Worley (1987) presented a model of the flow of water through a borehole that is drilled from the ground surface through two aquifers. The response variable y is the flow rate through the borehole in m3/year and is determined by the following equation:
rw = radius of borehole, 0.05 to 0.15 m
r = radius of influence, 100 to 50,000 m
Hu = potentiometric head of upper aquifer, 990 to 1100 m
Hl = potentiometric head of lower aquifer, 700 to 820 m
L = length of borehole, 1120 to 1680 m
Kw = hydraulic conductivity of borehole, 9855 to 12,045 m/year
1.

Select DOE > Space Filling Design.

2.

3.

Select Help > Sample Data Library and open Design Experiment/Borehole Factors.jmp (Factors Panel with Factor Values Loaded for Borehole Example).

4.

Click Continue.

5.

6.

Click the Sphere Packing button to produce the design.

7.

Click Make Table to make a table showing the design settings for the experiment.

To see a completed data table for this example, select Help > Sample Data Library and open Design Experiment/ Borehole Sphere Packing.jmp. Because the designs are generated from a random seed, the settings that you obtain will differ from those shown in the completed table.
The Borehole Sphere Packing.jmp data table contains a Fit Model script that you can use to analyze the data. Columns containing the true model, the prediction formula, and the prediction bias are included in the data table.
It is important to remember that deterministic data have no random component. As a result, pvalues from fitted statistical models do not have their usual meanings. A large F statistic (low pvalue) is an indication of an effect due to a model term. However, you cannot make valid confidence intervals about the size of the effects or about predictions made using the model.
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A stepwise regression of the response, log y, versus the full quadratic model in the eight factors, led to the prediction formula column.

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The prediction bias column is the difference between the true model column and the prediction formula column.

In this case, the true model column contains a formula that allows profiling the prediction bias to find its value anywhere in the region of the data. To understand the prediction bias in this example:
1.

Select Graph > Profiler.

2.

3.

Check the Expand Intermediate Formulas box, as shown at the bottom on the Profiler dialog in Profiler Dialog and Profile of the Prediction Bias in the Borehole SpherePacking Data. The prediction bias formula is a function of columns that are also created by formulas.

4.

Click OK.

The profile plots at the bottom in Profiler Dialog and Profile of the Prediction Bias in the Borehole SpherePacking Data show the prediction bias at the center of the design region. If there were no bias, the profile traces would be constant between the value ranges of each factor. In this example, the variables Hu and Hl show nonlinear effects.
The range of the prediction bias on the data is smaller than the range of the prediction bias over the entire domain of interest. To see this, look at the distribution analysis (Analyze > Distribution) of the prediction bias in Distribution of the Prediction Bias. Note that the maximum bias is 1.826 and the minimum is –0.684 (the range is 2.51).
The top plot in Prediction Plots Showing Maximum and Minimum Bias over Factor Domains shows the maximum bias (2.91) over the entire domain of the factors. The plot at the bottom shows the comparable minimum bias (–4.84). This gives a range of 7.75. This is more than three times the size of the range over the observed data.