How many details to include in a chart

This graphic by Bloomberg provides the context for understanding the severity of the Atlantic storm season. (link)

At this point of the season, 2020 appears to be one of the most severe in history.

I was momentarily fascinated by a feature of modern browser-based data visualization: the death of the aspect ratio. When the browser window is stretched sufficiently wide, the chart above is transformed to this look:

The chart designer has lost control of the aspect ratio.

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This Bloomberg chart is an example of the spaghetti-style plots that convey variability by displaying individual units of data (here, storm years). The envelope of the growth curves gives the range of historical counts while the density of curves roughly offers some sense of the most likely counts at different points of the season.

But these spaghetti-style plots are not precise at conveying the variability because the density is hard to gauge. That's where aggregating the individual units helps.

The following chart does not show individual storm years. It shows the counts for the median season at selected points in time, and also a band of variability (for example, you'd include say 90 or 95% of the seasons).

I don't have the raw data so the aggregating is done by eyeballing the spaghetti.

I prefer this presentation even though it does not plot every single data point one has in the dataset.

 

 

Everything in Texas is big, but not this BIG

Long-time reader John forwarded the following chart via Twitter.

The chart shows the recent explosive growth in deaths due to Covid-19 in Texas. John flagged this graphic as yet another example in which the data are encoded to the lengths of the squares, not their areas.

Fixing this chart just requires fixing the length of one side of the square. I also flipped it to make a conventional column chart.

The final product:

An important qualification lurks in the footnote; it is directly applied to the label of July.

How much visual distortion is created when data are encoded to the lengths and not the areas? The following chart shows what readers see, assuming they correctly perceive the areas of those squares. The value for March is held the same as above while the other months show the death counts implied by the relative areas of the squares.

Owing to squaring, the smaller counts are artificially compressed while the big numbers are massively exaggerated.

On data volume, reliability, uncertainty and confidence bands

This chart from the Economist caught my eye because of the unusual use of color-coded hexagonal tiles.

The basic design of the chart is easy to grasp: It relates people's "happiness" to national wealth. The thick black line shows that the average citizen of wealthier countries tends to rate their current life situation better.

For readers alert to graphical details, things can get a little confusing. The horizontal "wealth" axis is shown in log scale, which means that the data on the right side of the chart have been compressed while the data on the left side of the chart have been stretched out. In other words, the curve in linear scale is much flatter than depicted.

One thing you might notice is how poor the fit of the line is at both ends. Singapore and Afghanistan are clearly not explained by the fitted line. (That said, the line is based on many more dots than those eight we can see.) Moreover, because countries are widely spread out on the high end of the wealth axis, the fit is not impressive. Log scales tend to give a false impression of the tightness of fit, as I explained before when discussing coronavirus case curves.

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The hexagonal tiles replace the more typical dot scatter or contour shading. The raw data consist of results from polls conducted in different countries in different years. For each poll, the analyst computes the average life satisfaction score for that country in that year. From national statistics, the analyst pulls out that country's GDP per capita in that year. Thus, each data point is a dot on the canvass. A few data points are shown as black dots. Those are for eight highlighted countries for the year 2018.

The black line is fitted to the underlying dot scatter and summarizes the correlation between average wealth and average life satisfaction. Instead of showing the scatter, this Economist design aggregates nearby dots into hexagons. The deepest red hexagon, sandwiched between Finland and the US, contains about 60-70 dots, according to the color legend.

These details are tough to take in. It's not clear which dots have been collected into that hexagon: are they all Finland or the U.S. in various years, or do they include other countries? Each country is represented by multiple dots, one for each poll year. It's also not clear how much variation there exists within a country across years.

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The hexagonal tiles presumably serve the same role as a dot scatter or contour shading. They convey the amount of data supporting the fitted curve along its trajectory. More data confers more reliability.

For this chart, the hexagonal tiles do not add any value. The deepest red regions are those closest to the black line so nothing is actually lost by showing just the line and not the tiles.

Using the line chart obviates the need for readers to figure out the hexagons, the polls, the aggregation, and the inevitable unanswered questions.

***

An alternative concept is to show the "confidence band" or "error bar" around the black line. These bars display the uncertainty of the data. The wider the band, the less certain the analyst is of the estimate. Typically, the band expands near the edges where we have less data.

Here is conceptually what we should see (I don't have the underlying dataset so can't compute the confidence band precisely)

The confidence band picture is the mirror image of the hexagonal tiles. Where the poll density is high, the confidence band narrows, and where poll density is low, the band expands.

A simple way to interpret the confidence band is to find the country's wealth on the horizontal axis, and look at the range of life satisfaction rating for that value of wealth. Now pick any number between the range, and imagine that you've just conducted a survey and computed the average rating. That number you picked is a possible survey result, and thus a valid value. (For those who know some probability, you should pick a number not at random within the range but in accordance with a Bell curve, meaning picking a number closer to the fitted line with much higher probability than a number at either edge.)

Visualizing data involves a series of choices. For this dataset, one such choice is displaying data density or uncertainty or neither.

This chart shows why the PR agency for the UK government deserves a Covid-19 bonus

The Economist illustrated some interesting consumer research with this chart (link):

The survey by Dalia Research asked people about the satisfaction with their country's response to the coronavirus crisis. The results are reduced to the "Top 2 Boxes", the proportion of people who rated their government response as "very well" or "somewhat well".

This dimension is laid out along the horizontal axis. The chart is a combo dot and bubble chart, arranged in rows by region of the world. Now what does the bubble size indicate?

It took me a while to find the legend as I was expecting it either in the header or the footer of the graphic. A larger bubble depicts a higher cumulative number of deaths up to June 15, 2020.

The key issue is the correlation between a country's death count and the people's evaluation of the government response.

Bivariate correlation is typically shown on a scatter plot. The following chart sets out the scatter plots in a small multiples format with each panel displaying a region of the world.

The death tolls in the Asian countries are low relative to the other regions, and yet the people's ratings vary widely. In particular, the Japanese people are pretty hard on their government.

In Europe, the people of Greece, Netherlands and Germany think highly of their government responses, which have suppressed deaths. The French, Spaniards and Italians are understandably unhappy. The British appears to be the most forgiving of their government, despite suffering a higher death toll than France, Spain or Italy. This speaks well of their PR operation.

Cumulative deaths should be adjusted by population size for a proper comparison across nations. When the same graphic is produced using deaths per million (shown on the right below), the general story is preserved while the pattern is clarified:

The right chart shows deaths per million while the left chart shows total deaths.

***

In the original Economist chart, what catches our attention first is the bubble size. Eventually, we notice the horizontal positioning of these bubbles. But the star of this chart ought to be the new survey data. I swapped those variables and obtained the following graphic:

Instead of using bubble size, I switched to using color to illustrate the deaths-per-million metric. If ratings of the pandemic response correlate tightly with deaths per million, then we expect the color of these dots to evolve from blue on the left side to red on the right side.

The peculiar loss of correlation in the U.K. stands out. Their PR firm deserves a bonus!

Working with multiple dimensions, an example from Germany

An anonymous reader submitted this mirrored bar chart about violent acts by extremists in the 16 German states.

At first glance, this looks like a standard design. On a second look, you might notice what the reader discovered- the chart used two different scales, one for each side. The left side (red) depicting left-wing extremism is artificially compressed relative to the right side (blue). Not sure if this reflects the political bias of the publication - but in any case, this distortion means the only way to consume this chart is to read the numbers.

Even after fixing the scales, this design is challenging for the reader. It's unnatural to compare two years by looking first below then above. It's not simple to compare across states, and even harder to compare left- and right-wing extremism (due to mirroring).

The chart feels busy because the entire dataset is printed on it. I appreciate not including a redundant horizontal axis. (I wonder if the designer first removed the axis, then edited the scale on one side, not realizing the distortion.) Another nice touch, hidden in the legend, is the country totals.

I present two alternatives.

The first is a small-multiples "bumps chart".

Each plot presents the entire picture within a state. You can see the general level of violence, the level of left- and right-wing extremism, and their year-on-year change. States can be compared holistically.

Several German state names are rather long, so I explored a horizontal orientation. In this case, a connected dot plot may be more appropriate.

The sign of a good multi-dimensional visual display is whether readers can easily learn complex relationships. Depending on the question of interest, the reader can mentally elevate parts of this chart. One can compare the set of blue arrows to the set of red arrows, or focus on just blue arrows pointing right, or red arrows pointing left, or all arrows for Berlin, etc.

 

[P.S. Anonymous reader said the original chart came from the Augsburger newspaper. This link in German contains more information.]

Cornell must remove the logs before it reopens the campus in the fall

Against all logic, Cornell announced last week it would re-open in the fall because a mathematical model under development by several faculty members and grad students predicts that a "full re-opening" would lead to 80 percent fewer infections than a scenario of full virtual instruction. That's what was reported by the media.

The model is complicated, with loads of assumptions, and the report is over 50 pages long. I will put up my notes on how they attained this counterintuitive result in the next few days. The bottom line is - and the research team would agree - it is misleading to describe the analysis as "full re-open" versus "no re-open". The so-called full re-open scenario assumes the entire community including students, faculty and staff submit to a full program of test-trace-isolate, including (mandatory) PCR diagnostic testing once every five days throughout the 16-week semester, and immediate quarantine and isolation of new positive cases, as well as those in contact with such persons, plus full compliance with this program. By contrast, it assumes students do not get tested in the online instruction scenario. In other words, the researchers expect Cornell to get done what the U.S. governments at all levels failed to do until now.

[7/8/2020: The post on the Cornell model is now up on the book blog. Here.]

The report takes us back to the good old days of best-base-worst-case analysis. There is no data for validating such predictions so they performed sensitivity analyses, defined as changing one factor at a time assuming all other factors are fixed at "nominal" (i.e. base case) values. In a large section of the report, they publish a series of charts of the following style:

Each line here represents one of the best-base-worst cases (respectively, orange-blue-green). Every parameter except one is given the "nominal" value (which represents the base case). The parameter that is manpulated is shown on the horizontal axis, and for the above chart, the variable is the assumption of average number of daily contacts per person. The vertical axis shows the main outcome variable, which is the percentage of the community infected by the end of term.

This flatness of the lines in the above chart appears to say that the outcome is quite insensitive to the change in the average daily contact rate under all three scenarios - until the daily contact rises above 10 per person per day. It also appears to show that the blue line is roughly midway between the orange and the green so the percent infected is slightly less-than halved under the optimistic scenario, and a bit more than doubled under the pessimistic scenario, relative to the blue line.

Look again.

The vertical axis is presented in log scale, and only labeled at values 1% and 10%. About midway between 1 and 10 on the horizontal axis, the outcome value has already risen above 10%. Because of the log transformation, above 10%, each tick represents an increase of 10% in proportion. So, the top of the vertical axis indicates 80% of the community being infected! Nothing in the description or labeling of the vertical axis prepares the reader for this.

The report assumes a fixed value for average daily contacts of 8 (I rounded the number for discussion), which is invariable across all three scenarios. Drawing a vertical line about eight-tenths of the way towards 10 appears to signal that this baseline daily contact rate places the outcome in the relatively flat part of the curve.

Look again.

The horizontal axis too is presented in log scale. To birth one log-scale may be regarded as a misfortune; to birth two log scales looks like carelessness. 

Since there exists exactly one tick beyond 10 on the horizontal axis, the right-most value is 20. The model has been run for values of average daily contacts from 1 to 20, with unit increases. I can think of no defensible reason why such a set of numbers should be expressed in a log scale.

For the vertical axis, the outcome is a proportion, which is confined to within 0 percent and 100 percent. It's not a number that can explode.

***

Every log scale on a chart is birthed by its designer. I know of no software that automatically performs log transforms on data without the user's direction. (I write this line with trepidation wishing that I haven't planted a bad idea in some software developer's head.)

Here is what the shape of the original data looks like - without any transformation. All software (I'm using JMP here) produces something of this type:

At the baseline daily contact rate value of 8, the model predicts that 3.5% of the Cornell community will get infected by the end of the semester (again, assuming strict test-trace-isolate fully implemented and complied).  Under the pessimistic scenario, the proportion jumps to 14%, which is 4 or 5 times higher than the base case. In this worst-case scenario, if the daily contact rate were about twice the assumed value (just over 16), half of the community would be infected in 16 weeks!

I actually do not understand how there could only be 8 contacts per person per day when the entire student body has returned to 100% in-person instruction. (In the report, they even say the 8 contacts could include multiple contacts with the same person.) I imagine an undergrad student in a single classroom with 50 students. This assumption says the average student in this class only comes into contact with at most 8 of those. That's one class. How about other classes? small tutorials? dining halls? dorms? extracurricular activities? sports? parties? bars?

Back to graphics. Something about the canonical chart irked the report writers so they decided to try a log scale. Here is the same chart with the vertical axis in log scale:

The log transform produces a visual distortion. On the right side, where the three lines are diverging rapidly, the log transform pulls them together. On the left side, where the three lines are close together, the log transform pulls them apart.

Recall that on the log scale, a straight line is exponential growth. Look at the green line (worst case). That line is approximately linear so in the pessimistic scenario, despite assuming full compliance to a strict test-trace-isolate regimen, the cases are projected to grow exponentially.

Something about that last chart still irked the report writers so they decided to birth a second log scale. Here is the chart they ultimately settled on:

As with the other axis, the effect of the log transform is to squeeze the larger values (on the right side) and spread out the smaller values (on the left side). After this cosmetic surgery, the left side looks relatively flat while the right side looks steep.

In the next version of the Cornell report, they should replace all these charts with ones using linear scales.

***

Upon discovering this graphical mischief, I wonder if the research team received a mandate that includes a desired outcome.

 

[P.S. 7/8/2020. For more on the Cornell model, see this post.]

Presented without comment

Weekend assignment - which of these tells the story better?

Or:

The cop-out answer is to say both. If you must pick one, which one?

***

When designing a data visualization as a living product (not static), you'd want a design that adapts as the data change.

When the pie chart is more complex than the data

The trading house, Charles Schwab, included the following graphic in a recent article:

This graphic is more complicated than the story that it illustrates. The author describes a simple scenario in which an investor divides his investments into stocks, bonds and cash. After a stock crash, the value of the portfolio declines.

The graphic is a 3-D pie chart, in which the data are encoded twice, first in the areas of the sectors and then in the heights of the part-cylinders.

As readers, we perceive the relative volumes of the part-cylinders. Volume is the cross-sectional area (i.e. of the base) multipled by the height. Since each component holds the data, the volumes are proportional to the squares of the data.

Here is a different view of the same data:

This "bumps chart" (also called a slopegraph) shows clearly the only thing that drives the change is the drop in stock prices. Because the author assumes no change in bonds or cash, the drop in the entire portfolio is completely accounted for by the decline in stocks. Of course, this scenario seems patently unrealistic - different investment asset classes tend to be correlated.

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A cardinal rule of data visualization is that the visual should be less complex than the data.

What is the price for objectivity

I knew I had to remake this chart.

The simple message of this chart is hidden behind layers of visual complexity. What the analyst wants readers to focus on (as discerned from the text on the right) is the red line, the seven-day moving average of new hospital admissions due to Covid-19 in Texas.

My eyes kept wandering away from the line. It's the sideway data labels on the columns. It's the columns that take up vastly more space than the red line. It's the sideway date labels on the horizontal axis. It's the redundant axis labels for hospitalizations when the entire data set has already been printed. It's the two hanging diamonds, for which the clues are filed away in the legend above.

Here's a version that brings out the message: after Phase 2 re-opening, the number of hospital admissions has been rising steadily.

Dots are used in place of columns, which push these details to the background. The line as well as periods of re-opening are directly labeled, removing the need for a legend.

Here's another visualization:

This chart plots the weekly average new hospital admissions, instead of the seven-day moving average. In the previous chart, the raggedness of moving average isn't transmitting any useful information to the average reader. I believe this weekly average metric is easier to grasp for many readers while retaining the general story.

***

On the original chart by TMC, the author said "the daily hospitalization trend shows an objective view of how COVID-19 impacts hospital systems." Objectivity is an impossible standard for any kind of data analysis or visualization. As seen above, the two metrics for measuring the trend in hospitalizations have pros and cons. Even if one insists on using a moving average, there are choices of averaging methods and window sizes.

Scientists are trained to believe in objectivity. It frequently disappoints when we discover that the rest of the world harbors no such notion. If you observe debates between politicians or businesspeople or social scientists, you rarely hear anyone claim one analysis is more objective - or less subjective - than another. The economist who predicts Dow to reach a new record, the business manager who argues for placing discounted products in the front not the back of the store, the sportscaster who maintains Messi is a better player than Ronaldo: do you ever hear these people describe their methods as objective?

Pursuing objectivity leads to the glorification of data dumps. The scientist proclaims disinterest in holding an opinion about the data. This is self-deception though. We clearly have opinions because when someone else  "misinterprets" the data, we express dismay. What is the point of pretending to hold no opinions when most of the world trades in opinions? By being "objective," we never shape the conversation, and forever play defense.

Visualizing black unemployment in the U.S.

In a prior post, I explained how the aggregate unemployment rate paints a misleading picture of the employment situation in the United States. Even though the U3 unemployment rate in 2019 has returned to the lowest level we have seen in decades, the aggregate statistic hides some concerning trends. There is an alarming rise in the proportion of people considered "not in labor force" by the Bureau of Labor Statistics - these forgotten people are not counted as "employable": when a worker drops out of the labor force, the unemployment rate ironically improves.

In that post, I looked at the difference between men and women. This post will examine the racial divide, whites and blacks.

I did not anticipate how many obstacles I'd encounter. It's hard to locate a specific data series, and it's harder to know whether the lack of search results indicates the non-existence of the data, or the incompetence of the search engine. Race-related data tend not to be offered in as much granularity. I was only able to find quarterly data for the racial analysis while I had monthly data for the gender analysis. Also, I only have data from 2000, instead of 1990.

***

As before, I looked at the official unemployment rate first, this time presented by race. Because whites form the majority of the labor force, the overall unemployment rate (not shown) is roughly the same as that for whites, just pulled up slightly toward the line for blacks.

The racial divide is clear as day. Throughout the past two decades, black Americans are much more likely to be unemployed, and worse during recessions.

The above chart determines the color encoding for all the other graphics. Notice that the best employment situations occurred on either end of this period, right before the dotcom bust in 2000, and in 2019 before the Covid-19 pandemic. As explained before, despite the headline unemployment rate being the same in those years, the employment situation was not the same.

***

Here is the scatter plot for white Americans:

Even though both ends of the trajectory are marked with the same shade of blue, indicating almost identical (low) rates of unemployment, we find that the trajectory has failed to return to its starting point after veering off course during the recession of the early 2010s. While the proportion of part-time workers (counted as employed) returned to 17.5% in 2019, as in 2000, about 15 percent more whites are now excluded from the unemployment rate calculation.

The experience of black Americans appears different:

During the first decade, the proportion of black Americans dropping out of the labor force accelerated while among those considered employed, the proportion holding part-time jobs kept increasing. As the U.S. recovered from the Great Recession, we've seen a boomerang pattern. By 2019, the situation was halfway back to 2000. The last available datum for the first quarter of 2020 is before Covid-19; it actually showed a halt of the boomerang.

If the pattern we saw in the prior post holds for the Covid-19 world, we would see a marked spike in the out-of-labor-force statistic, coupled with a drop in part-time employment. It appeared that employers were eliminating part-time workers first.

***

One reader asked about placing both patterns on the same chart. Here is an example of this:

This graphic turns out okay because the two strings of dots fit tightly into the grid while not overlapping. There is a lot going on here; I prefer a multi-step story than throwing everything on the wall.

There is one insight that this chart provides that is not easily observed in two separate plots. Over the two decades, the racial gap has narrowed in these two statistics. Both groups have traveled to the top right corner, which is the worst corner to reside -- where more people are classified as not employable, and more of the employed are part-time workers.

The biggest challenge with making this combined scatter plot is properly controlling the color. I want the color to represent the overall unemployment rate, which is a third data series. I don't want the line for blacks to be all red, and the line for whites to be all blue, just because black Americans face a tough labor market always. The color scheme here facilitates cross-referencing time between the two dot strings.