Home / DOE / Charts / Surface Plot / Surface Plot How-To
Surface Plot How-To¶
This walkthrough draws one surface over Temp and Press, checks its four corners against numbers you can get another way, and shows what the 41 point grid means in the units of this design.
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 one surface
QXL DOE New > Charts > Analysis > Surface Plot.
In the Outputs tree tick Strength Y-Hat. In Chart layout choose One specific pair. Set Horizontal axis (X) to Temp and Series axis (Y) to Press. Leave both From and to pairs at the values they opened with, which are each factor's own low and high. Leave the Time set point at its default. 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 Surface Plot - Strength Y-Hat holding one three dimensional surface.
The grid behind it is 41 by 41, which is 1681 predictions. The count is fixed in the engine and no control changes it. In the units of this design that works out as:
| Axis | From | to | Step between positions |
|---|---|---|---|
| Temp | 180 | 220 | 1.0 |
| Press | 40 | 60 | 0.5 |
Forty steps, not forty one, because 41 positions have 40 gaps between them. Temp spans 40 degrees, so each step is one degree; Press spans 20 psi, so each step is half a psi.
The four corners are numbers you have already seen.
| Temp | Press | Height |
|---|---|---|
| 180 | 40 | 50.9500 |
| 220 | 40 | 67.4000 |
| 180 | 60 | 61.4000 |
| 220 | 60 | 68.9000 |
Those are the same four values the Temp against Press cell of Interaction Plots How-To plots. They must be: both are the fitted model evaluated at the same four settings with Time held at the same place. A surface simply fills in the 1677 predictions between them.
The surface spans 50.9500 to 68.9000, and the height axis is rounded out from that range rather than starting at zero.
It is a twisted plane, not a curved one. This model has no squared term, because a two level design cannot estimate one, so the only curvature available is the twist the AB interaction puts in. Read along the Press 40 edge and the surface climbs steeply; read along the Press 60 edge and it barely climbs at all. That difference is the interaction, and the surface is the shape it makes.
Things to try next¶
- Zoom in. Set the Temp From and to to 200 and 220. You still get 41 positions, now half a degree apart, over that narrower span. Narrowing does not add information; it spreads the same grid over less ground.
- Extrapolate on purpose. Set the Temp to value to 260, past the 220 the experiment reached. A prompt names the factor, the range you typed and the range the data covers, and asks whether to draw it anyway. It is a warning, not a refusal.
- Look at it from above. Contour Plot How-To draws the same 1681 predictions as bands.
- Slice it. Trellis How-To repeats this surface at three values of Time.