.  The rows in yellow are the center point replicates and the rows in light blue/gray are the star points.

    From my earlier post you know what the expressions for the reduced models look like and you know how to go about generating the full models.  So now you are in a position to run a lot of what-if scenarios.  For example:

    1. See what happens to the model coefficients when you reduce the number of center points to just 2.

    2. See what happens to the model form when you fractionate the full 2 level factorial design.

    3. See what happens when you randomly drop one or more of the factorial experiments (this happens all the time in real life – one or more of the experiments won’t run, can’t be run, was not run at the levels indicated, etc.) and see what impact that has on the final model, on the coefficients of the remaining model terms, and how severe the loss has to be (how many horses have to die) before backward elimination and stepwise regression converge to different models.

    4. In every one of the above situations you would want to run a complete regression analysis which would mean, among other things, a thorough graphical assessment of the residuals – I would recommend you borrow Chatterjee and Price’s book Regression Analysis by Example to guide you with respect to the world of regression analysis.

    If you try these things and do some reading in the recommended book you should be able to gain a good understanding of DOE,

    Switching subjects.

    In the presentation you provided the author indicated the minimum and maximum values for the factorial part of the design were going to be coded as -1 and 1.  In real life it is rare to have a situation where, for each experiment in the design, you run at exactly the stated minimum and maximum.  What this could mean is the following:  it may be the situation that the data for Case 2 consists of the measurements of the output of each experiment but only records the ideal settings of the X variables in the design.  If this is the case then that would be another reason for my inability to exactly match the author’s predicted settings for minimum and maximum response output.

    Attachments:
    1. Goodrich1.xlsx
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    #249869

    Fausto Galetto
    Participant

    @rbutler

    Fantastic Idea Robert (in the file)

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    #249938

    Fausto Galetto
    Participant

    The TWO responses cannot be analysed AS THOUGH they were independent.

    MANOVA must be used

     

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    #250087

    Fausto Galetto
    Participant

    Using MANOVA you will find that some interactions are NOT significant, while they are significant fo Y1, analysing separately

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    #250113

    Fausto Galetto
    Participant

    Using MANOVA the quadratic effects are significant for both the responses …

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