Explore Retaining a Factor in Generalized Regression

In this example, a pharmaceutical manufacturer has historical information on the dissolution of a tablet inside the body and various factors that may affect the dissolution rate. A tablet with a dissolution rate below 70 is considered defective. You want to understand which factors affect dissolution rate.

In this example, you will do the following:

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Construct a generalized regression model.

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Fit a reduced model using the non-zeroed terms.

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Based on the reduced model, use simulation to explore the likelihood that one of the factors is included in the model.

Fit the Model

In this section, you fit a model using generalized regression. If you prefer not to work through the steps in this section, click the green triangle next to the Generalized Regression script in the Tablet Production.jmp data table to obtain the model.

Select Help > Sample Data Library and open Tablet Production.jmp.

Select Analyze > Fit Model.

Click Dissolution and click Y.

Select Mill Time through Atomizer Pressure and click Add.

From the Personality menu, select Generalized Regression.

Click Run.

In the Model Launch panel, click Go.

Figure 9.116 Model Based on Adaptive Lasso

You are interested in the parameter estimates shown in the Adaptive Lasso with AICc Validation report. Based on the non-zero parameter estimates, the model suggests that Mill Time, Screen Size, Blend Time, Blend Speed, Compressor, Coating Viscosity, and Spray Rate are related to Dissolution.

Reduce the Model

Before reducing the model, ensure that no columns are selected in the Tablet Production.jmp data table. Selected columns are not deselected in the first step below. Ensuring that no columns are selected prevents the inadvertent inclusion of columns with zeroed terms.

If you prefer not to work through the steps in this section, click the green triangle next to the Generalized Regression Reduced Model script in the Tablet Production.jmp data table to obtain the reduced model.

Click the red triangle next to Adaptive Lasso with AICc Validation and select Relaunch with Active Effects.

This opens a Fit Model window that places the terms with nonzero coefficient estimates in the Parameter Estimates reports into the Construct Model Effects list. The response is entered as Y. The Generalized Regression personality is selected.

Click Run.

In the Model Launch panel, click Go.

Figure 9.117 Reduced Model Using Adaptive Lasso

Notice that the estimate for Blend Speed has a confidence interval (Lower 95%) that comes very close to including zero. Next, perform a simulation study to see how often Blend Speed would be included in the model had other data values from the dissolution distribution been observed.

Explore the Inclusion of Blend Speed in the Model

Use the report for the reduced model (Figure 9.117) in the steps below.

Click the red triangle next to Adaptive Lasso with AICc Validation and select Save Columns > Save Simulation Formula.

This adds a new column called Dissolution Simulation Formula to the Tablet Production.jmp data table.

(Optional) In the data table Columns panel, click the plus sign to the right of Dissolution Simulation Formula.

Figure 9.118 Simulation Formula

For each row, this formula simulates a value that could be obtained given the model and the distribution of Dissolution, which is estimated to be Normal with standard deviation about 1.998.

Click Cancel.

Go back to the reduced model report window. In the Parameter Estimates for Original Predictors report, right-click in the Estimate column and select Simulate.

Make sure that Dissolution is selected in the Column to Switch Out list.

Next to Number of Samples, enter 300.

For the simulation, you ask JMP to replace the Dissolution column in each of 300 analyses with values simulated using the Dissolution Simulation Formula column.

(Optional) Set the Random Seed to 123.

This reproduces the values in this example.

Figure 9.119 Completed Simulation Window

Click OK.

The first row of the table contains the initial values of the Estimates and is excluded. The remaining rows contain simulated values.

Run the Distribution script.

Press the Ctrl key, click the Intercept red triangle menu, and select Display Options > Customize Summary Statistics.

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Select N Zero.

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Click OK.

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Scroll to the Distribution report for Blend Speed.

Figure 9.120 Histogram of Simulated Blend Speed Coefficient Estimates

The Summary Statistics report shows that for 103/300 = 34.3% of the simulations, the Blend Speed estimates are zero.