In some applications, the only measurement type available is a pass/fail (binomial) measurement. In this example, two factors are of interest, X1 and X2, which you will vary between -1 and 1. You will construct a nonlinear design for the binomial response and then view it in the context of your proposed nonlinear model.
Model the probability of success for your binomial response (Y) using a logistic model:
This model is nonlinear in the unknown parameters β0, β1, and β2. Your goal is to estimate these parameters using an experimental design.
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β0 is 0, but might range from -2 to 2
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β1 is 5, but might range from 0 to 10
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β2 is 5, but might range from 0 to 10
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To construct a nonlinear design, you must first have a data table containing columns for the predictors and a column containing a formula that represents the nonlinear model that you are fitting. The Binomial Optimal Start.jmp data table, found in the Design Experiment folder, contains the following:
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Columns X1 and X2 for the two predictors. The Coding property defined for each of these columns causes the initial factor settings to be -1 and 1.
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A column called Logistic Model that contains a formula relating the predictors to the response. To view the formula, click on the plus sign to the right of Logistic Model in the Columns panel. See Formula Relating Predictors to Binomial Probability.
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Your initial guesses for the parameters b0, b1, and b2. When you defined these parameters, you were asked to specify a value. You set this value to your initial guess. These values are shown in the formula element panel at the top left of the formula editor window. See Formula Relating Predictors to Binomial Probability.
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A column called Variance that contains the formula for the variance of the predicted value based on the assumed logistic model. When you construct your design, this column indicates which design points have comparatively high variances.
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Select DOE > Special Purpose > Nonlinear Design.
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Click OK.
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Enter the following under Values for the three parameters:
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b1: 0 and 10
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Click Make Design.
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Click Augment Table.
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This adds the 14 runs to Binomial Optimal Start.jmp. Your design table will be different because the optimization algorithm has a random component.
Now that you have constructed your design, proceed to examine where the design points are located relative to the proposed logistic model. The Variance column gives the prediction variance at each design point, based on the logistic model.
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Click Done.
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Because your design differs from the one in Augmentation of Binomial Optimal Start.jmp, your plot will differ from the one in Design Settings.
Notice that there are no points at X1 = -1. The only point on a corner of the design region corresponds to X1 = 1 (more precisely, 0.996) and X2 = -1. There are several points in the central part of the design region.
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Select Graph > Surface Plot.
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Click OK.
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In the Dependent Variables outline, locate Logistic Model under Formula. In the Point Response Column Style list, click on none and select Logistic Model.
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Right-click in the plot and select Settings.
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Drag the Marker Size indicator to the right.
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Click Done.
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Because your design differs from the one in Augmentation of Binomial Optimal Start.jmp, your plot will differ from the one in Prediction Model with Design Points.
