Hanging things on your charts

The Financial Times published the following chart that shows the rollout of vaccines in the U.K.

(I can't find the online link to the article. The article is titled "AstraZeneca and Oxford face setbacks and success as battle enters next phase", May 29/30 2021.)

This chart form is known as a "streamgraph", and it is a stacked area chart in disguise. 

The same trick can be applied to a column chart. See the "hanging" column chart below:

The two charts show exactly the same data. The left one roots the columns at the bottom. The right one aligns the middle of the columns. 

I have rarely found these hanging charts useful. The realignment makes it harder to compare the sizes of the different column segments. On the normal stacked column chart, the yellow segments are the easiest to compare because they share the same base level. Even this is taken away from the reader on the right side.

Note also that the hanging version does not admit a vertical axis. 

The same comments apply to the streamgraph.

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Nevertheless, I was surprised that the FT chart shown above actually works. The main message I learned was that initially U.K. primarily rolled out AstraZeneca and, to a lesser extent, Pfizer, shots while later, they introduced other vaccines, including Johnson & Johnson, Novavax, CureVac, Moderna, and "Other". 

I can also see that the supply of AstraZeneca has not changed much through the entire time window. Pfizer has grown to roughly the same scale as AstraZeneca. Moderna remains a small fraction of total shots. 

I can even roughly see that the total number of vaccinations has grown about six times from start to finish. 

That's quite a lot for one chart, so job well done!

There is one problem with the FT chart. It should have labelled end of May as "today". Half the chart is history, and the other half is the future.

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For those following Covid-19 news, the FT chart is informative in a different way.

There is a misleading statement going around blaming the U.K.'s recent surge in cases on the Astrazeneca vaccine, claiming that the U.K. mostly uses AZ. This chart shows that from the start, about a third of the shots administered in the U.K. are Pfizer, and Pfizer's share has been growing over time. 

U.K. compared to some countries mostly using mRNA vaccines

U.K. is almost back to the winter peak. That's because the U.K. is serious about counting cases. Look at the state of testing in these countries:

What's clear about the U.S. case count is that it is kept low by cutting the number of tests by two-thirds, thus, our data now is once again severely biased towards severe cases. 

We can do a back-of-the-envelope calculation. The drop in testing may directly lead to a proportional drop in reported cases, thus removing 500 (asymptomatic, or mild) cases per million from the case count. The case count goes below 250 per million so the additional 200 or so reduction is due to other reasons such as vaccinations.

Start at zero improves this chart but only slightly

The following chart was forwarded to me recently:

It's a good illustration of why the "start at zero" rule exists for column charts. The poor Indian lady looks extremely short in this women's club. Is the average Indian woman really half as tall as the average South African woman? (Surely not!)

The problem is only superficially fixed by starting the vertical axis at zero. Doing so highlights the fact that the difference in average heights is but a fraction of the average heights themselves. The intra-country differences are squashed in such a representation - which works against the primary goal of the data visualization itself.

Recall the Trifecta Checkup. At the top of the trifecta is the Question. The designer obviously wants to focus our attention on the difference of the averages. A column chart showing average heights fails the job!

This "proper" column chart sends the message that the difference in average heights is noise, unworthy of our attention. But this is a bad take of the underlying data. The range of average heights across countries isn't that wide, by virtue of large population sizes.

According to Wikipedia, they range from 4 feet 10.5 to 5 feet 6 (I'm ignoring several entries in the table based on non representative small samples.) How do we know that the difference of 2 inches between averages of South Africa and India is actually a sizable difference? The Wikipedia table has the average heights for most of the world's countries. There are perhaps 200 values. These values are sprinkled inside the range of about 8 inches top to bottom. If we divide the full range into 10 equal bins, that's roughly 0.8 inches per bin. So if we have two numbers that are 2 inches apart, they almost span 2 bins. If the data were evenly distributed, that's a huge shift.

(In reality, the data should be normally distributed, bell-shaped, with much more at the center than on the edges. That makes a difference of 2 inches even more significant if these are normal values near the center but less significant if these are extreme values on the tails. Stats students should be able to articulate why we are sure the data are normally distributed without having to plot the data.)

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The original chart has further problems.

Another source of distortion comes from the scaling of the stick figures. The aspect ratio is being preserved, which means the area is being scaled. Given that the heights are scaled as per the data, the data are encoded twice, the second time in the widths. This means that the sizes of these figures grow at the rate of the square of the heights. (Contrast this with the scaling discussed in my earlier post this week which preserves the relative areas.)

At the end of that last post, I discuss why adding colors to a chart when the colors do not encode any data is a distraction to the reader. And this average height chart is an example.

From the Data corner of the Trifecta Checkup, I'm intrigued by the choice of countries. Why is Scotland highlighted instead of the U.K.? Why Latvia? According to Wikipedia, the Latvia estimate is based on a 1% sample of only 19 year olds.

Some of the data appear to be incorrect (or the designer used a different data source). Wikipedia lists the average height of Latvian women as 5 ft 6.5 while the chart shows 5 ft 5 in. Peru's average height of females is listed as 4 ft 11.5 and of males as 5 ft 4.5. The chart shows 5 ft 4 in.

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Lest we think only amateurs make this type of chart, here is an example of a similar chart in a scientific research journal:

(link to original)

I have seen many versions of the above column charts with error bars, and the vertical axes not starting at zero. In every case, the heights (and areas) of these columns do not scale with the underlying data.

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I tried a variant of the stem-and-leaf plot:

The scale is chosen to reflect the full range of average heights given in Wikipedia. The chart works better with more countries to fill out the distribution. It shows India is on the short end of the scale but not quite the lowest. (As mentioned above, Peru actually should be placed close to the lower edge.)

 

Reading this chart won't take as long as withdrawing troops from Afghanistan

Art sent me the following Economist chart, noting how hard it is to understand. I took a look, and agreed. It's an example of a visual representation that takes more time to comprehend than the underlying data.

The chart presents responses to 3 questions on a survey. For each question, the choices are Approve, Disapprove, and "Neither" (just picking a word since I haven't seen the actual survey question). The overall approval/disapproval rates are presented, and then broken into two subgroups (Democrats and Republicans).

The first hurdle is reading the scale. Because the section from 75% to 100% has been removed, we are left with labels 0, 25, 50, 75, which do not say percentages unless we've consumed the title and subtitle. The Economist style guide places the units of data in the subtitle instead of on
the axis itself.

Our attention is drawn to the thick lines, which represent the differences between approval and disapproval rates. These differences are signed: it matters whether the proportion approving is higher or lower than the proportion disapproving. This means the data are encoded in the order of the dots plus the length of the line segment between them.

The two bottom rows of the Afghanistan question demonstrates this mental challenge. Our brains have to process the following visual cues:

1) the two lines are about the same lengths

2) the Republican dots are shifted to the right by a little

3) the colors of the dots are flipped

What do they all mean?

A chart runs in trouble when you need a paragraph to explain how to read it.

It's sometimes alright to make complicated data visualization that illustrates complicated concepts. What justifies it is the payoff. I wrote about the concept of return on effort in data visualization here.

The payoff for this chart escaped me. Take the Democratic response to troop withdrawal. About 3/4 of Democrats approve while 15% disapprove. The thick line says 60% more Democrats approve than disapprove.

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Here, I show the full axis, and add a 50% reference line

Small edits but they help visualize "half of", "three quarters of".

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Next, I switch to the more conventional stacked bars.

This format reveals some of the hidden data on the chart - the proportion answering neither approve/disapprove, and neither yes/no.

On the stacked bars visual, the proportions are counted from both ends while in the dot plot above, the proportions are measured from the left end only.

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Read all my posts about Economist charts here

 

The time has arrived for cumulative charts

Long-time reader Scott S. asked me about this Washington Post chart that shows the disappearance of pediatric flu deaths in the U.S. this season:

The dataset behind this chart is highly favorable to the designer, because the signal in the data is so strong. This is a good chart. The key point is shown clearly right at the top, with an informative title. Gridlines are very restrained. I'd draw attention to the horizontal axis. The master stroke here is omitting the week labels, which are likely confusing to all but the people familiar with this dataset.

Scott suggested using a line chart. I agree. And especially if we plot cumulative counts, rather than weekly deaths. Here's a quick sketch of such a chart:

(On second thought, I'd remove the week numbers from the horizontal axis, and just go with the month labels. The Washington Post designer is right in realizing that those week numbers are meaningless to most readers.)

The vaccine trials have brought this cumulative count chart form to the mainstream. For anyone who have seen the vaccine efficacy charts, the interpretation of the panel of line charts should come naturally.

Instead of four plots, I prefer one plot with four superimposed lines. Like this:

 

 

 

Vaccine researchers discard the start-at-zero rule

I struggled to decide on which blog to put this post. The reality is it bridges the graphical and analytical sides of me. But I ultimately placed it on the dataviz blog because that's where today's story starts.

Data visualization has few set-in-stone rules. If pressed for one, I'd likely cite the "start-at-zero" rule, which has featured regularly on Junk Charts (here, here, and here, for example). This rule only applies to a bar chart, where the heights (and thus, areas) of the bars should encode the data.

Here is a stacked column chart that earns boos from us:

I made it so I'm downvoting myself. What's wrong with this chart? The vertical axis starts at 42 instead of zero. I've cropped out exactly 42 units from each column. Therefore, the column areas are no longer proportional to the ratio of the data. Forty-two is 84% of the column A while it is 19% of column B. By shifting the x-axis, I've made column B dwarf column A. For comparison, I added a second chart that has the x-axis start at zero.

On the right side, Column B is 22 times the height of column A. On the left side, it is 4 times as high. Both are really the same chart, except one has its legs chopped off.

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Now, let me reveal the data behind the above chart. It is a re-imagination of the famous cumulative case curve from the Pfizer vaccine trial.

I transferred the data to a stacked column chart. Each column block shows the incremental cases observed in a given week of the trial. All the blocks stacked together rise to the total number of cases observed by the time the interim analysis was presented to the FDA.

Observe that in the cumulative cases chart, the count starts at zero on Day 0 (first dose). This means the chart corresponds to the good stacked column chart, with the x-axis starting from zero on Day 0.

The Pfizer chart above is, however, disconnected from the oft-chanted 95% vaccine efficacy number. You can't find this number on there. Yes, everyone has been lying to you. In a previous post, I did the math, and if you trace the vaccine efficacy throughout the trial, you end up at about 80% toward the right, not 95%.

How can they conclude VE is 95% but show a chart that never reaches that level? The chart was created for a "secondary" analysis included in the report for completeness. The FDA and researchers have long ago decided, before the trials started enrolling people, that they don't care about the cumulative case curve starting on Day 0. The "primary" analysis counts cases starting 7 days after the second shot, which means Day 29.

The first week that concerns the FDA is Days 29-35 (for Pfizer's vaccine). The vaccine arm saw 41 cases in the first 28 days of the trial. In effect, the experts chop the knees off the column chart. When they talk about 95% VE, they are looking at the column chart with the axis starting at 42.

Yes, that deserves a boo.

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It's actually even worse than that, if you could believe it.

The most commonly cited excuse for the knee-chop is that any vaccine is expected to be useless in the first X days (X being determined after the trial ends when they analyze the data). A recently published "real world" analysis of the situation in Israel contains a lengthy defense of this tactic, in which they state:

Strictly speaking, the vaccine effectiveness based on this risk ratio overestimates the overall vaccine effectiveness in our study because it does not include the early follow-up period during which the vaccine has no detectable effect (and thus during which the ratio is 1). [Appendix, Supplement 4]

Assuming VE = 0 prior to day X is equivalent to stipulating that the number of cases found in the vaccine arm is the same (within margin of error) as the number of cases in the placebo arm during the first X days.

That assumption is refuted by the Pfizer trial (and every other trial that has results so far.)

The Pfizer/Biontech vaccine was not useless during the first week. It's not 95% efficacious, more like 16%. In the second week, it improves to 33%, and so on. (See the VE curve I plotted above for the Pfizer trial.)

What happened was all the weeks before which the VE has not plateaued were dropped.

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So I was simplifying the picture by chopping same-size blocks from both columns in the stacked column chart. Contrary to the no-effect assumption, the blocks at the bottom of each column are of different sizes. Much more was chopped from the placebo arm than from the vaccine arm.

You'd think that would unjustifiably favor the placebo. Not true! As almost all the cases on the vaccine arm were removed, the remaining cases on the placebo arm are now many multiples of those on the vaccine arm.

The following shows what the VE would have been reported if they had started counting cases from day X. The first chart counts all cases from first shot. The second chart removes the first two weeks of cases, corresponding to the analysis that other pharmas have done, namely, evaluate efficacy from 14 days after the first dose. The third chart removes even more cases, and represents what happens if the analysis is conducted from second dose. The fourth chart is the official Pfizer analysis, which began days after the second shot. Finally, the fifth chart shows analysis begining from 14 days after the second shot, the window selected by Moderna and Astrazeneca.

The premise that any vaccine is completely useless for a period after administration is refuted by the actual data. By starting analysis windows at some arbitrary time, the researchers make it unnecessarily difficult to compare trials. Selecting the time of analysis based on the results of a single trial is the kind of post-hoc analysis that statisticians have long warned leads to over-estimation. It's equivalent to making the vertical axis of a column chart start above zero in order to exaggerate the relative heights of the columns.

 

P.S. [3/1/2021] See comment below. I'm not suggesting vaccines are useless. They are still a miracle of science. I believe the desire to report a 90% VE number is counterproductive. I don't understand why a 70% or 80% effective vaccine is shameful. I really don't.

A note to science journal editors: require better visuals

In reviewing a new small-scale study of the Moderna vaccine, I found this chart:

This style of charts is quite common in scientific papers. And they are horrible. It irks me to think that some authors are forced to adopt such styles.

The study's main goal is to compare two half doses to two full doses of the Moderna vaccine. (To understand the science, read the post on my book blog.) The participants were stratified by age group. The vaccine is expected to work better for younger people than for older people. The point of the study isn't to measure the difference by age group, and so the age-group dimension is secondary.

Upon recognizing that, I reduce the number of colors from 4 to 2:

Halving the number of colors presents no additional difficulty. The reader spends less time cross-referencing.

The existence of the Pbo (placebo) and Conv (convalescent plasma) columns on the sides is both unsightly and suboptimal. The "Conv" serves as a reference level for the amount of antibodies the vaccine stimulates in people. A better way to display reference levels is using reference lines.

The biggest problem with the chart is the log scale on the vertical axis. This isn't even a log-10 but a log-2. (Each tick is a doubling of value.)

Take the first set of columns as an example. The second column is clearly less than twice the height of the first column, and yet 25 is 3.5 times bigger than 7.  The third column is also visually less than double the size of the second column, and yet 189 is 7.5 times bigger than 25. The areas (heights) of the columns do not convey the right information about relative sizes of the underlying data.

Here's an amusing observation. The brown area shaded below is half of the entire area of the chart - if we reverted it to a linear scale. And yet there is not a single data point above 250 in the data so the brown area is entirely empty.

An effect of a log scale is to compress the larger values of a dataset. That's what you're seeing here.

I now revisualize using dotplots:

The version on the left retains the log scale while the right one (pun intended) reverts to the linear scale.

The biggest effect by far is the spike of antibodies between day 29 and 43 - which is after the second shot is administered. (For Moderna, the second shot is targeted for day 28.) In fact, it is during that window that the level of antibodies went from below the "conv" level (i.e. from natural infection) to far above.

The log-scale version buries this finding because it squeezes the large numbers on the chart. In addition, it artificially pulls the small numbers toward the "Conv" level. On the right chart, the second dot for 18-54, full doses is only at half the level of "Conv"  but it looks tantalizing close to the "Conv" level on the left chart.

The authors of the study also claim that there is negligible dropoff by 30 days after the second dose, i.e. between the third and fourth dots in each set. That may be so on the log-scale chart but on the linear chart, we see a moderate reduction. I don't believe the size of this study allows us to make a stronger conclusion but the claim of no dropoff is dubious.

The left chart also obscures the age-group differences. It appears as if all four sets show roughly the same pattern. With the linear scale, we notice that the vaccine clearly works better for the younger subgroup. As I discussed on the book blog, no one actually knows what level of antibodies constitutes "protection," and so I can't say whether that age-group difference has practical significance.

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I recommend using log scales sparingly and carefully. They are a source of much mischief and misadventure.

 

 

 

These are the top posts of 2020

It's always very interesting as a writer to look back at a year's of posts and find out which ones were most popular with my readers.

Here are the top posts on Junk Charts from 2020:

How to read this chart about coronavirus risk

This post about a New York Times scatter plot dates from February, a time when many Americans were debating whether Covid-19 was just the flu.

Proportions and rates: we are no dupes

This post about a ArsTechnica chart on the effects of Covid-19 by age is an example of designing the visual to reflect the structure of the data.

When the pie chart is more complex than the data

This post shows a 3D pie chart which is worse than a 2D pie chart.

Twitter people upset with that Covid symptoms diagram

This post discusses some complicated graphics designed to illustrate complicated datasets on Covid-19 symptoms.

Cornell must remove the logs before it reopens in the fall

This post is another warning to think twice before you use log scales.

What is the price of objectivity?

This post turns an "objective" data visualization into a piece of visual story-telling.

The snake pit chart is the best election graphic ever

This post introduces my favorite U.S. presidential election graphic, designed by the FiveThirtyEight team.

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Here is a list of posts that deserve more attention:

Locating the political center

An example of bringing readers as close to the insights as possible

Visualizing change over time

An example of designing data visualization to reflect the structure of multivariate data

Bloomberg made me digest these graphics slowly

An example of simple and thoughtful graphics

The hidden bad assumption behind most dual-axis time-series charts

Read this before you make a dual-axis chart

Pie chart conventions

Read this before you make a pie chart

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Looking forward to bring you more content in 2021!

Happy new year.

Atypical time order and bubble labeling

This chart appeared in a Charles Schwab magazine in Summer, 2019.

This bubble chart does not print any data labels. The bubbles take our attention but the designer realizes that the actual values of the volatility are not intuitive numbers. The same is true of any standard deviation numbers. If you're told SD of a data series is 3, it doesn't tell you much by itself.

I first transformed this chart into the equivalent column chart:

Two problems surface on the axes.

For the time axis, the years are jumbled. Readers experience vertigo, as we try to figure out how to read the chart. Our expectation that time moves left to right is thwarted. This ordering also requires every single year label to be present.

For the vertical axis, I could have left out the numbers completely. They are not really meaningful. These represent the areas of the bubbles but only relative to how I measured them.

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In the next version, I sorted time in the conventional manner. Following Tufte's classic advice, only the tops of the columns are plotted.

What you see is that this ordering is much easier to comprehend. Figuring out that 2018 is an average year in terms of volatility is not any harder than in the original. In fact, we can reproduce the order of the previous chart just by letting our eyes sweep top to bottom.

To make it even easier to read the vertical axis, I converted the numbers into an index, with the average volatility as 100 %  (assigned to 0% on the chart) .

Now, you can see that 2018 is roughly at the average while 2008 is 400% above the average level. (How should we interpret this statement? That's a question I pose to my statistics students. It's not intuitive how one should interpret the statement that the standard deviation is 5 times higher.)

 

 

Is this an example of good or bad dataviz?

This chart is giving me feelings:

I first saw it on TV and then a reader submitted it.

Let's apply a Trifecta Checkup to the chart.

Starting at the Q corner, I can say the question it's addressing is clear and relevant. It's the relationship between Trump and McConnell's re-election. The designer's intended message comes through strongly - the chart offers evidence that McConnell owes his re-election to Trump.

Visually, the graphic has elements of great story-telling. It presents a simple (others might say, simplistic) view of the data - just the poll results of McConnell vs McGrath at various times, and the election result. It then flags key events, drawing the reader's attention to those. These events are selected based on key points on the timeline.

The chart includes wise design choices, such as no gridlines, infusing the legend into the chart title, no decimals (except for last pair of numbers, the intention of which I'm not getting), and leading with the key message.

I can nitpick a few things. Get rid of the vertical axis. Also, expand the scale so that the difference between 51%-40% and 58%-38% becomes more apparent. Space the time points in proportion to the dates. The box at the bottom is a confusing afterthought that reduces rather than assists the messaging.

But the designer got the key things right. The above suggestions do not alter the reader's expereince that much. It's a nice piece of visual story-telling, and from what I can see, has made a strong impact with the audience it is intended to influence.

This chart is proof why the Trifecta Checkup has three corners, plus linkages between them. If we just evaluate what the visual is conveying, this chart is clearly above average.

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In the D corner, we ask: what the Data are saying?

This is where the chart runs into several problems. Let's focus on the last two sets of numbers: 51%-40% and 58%-38%. Just add those numbers and do you notice something?

The last poll sums to 91%. This means that up to 10% of the likely voters responded "not sure" or some other candidate. If these "shy" voters show up at the polls as predicted by the pollsters, and if they voted just like the not shy voters, then the election result would have been 56%-44%, not 51%-40%. So, the 58%-38% result is within the margin of error of these polls. (If the "shy" voters break for McConnell in a 75%-25% split, then he gets 58% of the total votes.)

So, the data behind the line chart aren't suggesting that the election outcome is anomalous. This presents a problem with the Q-D and D-V green arrows as these pairs are not in sync.

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In the D corner, we should consider the totality of the data available to the designer, not just what the designer chooses to utilize. The pivot of the chart is the flag annotating the "Trump robocall."

Here are some questions I'd ask the designer:

What else happened on October 31 in Kentucky?

What else happened on October 31, elsewhere in the country?

Was Trump featured in any other robocalls during the period portrayed?

How many robocalls were made by the campaign, and what other celebrities were featured?

Did any other campaign event or effort happen between the Trump robocall and election day?

Is there evidence that nothing else that happened after the robocall produced any value?

The chart commits the XYopia (i.e. X-Y myopia) fallacy of causal analysis. When the data analyst presents one cause and one effect, we are cued to think the cause explains the effect but in every scenario that is not a designed experiment, there are multiple causes at play. Sometimes, the more influential cause isn't the one shown in the chart.

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Finally, let's draw out the connection between the last set of poll numbers and the election results. This shows why causal inference in observational data is such a beast.

Poll numbers are about a small number of people (500-1,000 in the case of Kentucky polls) who respond to polling. Election results are based on voters (> 2 million). An assumption made by the designer is that these polls are properly conducted, and their results are credible.

The chart above makes the claim that Trump's robocall gave McConnell 7% more votes than expected. This implies the robocall influenced at least 140,000 voters. Each such voter must fit the following criteria:

• Was targeted by the Trump robocall
• Was reached by the Trump robocall (phone was on, etc.)
• Responded to the Trump robocall, by either picking up the phone or listening to the voice recording or dialing a call-back number
• Did not previously intend to vote for McConnell
• If reached by a pollster, would refuse to respond, or say not sure, or voting for McGrath or a third candidate
• Had no other reason to change his/her behavior

Just take the first bullet for example. If we found a voter who switched to McConnell after October 31, and if this person was not on the robocall list, then this voter contributes to the unexpected gain in McConnell votes but weakens the case that the robocall influenced the election.

As analysts, our job is to find data to investigate all of the above. Some of these are easier to investigate. The campaign knows, for example, how many people were on the target list, and how many listened to the voice recording.

 

 

 

 

Aligning the visual and the data

The Washington Post reported a surge in donations to the Democrats after the death of Justice Ruth Ginsberg (link). A secondary effect, perhaps unexpected, was that donors decided to spread the money around; the proportion of donors who gave to six or more candidates jumped to 65%, where normally it is at 5%.

The text tells us what to look for, and the axis labels are commendably restrained. The color scheme is also intuitive.

There is something frustrating about this chart, though. It's that the spike is shown upside down. The level that the arrow points at is 45%, which is the total of the blue columns. The visual suggests the proportion of multiple beneficiaries (2 or more) should be 55%. There is a divergence between what the visual is saying and what the data are saying. Whichever number is correct, the required proportion is the inverse of the level shown on the percentage axis!

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This is the same chart flipped over.

Now, the number we need can be read off the vertical axis.

I also moved the color legend to the right side so that the entries can be printed vertically, in the same direction as the data. This is one of the unspoken rules of data visualization I featured in my feature for DataJournalism.com.

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In the Trifecta Checkup (link), the issue is with the green arrow between the D corner and the V corner. The data and the visual are not in sync.