 5. Process Improvement
5.5.9. An EDA approach to experimental design

## DOE Youden plot

Purpose The DOE (design of experiments) Youden plot answers the following question:
What are the important factors (including interactions)?
In its original interlab rendition, the Youden plot was a graphical technique developed in the 1960's by Jack Youden of NIST for assessing between-lab biases and within-lab variation problems in the context of interlab experimentation. In particular, it was appropriate for the analysis of round-robin data when exactly two materials, batches, etc. were used in the design.

In a design of experiments context, we borrow this duality emphasis and apply it to 2-level designs. The 2-component emphasis of the Youden plot makes it a natural to be applied to such designs.

Output The DOE Youden plot provides specific information on
1. Ranked list of factors (including interactions); and
2. Separation of factors into two categories: important and unimportant.
The primary output from a DOE Youden plot is the ranked list of factors (out of the k factors and interactions). For full factorial designs, interactions include the full complement of interactions at all orders; for fractional factorial designs, interactions include only some, and occasionally none, of the actual interactions. Further, the DOE Youden plot yields information identifying which factors/interactions are important and which are unimportant.
Definition The DOE Youden plot consists of the following:
• Vertical Axis: Mean response at the "+" setting for each factor and each interaction. For a given factor or interaction, n/2 response values will go into computing the "+" mean.

• Horizontal Axis: Mean response at the "-" setting for each factor and each interaction. For a given factor or interaction, n/2 response values will go into computing the "-" mean.

• Plot Character: Factor/interaction identification for which

1 indicates factor X1;
2 indicates factor X2;
...
12 indicates the 2-factor X1*X2 interaction
123 indicates the 3-factor X1*X2*X3 interaction
etc.
In essence, the DOE Youden plot is a scatter plot of the "+" average responses versus the "-" average responses. The plot will consist of n - 1 points with one point for each factor and one point for each (available) interaction. Each point on the plot is annotated to identify which factor or interaction is being represented.
Motivation Definitionally, if a factor is unimportant, the "+" average will be approximately the same as the "-" average, and if a factor is important, the "+" average will be considerably different from the "-" average. Hence a plot that compares the "+" averages with the "-" averages directly seems potentially informative.

From the definition above, the DOE Youden plot is a scatter plot with the "+" averages on the vertical axis and the "-" averages on the horizontal axis. Thus, unimportant factors will tend to cluster in the middle of the plot and important factors will tend to be far removed from the middle.

Because of an arithmetic identity which requires that the average of any corresponding "+" and "-" means must equal the grand mean, all points on a DOE Youden plot will lie on a -45 degree diagonal line. Or to put it another way, for each factor

average (+) + average (-) = constant (with constant = grand mean)
So
average (+) = constant - average (-)
Therefore, the slope of the line is -1 and all points lie on the line. Important factors will plot well-removed from the center because average (+) = average (-) at the center.
Plot for defective springs data Applying the DOE Youden plot for the defective springs data set yields the following plot.
How to interpret In the DOE Youden plot, we look for the following:
1. A ranked list of factors (including interactions). The intersecting dotted lines at the center of the plot are the value of the grand mean on both the vertical and horizontal axes. Scan the points along the negative-slope diagonal line and note as to whether such points are clustered around the grand mean or are displaced up or down the diagonal line.

1. Which point is farthest away from the center? This defines the "most important" factor.
2. Which point is next farthest away from the center? This defines the "second most important" factor.
3. Continue in a similar manner for the remaining points. The points closest to the center define the "least important" factors.

2. Separation of factors into important/unimportant categories. Interpretationally, if a factor is unimportant, the "+" average will be about the same as the "-" average, so the plot of "+" vertically and "-" horizontally will be near the grand mean of all n - 1 data points.

Conversely, if a factor is important, the "+" average will differ greatly from the "-" average, and so the plot of "+" vertically and "-" horizontally will be considerably displaced up into the top left quadrant or down into the bottom right quadrant.

The separation of factors into important/unimportant categories is thus done by answering the question:

Which points visually form a cluster around the center? (these define the "unimportant factors"--all remaining factors are "important").
This ranked list of important factors derived from the DOE Youden plot is to be compared with the ranked lists obtained from previous steps. Invariably, there will be a large degree of consistency exhibited across all/most of the techniques.
Conclusions for the defective springs data The application of the DOE Youden plot to the defective springs data set results in the following conclusions:
1. Ranked list of factors (including interactions):

1. X1 (most important)
2. X1*X3 (next most important)
3. X2
4. other factors are of lesser importance

2. Separation of factors into important/unimportant categories:

• "Important": X1, X1*X3, and X2
• "Unimportant": the remainder 