Home / DOE / Charts / Contour Plot / Contour Plot How-To
Contour Plot How-To¶
This walkthrough draws a contour over Temp and Press from the same grid a surface uses, and shows what the fixed 41 by 41 grid means for the lines you see.
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 contour
QXL DOE New > Charts > Analysis > Contour 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, and leave both From and to pairs alone. 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 Contour Plot - Strength Y-Hat holding one contour chart: the same fitted response a surface would show, seen from directly above, with bands of colour between lines of equal Strength.
It is the same 1681 predictions as the surface. Both charts run one compute; only the chart Excel is asked to draw differs.
| Axis | From | to | Step between positions |
|---|---|---|---|
| Temp | 180 | 220 | 1.0 |
| Press | 40 | 60 | 0.5 |
The corners are the same four numbers, 50.9500 at Temp 180 with Press 40, 67.4000 at Temp 220 with Press 40, 61.4000 at Temp 180 with Press 60, and 68.9000 at Temp 220 with Press 60. The whole plot lies between 50.9500 and 68.9000.
Where the bands crowd together the response is changing fastest. Along the bottom edge, at Press 40, Strength moves 16.4500 across the Temp range; along the top edge, at Press 60, it moves 7.5000. So the bands are close together low down and spread out higher up, and that spacing is the AB interaction seen as a picture.
Read the crowding, not the count. How many bands Excel renders is Excel's decision from the axis it was given, so the number of bands is not a quantity to read off. What the chart tells you is where they bunch.
A contour is stricter about its axes than a surface¶
Both pickers on a contour offer quantitative factors only, because a contour needs both axes continuous. On this design all three factors are quantitative, so all three are offered and you would not notice. Add a categorical factor to a design and it simply will not appear in either list.
The same rule bites harder in grid mode: a cell of a contour grid is drawn only when both of its factors are quantitative, where a surface grid needs only the horizontal one to be.
Things to try next¶
- See it in three dimensions. Surface Plot How-To draws the same grid as a surface, where a ridge can hide what is behind it and a contour cannot.
- Draw the whole grid. Switch Chart layout back to All factor pairs (grid) and you get six contours on one sheet, one per ordered pair. Each cell gets its own value scale, computed from that cell alone, so two cells are not to be compared by colour.
- Slice it. Trellis How-To repeats one contour at three values of Time, and that chart does force a single shared scale across its panels.