By: Rees Morrison
This article introduces a type of graph that shows multiple metrics of law departments on the same scale – probably a new graph for most readers – a parallel coordinate plot (PCP). We will explain the PCP1 using data from the General Counsel Metrics LLC benchmarking survey sponsored by Major, Lindsey & Africa.
To create the PCP for this article, we extracted data of law departments that participated one or more years over the past six years of the benchmark survey. More specifically, the subset consists of U.S.-based companies in technology or telecommunications. We then narrowed the group to companies that reported fiscal year revenue between $500 million and $5 billion. The resulting 62 companies formed this article’s “tech” industry.
Establishing the tech law department data set is one step, but analyzing and presenting conclusions about its metrics is another. One of the challenges of displaying benchmark data in graphs is organizing, on one chart, multiple metrics for each law department. Making the challenge even harder is when the data comes in different scales such as single or double digits for head-count metrics, hundreds or thousands for patent records, millions for spend data, and billions for revenue.
To bring this challenge home, in situations of multiple metrics on multiple orders of magnitude, how can we show something informative about the number of lawyers in these law departments at the same time as we show their number of paralegals and number of other support staff (such as file clerks, receptionists, secretaries, administrators, and others), as well as legal spending and company revenue? Let’s tackle the display of head-count metrics.
A parallel coordinate graph displays as many variables as you want, and links each law department’s variables to each other on a segmented line.
Let’s explain the pieces of this impressive (but probably imposing) PCP graph. Across the bottom are the three head-count categories (lawyers, paralegals, and support). Above each category are 62 points (think of them as coordinates on the Cartesian grid), one for each law department in the data set, that represent its number of lawyers, paralegals, and support. The light blue shading simply makes it easier to see the three head-count categories.
The R programming language’s package GGally created the PCP. R is free, open-source software that is fantastic for analyzing and graphing numbers. The software scales each law department’s metric, and for each tech law department it draws a line from its lawyer scaled number in the first shaded area to the right, to its paralegal scaled number in the middle shaded area, and then over to the right to its support staff scaled number. With this design, the coordinates (points) are parallel to each other in the plot (PCP).
Scaling deserves a fuller explanation. Scaling is a statistical transformation that puts all scaled data on the same footing, so to speak. The R package mentioned that drew the graph, handles six methods with each displaying results somewhat differently. This article chose the default, simple one (the method’s name isn’t simple, “uniminimax,” which roughly translates as “a distribution between the smallest [minimum] and largest figure [maximum] using zero to one [uni]).”
Thus, this method transforms each law department’s number of lawyers, and converts it to a five-digit decimal from 0 to 1. The smallest department, with only one lawyer, became 0.000 on the scale, whereas the largest department, with 74 lawyers, became 1.000. Their lines are the lowest and highest ones respectively on the vertical shaded area for the number-of-lawyers coordinates. The second largest department had 72 lawyers and a scaled value of .940, so its line starts just below the largest one. Since relatively many more law departments are small than large, the lines for scaled lawyers cluster at the bottom of the plot. Using the same method, the paralegal numbers are scaled, as are the numbers of support staff.
The other ways of scaling available in this software rely on standardizing data using means or medians (both divided by the standard deviation of the category’s distribution) or standardizing against an assigned midpoint. Each scaling method has its virtues and drawbacks.
The advantage of scaling values is that different categories can be viewed the same way. For instance, if an additional category were internal legal spending, which is measured in millions of dollars, an absolute scale would mean that tiny metrics like numbers of paralegals would barely appear at the very bottom. But when you scale the millions of dollars on a 0 to 1 range, you can compare both figures equivalently. This is why a PCP can show head-count, legal spend and revenue all in comparable ranges.
As said, a segmented line passes from left to right for each department. They are “parallel” lines in the sense that they follow a similar structure. The “coordinates” can be thought of as the categories – hence a parallel coordinate chart that can display data on many more categories than can other kinds of plots (scatterplots, bar charts, pie charts, etc.).
With this method of visual presentation, you can see whether there are any patterns in clustering, according to those three categories. For example, the largest law department is at the top of the charts, but its scaled number of paralegals is not also the largest nor is its support staff metric. Stepping back, you can see that all three categories tend to increase or decrease generally in sync – as departments add lawyers they add paralegals and support staff.
Another observation that jumps out from a PCP is whether there are outliers. Outliers have very unusual combinations of category metrics. For example, the third and fourth largest law departments by number of paralegals are much lower by scaled number of lawyers. Outliers may simply have unusual numbers in some head-count category, or they
may have mistakes in their data.
When you interpret the PCP, think of each line as representing a single law department where it crosses the lawyer, paralegal, or support interval, its number has been scaled against all of the other law departments of that industry. This plot therefore, compactly represents three categories of variables and conveys information about outliers, clusters and obvious patterns. We should point out that the plot designer can choose different orders of the categories and can color the lines to represent, for example, publicly traded companies or privately held companies. The designer can also flip the PCP so that the categories are on the left, vertical axis.
Other kinds of plots can represent multiple categories. For example, a radar plot could compare each tech law department on these categories and would also scale the data. Radar plots are harder to interpret than PCPs with large numbers of law departments. With 62 companies, a radar plot would be well nigh unreadable. Plots such as sunflower plots can also handle multiple categories of data per law department, but most people find them a challenge to interpret. Additionally, they represent one department per sunflower glyph. Self-organizing maps (SOMs) tackle the challenge in yet another way.
A parallel coordinate plot can be daunting, but it is a sophisticated graphical tool for what is often called exploratory data analysis. Law departments and data scientists in the legal field should include PCPs in their graphical repertoire.