1. Exploratory Data Analysis
1.3. EDA Techniques
1.3.3. Graphical Techniques: Alphabetic

1.3.3.11.

DEX Scatter Plot

Purpose:
Determine Important Factors with Respect to Location and Scale

The dex scatter plot shows the response values for each level of each factor (i.e., independent) variable. This graphically shows how the location and scale vary for both within a factor variable and between different factor variables. This graphically shows which are the important factors and can help provide a ranked list of important factors from a designed experiment. The dex scatter plot is a complement to the traditional analyis of variance of designed experiments.

Dex scatter plots are typically used in conjunction with the dex mean plot and the dex standard deviation plot. The dex mean plot replaces the raw response values with mean response values while the dex standard deviation plot replaces the raw response values with the standard deviation of the response values. There is value in generating all 3 of these plots. The dex mean and standard deviation plots are useful in that the summary measures of location and spread stand out (they can sometimes get lost with the raw plot). However, the raw data points can reveal subtleties, such as the presence of outliers, that might get lost with the summary statistics.

Sample Plot:
Factors 4, 2, 3, and 7 are the Important Factors.

Description of the Plot

For this sample plot, there are seven factors and each factor has two levels. For each factor, we define a distinct x coordinate for each level of the factor. For example, for factor 1, level 1 is coded as 0.8 and level 2 is coded as 1.2. The y coordinate is simply the value of the response variable. The solid horizontal line is drawn at the overall mean of the response variable. The vertical dotted lines are added for clarity.

Although the plot can be drawn with an arbitrary number of levels for a factor, it is really only useful when there are two or three levels for a factor.

Conclusions

This sample dex scatter plot shows that:

1. there does not appear to be any outliers;
2. the levels of factors 2 and 4 show distinct location differences; and
3. the levels of factor 1 show distinct scale differences.

Definition:
Response Values Versus Factor Variables

Dex scatter plots are formed by:

• Vertical axis: Value of the response variable
• Horizontal axis: Factor variable (with each level of the factor coded with a slightly offset x coordinate)

Questions

The dex scatter plot can be used to answer the following questions:

1. Which factors are important with respect to location and scale?
2. Are there outliers?

Importance:
Identify Important Factors with Respect to Location and Scale

The goal of many designed experiments is to determine which factors are important with respect to location and scale. A ranked list of the important factors is also often of interest. Dex scatter, mean, and standard deviation plots show this graphically. The dex scatter plot additionally shows if outliers may potentially be distorting the results.

Dex scatter plots were designed primarily for analyzing designed experiments. However, they are useful for any type of multi-factor data (i.e., a response variable with 2 or more factor variables having a small number of distinct levels) whether or not the data were generated from a designed experiment.

Extension for Interaction Effects

Using the concept of the scatterplot matrix, the dex scatter plot can be extended to display first order interaction effects.

Specifically, if there are k factors, we create a matrix of plots with k rows and k columns. On the diagonal, the plot is simply a dex scatter plot with a single factor. For the off-diagonal plots, we multiply the values of X_i and X_j. For the common 2-level designs (i.e., each factor has two levels) the values are typically coded as -1 and 1, so the multiplied values are also -1 and 1. We then generate a dex scatter plot for this interaction variable. This plot is called a dex interaction effects plot and an example is shown below.

Interpretation of the Dex Interaction Effects Plot

We can first examine the diagonal elements for the main effects. These diagonal plots show a great deal of overlap between the levels for all three factors. This indicates that location and scale effects will be relatively small.

We can then examine the off-diagonal plots for the first order interaction effects. For example, the plot in the first row and second column is the interaction between factors X1 and X2. As with the main effect plots, no clear patterns are evident.

Related Techniques

Dex mean plot
Dex standard deviation plot
Block plot
Box plot
Analysis of variance

Case Study

The dex scatter plot is demonstrated in the ceramic strength data case study.

Software

Dex scatter plots are available in some general purpose statistical software programs, although the format may vary somewhat between these programs. They are essentially just scatter plots with the X variable defined in a particular way, so it should be feasible to write macros for dex scatter plots in most statistical software programs. Dataplot supports a dex scatter plot.