Home / DOE / Charts / Cube Plot / Cube Plot How-To
Cube Plot How-To¶
This walkthrough draws all eight corners of a three factor cube twice, once from the fitted model and once from the raw runs, and reads the difference between them.
Every chart in this family is drawn from a fitted model, so the walkthrough starts by building one. The same experiment is used on all ten chart walkthroughs, so once you have the regression sheet you can work through any of the others without setting the data up again.
The data is on this page rather than in a file to download. Press Copy for Excel, then paste it into a blank worksheet.
The data¶
A three factor experiment on a bonding process, run twice through. Temp is in degrees C, Press in psi and Time in seconds, and the response Strength is the peel strength of the finished bond. Each of the eight factor combinations was run twice, giving 16 runs.
| Temp | Press | Time | Strength |
|---|---|---|---|
| 180 | 40 | 10 | 49.5 |
| 220 | 40 | 10 | 65.4 |
| 180 | 60 | 10 | 61.8 |
| 220 | 60 | 10 | 66.8 |
| 180 | 40 | 20 | 52.3 |
| 220 | 40 | 20 | 70.7 |
| 180 | 60 | 20 | 60.7 |
| 220 | 60 | 20 | 70.0 |
| 180 | 40 | 10 | 50.6 |
| 220 | 40 | 10 | 66.4 |
| 180 | 60 | 10 | 61.2 |
| 220 | 60 | 10 | 67.2 |
| 180 | 40 | 20 | 51.4 |
| 220 | 40 | 20 | 67.1 |
| 180 | 60 | 20 | 61.9 |
| 220 | 60 | 20 | 71.6 |
Each factor takes exactly two values, so this is a full factorial of the three at two levels each, replicated once. That matters for more than one chart: a factor with exactly two observed values is the only kind that can carry a Cube Plot axis, and a design with centre points would have failed that test.
Steps¶
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Put the data in Excel
Press Copy for Excel above the table. In Excel, open a blank worksheet, click cell A1, and press Ctrl+V. You should have headers in row 1 and the 16 runs in rows 2 through 17.
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Make a design sheet to hold it
From the Excel ribbon: QXL DOE New > Create Design > Special > Create Custom Design.
On the first page set the number of factors to 3 and the number of runs to 16. On the second page set the number of outputs to 1. That gives you an empty design sheet of the right shape, which is how data collected outside Quantum XL gets analysed.
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Fill in the design sheet
Copy the four columns of data into the three factor columns and the one output column. Name the factors Temp, Press and Time, and the output Strength, so the names on your charts match the ones quoted below.
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Check the model holds the three two-factor interactions
The charts below assume a model of the three main effects plus AB, AC and BC. Add any that are missing with QXL DOE New > Modify Design > Inputs > Add/Remove Interactions.
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Run the regression
QXL DOE New > Analyze Design > Run Regression. A worksheet called Regression is added after the design sheet. Every chart in this family reads that sheet.
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Draw the cube from the model
QXL DOE New > Charts > Analysis > Cube Plot.
In the Outputs tree tick Strength Y-Hat on the Regression sheet. In Cube axes set Horizontal axis (1) to Temp, Vertical axis (2) to Press, and Depth axis, optional (3) to Time. Press Create.
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Draw it again from the runs
Open the same dialog and tick Strength Y-bar on the design sheet instead. The Set points box disappears, because a design source averages the runs it has and reads no set point. Press Create.
Check the fit first¶
Before drawing anything, check the coefficient table on the Regression sheet against this. If these numbers match, every chart number quoted further down will match too; if they do not, the design was not entered the way this walkthrough assumes and nothing below will line up.
| Term | Coefficient | Standard error | t | p |
|---|---|---|---|---|
| Constant | 62.1625 | 0.2912 | 213.477 | 0.000000 |
| Temp (A) | 5.9875 | 0.2912 | 20.562 | 0.000000 |
| Press (B) | 2.9875 | 0.2912 | 10.260 | 0.000003 |
| Time (C) | 1.0500 | 0.2912 | 3.606 | 0.005696 |
| AB | -2.2375 | 0.2912 | -7.684 | 0.000030 |
| AC | 0.6500 | 0.2912 | 2.232 | 0.052506 |
| BC | -0.1500 | 0.2912 | -0.515 | 0.618877 |
The fitted equation is
Strength = 62.1625 + 5.9875 A + 2.9875 B + 1.0500 C - 2.2375 AB + 0.6500 AC - 0.1500 BC
in coded units, where each factor runs from -1 at its low value to +1 at its high value.
Every standard error is the same 0.2912. That is not a coincidence and it is a useful sign that the design was entered correctly: on a balanced full factorial every coded column is orthogonal to every other and carries the same amount of information, so every coefficient is estimated equally precisely.
What you should see¶
A worksheet called Cube Plot holding a cube drawn in projection, with a number at each of its eight corners. All three factors were available as axes because each takes exactly two values across the runs, which is the only kind of factor a cube axis accepts.
| Temp | Press | Time | From the model | From the runs | Difference |
|---|---|---|---|---|---|
| 180 | 40 | 10 | 50.4000 | 50.0500 | -0.3500 |
| 220 | 40 | 10 | 65.5500 | 65.9000 | 0.3500 |
| 180 | 60 | 10 | 61.1500 | 61.5000 | 0.3500 |
| 220 | 60 | 10 | 67.3500 | 67.0000 | -0.3500 |
| 180 | 40 | 20 | 51.5000 | 51.8500 | 0.3500 |
| 220 | 40 | 20 | 69.2500 | 68.9000 | -0.3500 |
| 180 | 60 | 20 | 61.6500 | 61.3000 | -0.3500 |
| 220 | 60 | 20 | 70.4500 | 70.8000 | 0.3500 |
All eight corners are filled either way. A regression source can predict anywhere, so it always fills every corner. A design source fills a corner only when runs were made there, and this full factorial ran all eight combinations, so it fills them too. A half fraction would have left four corners of the design-source cube blank.
Every difference is exactly 0.3500, alternating in sign. That is worth pausing on. The model has three main effects and three two factor interactions but no three factor ABC term, so it cannot reproduce eight cell means exactly; it has six terms plus a constant to fit eight corners. The part of the data it cannot reach is precisely the ABC effect, and because the design is balanced that leftover lands on every corner with the same size and alternating sign.
Each design-source corner is the mean of two runs. The 180, 40, 10 corner averages 49.5 and 50.6 to 50.05. Every corner here has exactly two runs behind it, because the whole design was replicated once.
Why this design can carry a cube at all¶
A cube axis needs a factor with exactly two positions, and nothing else will do. For a quantitative factor that means exactly two distinct values across the loaded runs. All three factors here qualify.
Add centre points and every one of them stops qualifying, because a centre point is a third distinct value. A design with centre points gets no cube plot at all, and the dialog says so rather than drawing a partial one.
Things to try next¶
- Drop to a square. Set Depth axis, optional (3) back to (None) and you get four corners instead of eight, with Time held at its set point rather than being an axis.
- Move the held set point. With Time off the cube, change its set point from 15 to 20 and redraw from the model. The four corners move to the values the eight corner cube has on its Time 20 face.
- Compare with the surface. Surface Plot How-To sweeps the same region continuously instead of sampling its corners, and its four corners at Time 15 are 50.9500, 67.4000, 61.4000 and 68.9000.