Home / DOE / Charts / Interaction Plots / Interaction Plots How-To
Interaction Plots How-To¶
This walkthrough draws the full grid of interaction charts, then reads one of them closely enough to recover the AB coefficient from the four plotted points.
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 grid
QXL DOE New > Charts > Analysis > Interaction Plots.
In the Outputs tree tick Strength Y-Hat. In Chart layout leave All factor pairs (grid) selected and leave all three factors ticked. 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¶
One worksheet holding a three by three grid. The diagonal carries the factor names rather than charts, so six charts are drawn, not three.
Every pair appears twice. Temp against Press and Press against Temp are both there, because which factor runs along the horizontal axis changes what the chart shows. Three ticked factors give 3 x 2 = 6 charts, and four would give twelve.
Find the cell with Temp along the bottom and one line per level of Press. Its four points are:
| Temp | Press | Plotted value |
|---|---|---|
| 180 | 40 | 50.9500 |
| 180 | 60 | 61.4000 |
| 220 | 40 | 67.4000 |
| 220 | 60 | 68.9000 |
The two lines are not parallel, and that is the interaction. Follow the Press 40 line from Temp 180 to Temp 220 and it climbs 16.4500. Follow the Press 60 line across the same span and it climbs 7.5000. Temp does more than twice as much at the low pressure as at the high one.
The gap between those two climbs is four times the AB coefficient.
16.4500 - 7.5000 = 8.9500 = 4 x 2.2375
That is the arithmetic relationship between a two factor interaction coefficient and the picture: the coefficient is a quarter of the difference between the two slopes measured end to end. If your chart gives those two climbs, the AB term is right.
The transposed cell shows the same fit differently. In the Press-along-the-bottom cell the same four numbers appear, arranged as one line per Temp level. Press gains 10.4500 at Temp 180 and 1.5000 at Temp 220. Those two differ by 8.9500 as well, which they must: it is one interaction seen from two directions.
The two sources agree here, and that is not general¶
Draw the grid again from the design sheet instead of the regression sheet and every point lands on the same value. That happens because this design is balanced and every held factor was left at its default set point, which is the midpoint of its range, so each held factor codes to exactly zero and drops out of the prediction. What is left is the mean of the runs in that cell, which is what the design source plots.
Move one set point off its midpoint and the two part company. The agreement holds at one particular arrangement rather than in general, so a design-source chart and a regression-source chart of the same pair are only comparable when you know where the held factors were put.
Things to try next¶
- Draw one pair on its own. Choose One specific pair and set the two pickers to Temp and Press. You get the same chart with the whole sheet to itself.
- Watch the curve appear. Each line carries a second series of 100 points spread across its own span. On this two level design the fitted line between the two settings is straight, so the curve lies on the segment. It is there so that a squared term or a logistic bend shows when the model has one.
- Compare with the main effects. Main Effects Plots How-To reports Temp rising 11.9750, which is the average of the 16.4500 and 7.5000 above.