 5. Process Improvement
5.5.9. An EDA approach to experimental design
5.5.9.10. DOE contour plot

## How to Interpret: Axes

What factors go on the two axes? For this first item, we choose the two most important factors in the experiment as the plot axes.

These are determined from the ranked list of important factors as discussed in the previous steps. In particular, the |effects| plot includes a ranked factor table. For the defective springs data, that ranked list consists of

Factor/Interaction Effect Estimate
X1 23
X1*X3 10
X2 -5
X3 1.5
X1*X2 1.5
X1*X2*X3 0.5
X2*X3 0
Possible choices In general, the two axes of the contour plot could consist of

• X1 and X2,
• X1 and X3, or
• X2 and X3.

In this case, since X1 is the top item in the ranked list, with an estimated effect of 23, X1 is the most important factor and so will occupy the horizontal axis of the contour plot. The admissible list thus reduces to

• X1 and X2, or
• X1 and X3.

To decide between these two pairs, we look to the second item in the ranked list. This is the interaction term X1*X3, with an estimated effect of 10. Since interactions are not allowed as contour plot axes, X1*X3 must be set aside. On the other hand, the components of this interaction (X1 and X3) are not to be set aside. Since X1 has already been identified as one axis in the contour plot, this suggests that the other component (X3) be used as the second axis. We do so. Note that X3 itself does not need to be important (in fact, it is noted that X3 is ranked fourth in the listed table with a value of 1.5).

In summary then, for this example the contour plot axes are:

Horizontal Axis: X1
Vertical Axis: X3
Four cases for recommended choice of axes Other cases can be more complicated. In general, the recommended rule for selecting the two plot axes is that they be drawn from the first two items in the ranked list of factors. The following four cases cover most situations in practice:

• Case 1:
1. Item 1 is a main effect (e.g., X3)
2. Item 2 is another main effect (e.g., X5)

Recommended choice:

1. Horizontal axis: item 1 (e.g., X3);
2. Vertical axis: item 2 (e.g., X5).

• Case 2:
1. Item 1 is a main effect (e.g., X3)
2. Item 2 is a (common-element) interaction (e.g., X3*X4)

Recommended choice:

1. Horizontal axis: item 1 (e.g., X3);
2. Vertical axis: the remaining component in item 2 (e.g., X4).

• Case 3:
1. Item 1 is a main effect (e.g., X3)
2. Item 2 is a (non-common-element) interaction (e.g., X2*X4)

Recommended choice:

1. Horizontal axis: item 1 (e.g., X3);
2. Vertical axis: either component in item 2 (e.g., X2, or X4), but preferably the one with the largest individual effect (thus scan the rest of the ranked factors and if the X2 |effect| > X4 |effect|, choose X2; otherwise choose X4).

• Case 4:
1. Item 1 is a (2-factor) interaction (e.g., X2*X4)
2. Item 2 is anything

Recommended choice:

1. Horizontal axis: component 1 from the item 1 interaction (e.g., X2);
2. Horizontal axis: component 2 from the item 1 interaction (e.g., X4). 