When and How to Use Conditional Plots (Co-Plots) for Data Analysis

In contrast to scatter-plot matrices, which always show all data points but project them onto different surfaces of the parameter space, co-plots (short for “conditional plots”) show various slices through the parameter space such that each slice contains only a subset of the data points. The slices are taken in a systematic manner, and we can form an image of the entire parameter space by mentally gluing the slices back together again (the salami principle). Because of the regular layout of the subplots, this technique is also known as a trellis plot.

Figure 5-5 shows a trivariate data set projected onto the two-dimensional xy plane. Although there is clearly structure in the data, no definite pattern emerges. In particular, the dependence on the third parameter is entirely obscured!






Figure 5-6 shows a co-plot of the same data set that is sliced or conditioned on the third parameter a. The bottom part of the graph shows six slices through the data corresponding to different ranges of a. (The slice for the smallest values of a is in the lower left, and the one for the largest values of a is in the upper righthand corner.) As we look at the slices, the structure in the data stands out clearly, and we can easily follow the dependence on the third parameter a.




The top part of Figure 5-6 shows the range of values that a takes on for each of the slices. If you look closely, you will find that there are some subtle issues hidden in (or rather revealed by) this panel, because it provides information on the details of the slicing operation.

Two decisions need to be made with regard to the slicing:

• By what method should the overall parameter range be cut into slices?
  
  
• Should slices overlap or not?

In many ways, the most “natural” answer to these questions would be to cut the entire parameter range into a set of adjacent intervals of equal width. It is interesting to observe (by looking at the top panel in Figure 5-6) that in the example graph, a different decision was made in regard to both questions! The slices are not of equal width in the range of parameter values that they span; instead, they have been made in such a way that each slice contains the same number of points. Furthermore, the slices are not adjacent but partially overlap each other.

The first decision (to have each slice contain the same number of points, instead of spanning the same range of values) is particularly interesting because it provides additional information on how the values of the parameter a are distributed. For instance, we can see that large values of a (larger than about a = –1) are relatively rare, whereas values of a between –4 and –2 are much more frequent. This kind of behavior would be much harder to recognize precisely if we had chopped the interval for a into six slices of equal width. The other decision (to make the slices overlap partially) is more important for small data sets, where otherwise each slice contains so few points that the structure becomes hard to see. Having the slices overlap makes the data “go farther” than if the slices were entirely disjunct.

Co-plots are especially useful if some of the variables in a data set are clearly “control” variables, because co-plots provide a systematic way to study the dependence of the remaining (“response”) variables on the controls.