A common problem among many professions is the inability of expert practitioners to communicate with their clients. Attorneys are often guilty of speaking legalese to the folks that they represent, unaware that it is unfamiliar to them. Medical doctors sometimes struggle in the same way, even though their effectiveness relies on their ability to communicate clearly with their patients. Statisticians struggle with this problem more than most. You can be the most advanced statistician in the world, but if you cannot clearly report your findings to decision makers, your work is wasted. Learning to express statistical findings in ways that non-statisticians can understand should be a fundamental requirement of statistical training. I suspect that this problem is often due, not to inability, but instead, to a lack of awareness. It is indeed difficult to refrain from using statistical speak once you’ve become fluent in it, but I think that most statisticians lose awareness of the fact that others don’t understand it, so they rarely even try to overcome the problem. The solution to this problem begins with awareness. I’ll use an example from the work of a talented statistician, Howard Wainer, to illustrate this problem and its solution.
On the inside cover of Howard Wainer’s newest book, Truth or Truthiness, appear the words, “This wise book is a must-read for anyone who’s ever wanted to challenge the pronouncements of authority figures.” Including “truthiness” in the title—a word that was coined by the comedian Stephen Colbert—further suggests that Wainer’s intended audience is broad; certainly not limited to statisticians. Over the course of a long and productive career, Wainer has contributed a tremendous amount to the fields of statistics and data visualization. I’ve learned a great deal from his books. When reading them, however, I have at times cringed in response to sections that general readers would find confusing or even misleading due to a lack of statistical training. I find this frustrating, because I want the basic concepts of statistics to be more broadly understood. I celebrate those rare statisticians who manage to speak of their craft in accessible ways. Charles Whelan, the author of Naked Statistics, and Nate Silver, the author of The Signal and the Noise, are two statisticians who haven’t lost touch with the world at large.
In Truth or Truthiness, Wainer critiques a graph that appeared in the New York Times and redesigns it in a way that, in his opinion, is more effective. Here’s the original graph:
This combination of a bubble plot and a bar graph tells the story of increases in China’s acquisitions outside of the country, both in the number of deals and in the costs of those deals in millions of dollars. Although Wainer believes that this could be displayed more effectively, as do I, he credits it with two positive characteristics.
The New York Times’s plot of China’s increasing acquisitiveness has two things going for it. It contains thirty-four data points, which by mass media standards is data rich, showing vividly the concomitant increases in the two data series over a seventeen-year period…And second, by using Playfair’s circle representation it allows the visibility of expenditures over a wide scale.
(Truth or Truthiness, Howard Wainer, Cambridge University Press, 2016, p. 105)
While it is true that the New York Times does a better job of basing their stories on sufficient data than most news publications, I wouldn’t cite their use of bubbles in the upper chart as a benefit. Bubbles, which encode values based on their areas, require less vertical space to show this wide range of values than bars, but this slight advantage is wiped out by the fact that people cannot judge the relative areas of circles easily or accurately, nor can they easily compare bars to bubbles to clearly see the relationship between these two variables as they change through time. Wainer points out that the use of bubbles was introduced by William Playfair, the great pioneer of graphical communication, but Playfair did not have the benefit of our knowledge of visual perception when he used this technique. Statisticians must learn what works perceptually as part of their training in data visualization. Part of understanding your audience is understanding a few things about how their brains work.
Let’s look now at the alternative display that Wainer proposes.
Before critiquing this ourselves, let’s hear what Wainer has to say.
Might other alternatives perform better? Perhaps. In Figure 9.14 is a two-paneled display in which each panel carries one of the data series. Panel 9.14a [the upper panel] is a straightforward scatter plot showing the linear increases in the number of acquisitions that China has made over the past seventeen years. The slope of the fitted line tells us that over those seventeen years China has, on average, increased its acquisitions by 5.5/year. This crucial detail is missing from the sequence of bars but is obvious from the fitted regression line in the scatter plot. Panel 9.14b [the lower panel] shows the increase in money spent on acquisitions over the same seventeen years. The plot is on a log scale, and its overall trend is well described by a straight line. That line has a slope of 0.12 in the log scale and hence translates to an increase of about 32 percent per year. Thus, the trend established over these seventeen years shows that China has both increased the number of assets acquired each year and also has acquired increasingly expensive assets.
The key advantage of using paired scatter plots with linearizing transformations and fitted straight lines is that they provide a quantitative measure of how China’s acquisitiveness has changed. This distinguishes Figure 9.14 from the New York Times plot, which, although it contained all the quantitative information necessary to do these calculations, had primarily a qualitative message.
(ibid., p. 105)
Wainer’s scatterplots and his explanation of them include several assumptions about his audience’s knowledge that miss the boat. Even if his readers all understand how to read scatterplots, a scatterplot is not a good choice for this information. Clearly, a central theme of this story is how China’s acquisitions changed through time, but this isn’t easy to see in a scatterplot. Merely by connecting the values in each graph with a line, the patterns of change through time and their comparisons would become clearly visible.
About the upper graph, Wainer says,
The slope of the fitted line tells us that over those seventeen years China has, on average, increased its acquisitions by 5.5/year. This crucial detail is missing from the sequence of bars but is obvious from the fitted regression line in the scatter plot.
This is a vivid example of the disconnection from the world at large that plagues many statisticians. Most people do not understand the meaning of the slope of a trend line in a scatterplot other than the fact that, in this case, it is trending upwards. Without the annotation that he included in the chart, the 5.5/year increase in deals per year on average would remain unknown. I also don’t think that pointing this 5.5/year increase out is an appropriate summary of the story, for it suggests greater consistency than we see in the data.
The lower scatterplot introduces a number of problems for typical readers. First of all, most people don’t know how to interpret log scales. In fact, many readers might not even notice that the scale is logarithmic. They certainly wouldn’t know what the slope of the trend line means, nor would they understand that this straight line of best fit with a log scale indicates an exponential rate of increase, which Wainer fails to mention. Most readers would be inclined to compare the trend lines and conclude that the patterns of change are nearly the same. Also, one of Wainer’s statements about the data isn’t entirely correct:
The trend established over these seventeen years shows that China has both increased the number of assets acquired each year and also has acquired increasingly expensive assets.
China did not increase the number of assets or the amount of money spent on those assets each year. There are many examples of years when these values decreased, which to me is an important part of the story.
In the final paragraph of his explanation, Wainer claims:
The key advantage of using paired scatter plots with linearizing transformations and fitted straight lines is that they provide a quantitative measure of how China’s acquisitiveness has changed.
This would only be an advantage if readers knew how to read these “paired scatter plots with linearizing transformations and fitted straight lines.” Unfortunately, most readers would not. In fact, phrases such as “linearizing transformations” might cause them to flee in horror.
The news story that the New York Times was attempting to tell could have covered all of the important facts in ways that were easily understood by a general audience. If the relationship between the number of acquisitions and the costs of those acquisitions was important to the story, a single scatterplot designed in the following way with a bit of text to explain it could have done the job.
I’ve intentionally used linear scales for both axes so that the trend line clearly exhibits the exponential nature of the correlation between the two variables. I wouldn’t rely on the graph alone to tell this part of the story, but would explain in words that when a line curves upwards in this fashion it exhibits an exponential rate of increase: the cost of the acquisitions does not increase in increments that are equal to the number of them, but instead increases by ever greater amounts as the number of acquisitions increases. In addition to the overall nature of the relationship, this graph also clearly exhibits the fact that the relationship varies somewhat, which is especially illustrated by the outlier that strays far from the trend line in the lower right corner, showing that in a particular year the number of acquisitions was not associated with an exponential increase in costs.
It is doubtful that the New York Times was particularly concerned with the nature of the relationship between the two variables, but mostly wanted to show how both variables increased through this period of time. To tell this story, I would suggest a couple of displays, starting with the paired line graphs below.
This would be easy for general readers to understand and it supports the basic message well. What it doesn’t do especially well, however, is clearly show the pattern of change in the value of acquisitions because to scale this graph to include the last two extremely high values, most of the values reside in the bottom 25% of the scale (i.e., from 0 to 40 billion dollars out of a total scale that extends to 160 billion dollars), resulting in a line that is looks a great deal flatter than it would if the graph were scaled to exclude the last two values. If this pattern of change should be displayed more clearly, and if we were assured that our readers understood logarithmic scales, rather than displaying the number of acquisitions on a linear scale and the value of acquisitions on a log scale, the patterns would be more comparable if both were scaled logarithmically. In fact, expressing the value of acquisitions in billions, rather than millions of dollars as Wainer did, would allow us to use the same exact log scales in both graphs, to make them fully comparable, as follows.
Let’s assume, however, that it is best to avoid log scales altogether to prevent confusion, which would be the case with a general audience, even with readers of the New York Times.
One potential improvement would be to place both lines in a single graph, but to do this without creating a confusing and potentially misleading dual-scaled graph. To do this, we must express both sets of values using the same unit of measure and scale. One simple and common way to do this is to express both time series as the percentage difference of each value compared to the initial value (i.e., the value for the year 1990). Another common expression of the values that is perhaps even easier for people to understand involves expressing each year’s value as its percentage of the total for the entire period, as follows:
Now that the two lines appear in the same graph, they are easier to compare. It is clear that the number of acquisitions and their dollar value trended upward during this period, but not always and not always together. In other words, the correlation between the number and dollar amounts of acquisitions is there, but it isn’t particularly strong. Even though we have the scaling problem caused by the extremely high dollar values in 2005 and 2006, patterns of change during 1990 through 2004 are relatively clear and easy to compare. If this were not the case, however, we could address the scaling problem by providing a second line graph that only includes data from 1990 through 2004, as follows:
Now, let’s return to the main point. Those who do the work of data analysis must know how to clearly present their findings to those who rely on that information to make decisions and take action. This is an essential skill. Highly skilled statisticians are incredibly valuable, but only if they can explain their findings in understandable terms. This requires communications skills, both in the use of words and in the use of graphics. Training in these skills is every bit as important as training in statistics.