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BLե2 2 mathcad/worksheet.xml ( Response surface modelingBox Behnken designs, P. Watté, piet.watté@skynet.beExample of a Box-Behnken experiment with 3 parameters at 3 levels A Box-Behnken experiment is an example of response surface modelling (RSM). It features the economics of experimentation since it allows to reduce a classical 3³ set up, requiring 27 replications to 15 replications. The lineair, quadratic effects and first order interactions can be elegantly estimated, which is in most cases satisfactory.The contrast coefficient matrix is prescribed and given hereafter as matrix X. The design points reside in the middle of the sides of a cube and at the corner points.The formalism outlined in Meyers and Montgomery [1] can be adopted using mathcad's powerful matrix calculation proceduresThe empirical model which is treated here, is a first order response surface model of the form(Eq.1)In order to find the coefficients of this response plane, experiments will be carried out with the settings of the parameters x1 and x2 at the levels given in the contrast matrix XFill here the name of your input parameters:Fill the column Y with the response data. This is the outcome of the experiment with the given settings of x1 and x2 as listed by the contrast table X.Y is the replica matrix; each element in a row contains the outcome of an experiment conducted with the parameter settings out of the corresponding coefficient matrix XEq 1 can be rewritten in matrix form as:We wish to find a vector of least squares estimators that minimizes:The least squares estimators must satisfy:In (1) it is shown that in that case:The response function which enables to generate the 3 dimensional plots is given by:Construction of the response surfacesResponse surface of x versus y at constant zthe value of z may be arbitrarily chosen between -1 and 1111110Response surface of y versus z at constant xthe value of x may be arbitrarily chosen between -1 and 111119Response surface of x versus z at constant ythe value of x may be arbitrarily chosen between -1 and 1111110Significance testing of the coefficients of the response surfaceFollowing calculations enable to find the significance of the various coefficients of the response surfacen is the number of replica's in the central composite design, which is in this case p is the number of coefficients to be estimatedFormer routine was set up in order to generate a matrix, containing three columns.The first column contains the coefficients of the response surface. The second one lists the t-values.For each separate coefficient a t-test is performed in order to verify whether the coefficient is worthwile considering. The third column is the power of the t-test. It yields the a value of the t-test. a is related to the confidence, with which one can state that the parameter under consideration is important. b |t0| power significant?Interpretation of the results tableThe first column contains the coefficients of the response surface.Columns 1 and 2 establish the t value and the subsidiairy power of a t-test revealing the significance of the corresponding coefficientColumn 3 shows whether the coefficient is significant or not on your chosen significance level. The coefficients (lineair, quadratic, or interaction) with the lowest power value are the most important. For optimising purposes, these variables should be controlled.Analysis of the residualslines1111-0.20.21. R. Myers and D. Montgomery, "Response surface modeling and process optimisation", Wiley, (1995)Response surface methodology, Process and product optimization using designed experiments, R.H. Myers and D.C. Montgomery, Wiley, (1995)PK
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