The most dangerous day

Our World in Data published this interesting chart about infant mortality in the U.S.

The article that sent me to this chart called the first day of life the "most dangerous day". This dot plot seems to support the notion, as the "per-day" death rate is the highest on the day of birth, and then drops quite fast (note log scale) over the the first year of life.

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Based on the same dataset, I created the following different view of the data, using the same dot plot form:

By this measure, a baby has 99.63% chance of surviving the first 30 days while the survival rate drops to 99.5% by day 180.

There is an important distinction between these two metrics.

The "per day" death rate is the chance of dying on a given day, conditional on having survived up to that day. The chance of dying on day 2 is lower partly because some of the more vulnerable ones have died on day 1 or day 0,  etc.

The survival rate metric is cumulative: it measures how many babies are still alive given they were born on day 0. The survival rate can never go up, so long as we can't bring back the dead.

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If we are assessing a 5-day-old baby's chance of surviving day 6, the "per-day" death rate is relevant since that baby has not died in the first 5 days.

If the baby has just been born, and we want to know the chance it might die in the first five days (or survive beyond day 5), then the cumulative survival rate curve is the answer. If we use the per-day death rate, we can't add the first five "per-day" death rates It's a more complicated calculation of dying on day 0, then having not died on day 0, dying on day 1, then having not died on day 0 or day 1, dying on day 2, etc.

 

The radial is still broken

It's puzzling to me why people like radial charts. Here is a recent set of radial charts that appear in an article in Significance magazine (link to paywall, currently), analyzing NBA basketball data.

This example is not as bad as usual (the color scheme notwithstanding) because the story is quite simple.

The analysts divided the data into three time periods: 1980-94, 1995-15, 2016-23. The NBA seasons were summarized using a battery of 15 metrics arranged in a circle. In the first period, all but 3 of the metrics sat much above the average level (indicated by the inner circle). In the second period, all 15 metrics reduced below the average, and the third period is somewhat of a mirror image of the first, which is the main message.

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The puzzle: why prefer this circular arrangement to a rectangular arrangement?

Here is what the same graph looks like in a rectangular arrangement:

One plausible justification for the circular arrangement is if the metrics can be clustered so that nearby metrics are semantically related.

Nevertheless, the same semantics appear in a rectangular arrangement. For example, P3-P3A are three point scores and attempts while P2-P2A are two-pointers. That is a key trend. They are neighborhoods in this arrangement just as they are in the circular arrangement.

So the real advantage is when the metrics have some kind of periodicity, and the wraparound point matters. Or, that the data are indexed to directions so north, east, south, west are meaningful concepts.

If you've found other use cases, feel free to comment below.

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I can't end this post without returning to the colors. If one can take a negative image of the original chart, one should. Notice that the colors that dominate our attention - the yellow background, and the black lines - have no data in them: yellow being the canvass, and black being the gridlines. The data are found in the white polygons.

The other informative element, as one learns from the caption, is the "blue dashed line" that represents the value zero (i.e. average) in the standardized scale. Because the size of the image was small in the print magazine that I was reading, and they selected a dark blue encroaching on black, I had to squint hard to find the blue line.

 

 

Adjust, and adjust some more

This Financial Times report illustrates the reason why we should adjust data.

The story explores the trend in economic statistics during 14 years of governing by conservatives. One of those metrics is so-called council funding (local governments). The graphic is interactive: as the reader scrolls the page, the chart transforms.

The first chart shows the "raw" data.

The vertical axis shows year-on-year change in funding. It is an index relative to the level in 2010. From this line chart, one concludes that council funding decreased from 2010 to around 2016, then grew; by 2020, funding has recovered to the level of 2010 and then funding expanded rapidly in recent years.

When the reader scrolls down, this chart is replaced by another one:

This chart contains a completely different picture. The line dropped from 2010 to 2016 as before. Then, it went flat, and after 2021, it started raising, even though by 2024, the value is still 10 percent below the level in 2010.

What happened? The data journalist has taken the data from the first chart, and adjusted the values for inflation. Inflation was rampant in recent years, thus, some of the raw growth have been dampened. In economics, adjusting for inflation is also called expressing in "real terms". The adjustment is necessary because the same dollar (hmm, pound) is worth less when there is inflation. Therefore, even though on paper, council funding in 2024 is more than 25 percent higher than in 2010, inflation has gobbled up all of that and more, to the point in which, in real terms, council funding has fallen by 20 percent.

This is one material adjustment!

Wait, they have a third chart:

It's unfortunate they didn't stabilize the vertical scale. Relative to the middle chart, the lowest point in this third chart is about 5 percent lower, while the value in 2024 is about 10 percent lower.

This means, they performed a second adjustment - for population change. It is a simple adjustment of dividing by the population. The numbers look worse probably because population has grown during these years. Thus, even if the amount of funding stayed the same, the money would have to be split amongst more people. The per-capita adjustment makes this point clear.

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The final story is much different from the initial one. Not only was the magnitude of change different but the direction of change reversed.

Whenever it comes to adjustments, remember that all adjustments are subjective. In fact, choosing not to adjust is also subjective. Not adjusting is usually much worse.

 

 

 

 

Excess delay

The hot topic in New York at the moment is congestion pricing for vehicles entering Manhattan, which is set to debut during the month of June. I found this chart (link) that purports to prove the effectiveness of London's similar scheme introduced a while back.

This is a case of the visual fighting against the data. The visual feels very busy and yet the story lying beneath the data isn't that complex.

This chart was probably designed to accompany some text which isn't available free from that link so I haven't seen it. The reader's expectation is to compare the periods before and after the introduction of congestion charges. But even the task of figuring out the pre- and post-period is taking more time than necessary. In particular, "WEZ" is not defined. (I looked this up, it's "Western Extension Zone" so presumably they expanded the area in which charges were applied when the travel rates went back to pre-charging levels.)

The one element of the graphic that raises eyebrows is the legend which screams to be read.

Why are there four colors for two items? The legend is not self-sufficient. The reader has to look at the chart itself and realize that purple is the pre-charging period while green (and blue) is the post-charging period (ignoring the distinction between CCZ and WEZ).

While we are solving this puzzle, we also notice that the bottom two colors are used to represent an unchanging quantity - which is the definition of "no congestion". This no-congestion travel rate is a constant throughout the chart and yet a lot of ink of two colors have been spilled on it. The real story is in the excess delay, which the congestion charging scheme was supposed to reduce.

The excess on the chart isn't harmless. The excess delay on the roads has been transferred to the chart reader. It actually distracts from the story the analyst is wanting to tell. Presumably, the story is that the excess delays dropped quite a bit after congestion charging was introduced. About four years later, the travel rates had creeped back to pre-charging levels, whereupon the authorities responded by extending the charging zone to WEZ (which as of the time of the chart, wasn't apparently bringing the travel rate down.)

Instead of that story, the excess of the chart makes me wonder... the roads are still highly congested with travel rates far above the level required to achieve no congestion, even after the charging scheme was introduced.

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I started removing some of the excess from the chart. Here's the first cut:

This is better but it is still very busy. One problem is the choice of columns, even though the data are found strictly on the top of each column. (Besides, when I chop off the unchanging sections of the columns, I created a start-not-from-zero problem.) Also, the labeling of the months leaves much to be desired, there are too many grid lines, etc.

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Here is the version I landed on. Instead of columns, I use lines. When lines are used, there is no need for month labels since we can assume a reader knows the structure of months within a year.

A priniciple I hold dear is not to have legends unless it is absolutely required. In this case, there is no need to have a legend. I also brought back the notion of a uncongested travel speed, with a single line (and annotation).

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The chart raises several questions about the underlying analysis. I'd interested in learning more about "moving car observer surveys". What are those? Are they reliable?

Further, for evidence of efficacy, I think the pre-charging period must be expanded to multiple years. Was 2002 a particularly bad year?

Thirdly, assuming WEZ indicates the expansion of the program to a new geographical area, I'm not sure whether the data prior to its introduction represents the travel rate that includes the WEZ (despite no charging) or excludes it. Arguments can be made for each case so the key from a dataviz perspective is to clarify what was actually done.

 

P.S. [6-6-24] On the day I posted this, NY State Governer decided to cancel the congestion pricing scheme that was set to start at the end of June.

What's a histogram?

Almost all graphing tools make histograms, and almost all dataviz books cover the subject. But I've always felt there are many unanswered questions. In my talk this Thursday in NYC, I'll provide some answers. You can reserve a spot here.

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Here's the most generic histogram:

Even Excel can make this kind of histogram. Notice that we have counts in the y-axis. Is this really a useful chart?

I haven't found this type of histogram useful ever, since I don't do analyses in which I needed to know the exact count of something - when I analyze data, I'm generalizing from the observed sample to a larger group.

Speaking of Excel, I felt that the developers have always hated histograms. Why is it much harder to make histograms than other basic charts?

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Another question. We often think of histograms as a crude approximation to a probability density function (PDF). An example of a PDF is the famous bell curve. Textbooks sometimes show the concept like this:

This is true of only some types of histograms (and not the one shown in the first section!) Instead, we often face the following situation:

This isn't a trick. The data in the histogram above were generated by sampling the pink bell curve.

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If you've used histograms, you probably also have run into strange issues. I haven't found much materials out there to address these questions, and they have been lingering in my mind, hidden, for a long time.

My Thursday talk will hopefully fill in some of these gaps.

My talk next week on histograms

Next Thursday (March 14), I'll be presenting at the Data Visualization New York Meetup, hosted by Naomi and Cameron. The event is in-person at Datadog's office. You can reserve your spot here.

This talk is brand new, based on some work inspired by a blog post by Andrew Gelman. One of Andrew's correspondents asked about a particular type of histogram. While exploring this topic, I filled some of my own gaps in knowledge about this deceptively simple chart form. I'll be sharing this story.

Bits and pieces have appeared before on my blog. See this, this, and this for background.

If you're attending the talk, come up and say hi.

To register, click here.

The cult of raw unadjusted data

Long-time reader Aleks came across the following chart on Facebook:

The author attached a message: "Let's look at raw, unadjusted temperature data from remote US thermometers. What story do they tell?"

I suppose this post came from a climate change skeptic, and the story we're expected to take away from the chart is that there is nothing to see here.

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What are we looking at, really?

"Nothing to see" probably refers to the patch of blue squares that cover the entire plot area, as time runs left to right from the 1910s to the present.

But we can't really see what's going on in the middle of the patch. So, "nothing to see" is effectively only about the top-to-bottom range of roughly 29.8 to 82.0. What does that range signify?

The blue patch is subdivided into vertical lines consisting of blue squares. Each line is a year's worth of temperature measurements. Each square is the average temperature on a specific day. The vertical range is the difference between the maximum and minimum daily temperatures in a given year. These are extreme values that say almost nothing about the temperatures in the other ~363 days of the year.

We know quite a bit more about the density of squares along each vertical line. They are broken up roughly by seasons. Those values near the top came from summers while the values near the bottom came from winters. The density is the highest near the middle, where the overplotting is so severe that we can barely see anything.

Within each vertical line, the data are not ordered chronologically. This is a very key observation. From left to right, the data are ordered from earliest to latest but not from top to bottom! Therefore, it is impossible for the human eye to trace the entire trajectory of the daily temperature readings from this chart. At best, you can trace the yearly average temperature – but only extremely roughly by eyeballing where the annual averages are inside the blue patch.

Indeed, there is "nothing to see" on this chart because its design has pulverized the data.

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In Numbersense (link), I wrote "not adjusting the raw data is to knowingly publish bad information. It is analogous to a restaurant's chef knowingly sending out spoilt fish."

It's a fallacy to think that "raw unadjusted" data are the best kind of data. It's actually the opposite. Adjustments are designed to correct biases or other problems in the data. Of course, adjustments can be subverted to introduce biases in the data as well. It is subversive to presume that all adjustments are of the subversive kind.

What kinds of adjustments are of interest in this temperature dataset?

Foremost is the seasonal adjustment. See my old post here. If we want to learn whether temperatures have risen over these decades, we can't do so without separating out the seasons.

The whole dataset can be simplified by drawing the smoothed annual average temperature grouped by season of the year, and when that is done, the trend of rising temperatures is obvious.

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The following chart by the EPA roughly implements the above:

The original can be found here. They made one adjustment which isn't the one I expected.

Note the vertical scale is titled "temperature anomaly". So, they are not plotting the actual recorded average temperatures, but the "anomalies", i.e. the difference between the recorded temperatures and some kind of "expected" temperature. This is a type of data adjustment as well. The purpose is to focus attention on the relative rather than absolute values. Think of this formula: recorded value = expected value + anomaly. The chart shows how many degrees above or below expectation, rather than how many degrees.

For a chart like this, there should be a required footnote that defines what "anomaly" is. Specifically, the reader should know about the model behind the "expectation". Typically, it's a kind of long-term average value.

For me, this adjustment is not necessary. Without the adjustment, the four panels can be combined into one panel with four lines. That's because the data nicely fit into four levels based on seasons.

The further adjustment I'd have liked to see is "smoothing". Each line above has a "smooth" trend, as well as some variability around this trend. The latter is not a big part of the story.

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It's weird to push back on climate change advocacy by attacking data adjustments. The more productive direction, in my view, is to ask whether the observed trend is caused by human activities or part of some long-term up-and-down cycle. That is a very challenging question to answer.

To a new year of pleasant surprises

Happy new year!

This year promises to be the year of AI. Already last year, we pretty much couldn't lift an eyebrow without someone making an AI claim. This year will be even noisier. Visual Capitalist acknowledged this by making the noisiest map of 2023:

I kept thinking they have a geography teacher on the team, who really, really wants to give us a lesson of where each country is on the world map.

All our attention is drawn to the guiding lines and the random scatter of numbers. We have to squint to find the country names. All this noise drowns out the attempt to make sense of the data, namely, the inset of the top 10 countries in the lower left corner, and the classification of countries into five colored groups.

A small dose of editing helps. Remove most data labels except for the countries for which they have a story. Provide a data table below for those who want details.

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In the Methodology section, the data analysts (possibly from a third party called ElectronicsHub) indicated that they used Google search volume of "over 90 of the most popular generative AI tools", calculating the "overall volume across all tools per 100k population". Then came a baffling line: "all search volumes were scaled up according to the search engine market share in each country, using figures from statscounter.com." (Note: in the following, I'm calling the data "AI-related search" for simplicity even though their measurement is restricted to the terms described above.)

It took me a while to comprehend what they could have meant by that line. I believe this is what that sentence means: Google is not the only search engine out there so by only researching Google search volume, they undercount the true search volume. How did they deal with the missing data problem? They "scaled up" so if Google is 80% of the search volume in a country, then they divide the Google volume by 80% to "scale up" to 100%.

Whenever we use heuristics like this, we should investigate its foundations. What is the implicit assumption behind this scaling-up procedure? It is that all search engines are effectively the same. The users of non-Google search engines behave exactly as the Google search engine users. If the analysts somehow could get their hands on the data of other search engines, they would discover that the proportion of search volume that is AI-related is effectively the same as seen on Google.

This is one of those convenient, and obviously wrong assumptions – if true, the market would have no need for more than one search engine. Each search engine's audience is just a random sample from the population of all users.

Let's make up some numbers. Let's say Google has 80% share of search volume in Country A, and AI-related search 10% of the overall Google search volume. The remaining search engines have 20% share. Scaling up here means taking the 8% of Google AI-related search volume, divide by 80%, which yields 10%. Since Google owns 8% of the 10%, the other search engines see 2% of overall search volume attributed to AI searches in Country A. Thus, the proportion of AI-related searches on those other search engines is 2%/20% = 10%.

Now, in certain countries, Google is not quite as dominant. Let's say Google only has 20% share of Country B's search volume. AI-related search on Google is 2%, which is 10% of its total. Using the same scaling-up procedure, the analysts have effectively assumed that the proportion of AI-related search volume in the dominant search engines in Country B to be also 10%.

I'm using the above calculations to illustrate a shortcoming of this heuristic. Using this procedure inflates the search volume in countries in which Google is less dominant because the inflation factor is the reciprocal of Google's market share. The less dominant Google is, the larger the inflation factor.

What's also true? The less dominant Google is, the smaller proportion of the total data the analysts are able to see, the lower the quality of the available information. So the heuristic is the most influential where it has the greatest uncertainty.

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Hope your new year is full of uncertainty, and your heuristics shall lead you to pleasant surprises.

If you like the blog's content, please spread the word. I'm looking forward to sharing more content as the world of data continues to evolve at an amazing pace.

Disclosure: This blog post is not written by AI.

Stranger things found on scatter plots

Washington Post published a nice scatter plot which deconstructs scores from the recent World Championships in Gymnastics. (link)

The chart presents the main message clearly - the winner Simone Biles scored the highest on both components of the score (difficulty and execution), by quite some margin.

What else can we learn from this chart?

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Every athlete who qualified for the final scored at or above average on both components.

Scoring below average on either component is a death knell: no athlete scored enough on the other component to compensate. (The top left and bottom right quadrants would have had some yellow dots otherwise.)

Several athletes in the top right quadrant presumably scored enough to qualify but didn't. The footnote likely explains it: each country can send at most two athletes to the final. It may be useful to mark out these "unlucky" athletes using a third color.

Curiously, it's not easy to figure out who these unlucky athletes were from this chart alone. We need two pieces of data: the minimum qualifying score, and the total score for each athlete. The scatter plot isn't the best chart form to show totals, but qualification to the final is based on the sum of the difficulty and execution scores. (Note also, neither axis starts at zero, compounding the challenge.)

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This scatter plot is most memorable for shattering one of my expectations about risk and reward in sports.

I expect risk-seeking athletes to suffer from higher variance in performance. The tennis player who goes for big serves tend to also commit more double faults. The sluggers who hit home runs tend to strike out more often. Similarly, I expect gymnasts who attempt more difficult skills to receive lower execution scores.

Indeed, the headline writer seemed to agree, suggesting that Biles is special because she's both high in difficulty and strong in execution.

The scatter plot, however, sends the opposite message - this should not surprise. The entire field shows a curiously strong positive correlation between difficulty and execution scores. The more difficult is the routine, the higher the excution score!

It's hard to explain such a pattern. My guesses are:

a) judges reward difficult routines, and subconsciously confound execution and difficulty scores. They use separate judges for excecution and difficulty. Paradoxically, this arrangement may have caused separation anxiety - the judges for execution might just feel the urge to reward high difficulty.

b) those athletes who are skilled enough to attempt more difficult routines are also those who are more consistent in execution. This is a type of self-selection bias frequently found in observational data.

Regardless of the reasons for the strong correlation, the chart shows that these two components of the total score are not independent, i.e. the metrics have significant overlap in what they measure. Thus, one cannot really talk about a difficult routine without also noting that it's a well-executed routine, and vice versa. In an ideal scoring design, we'd like to have independent components.

When words speak louder than pictures

I've been staring at this chart from the Wall Street Journal (link) about U.S. workers working remotely:

It's one of those offerings I think on which the designer spent a lot of effort, but ultimately didn't realize that the reader would spend equal if not more effort deciphering.

However, the following paragraph lifted straight from the article says exactly what needs to be said:

Workers overall spent an average of 5 hours and 25 minutes a day working from home in 2022. That is about two hours more than in 2019, the year before Covid-19 sent millions of workers scrambling to set up home oces, and down just 12 minutes from 2021, according to the Labor Department’s American Time Use Survey.

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Why is the chart so hard to read?

It's mostly because the visual is fighting the message. In the Trifecta Checkup (link), this is represented by a disconnect between the Q(uestion) and the V(isual) corners - note the green arrow between these two corners.

The message concentrates on two comparisons: first, the increase in amount of remote work after the pandemic; and second, the mild decrease in 2022 relative to 2021.

On the chart, the elements that grab my attention are (a) the green and orange columns (b) the shading in the bottom part of those green and orange columns (c) the thick black line that runs across the chart (d) the indication on the left side that tells me one unit is an hour.

None of those visual elements directly addresses the comparisons. The first comparison - before and after the pandemic - is found by how much the green column spikes above the thick black line. Our comprehension is retarded by the decision to forego the typical axis labels in favor of chopping columns into one-hour blocks.

The second comparison - between 2022 and 2021 - is found in the white space above the top of the orange column.

So, in reality, the text labels that say exactly what needs to be said are carrying a lot of weight. A slight edit to the pointers helps connect those descriptions to the visual depiction, like this:

I've essentially flipped the tactics used in the various pointers. For the average level of remote work pre-pandemic, I dispense of any pointers while I'm using double-headed arrows to indicate differences across time.

Nevertheless, this modified chart is still too complex.

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Here is a version that aligns the visual to the message:

It's a bit awkward because the 2 hour 48 minutes calculation is the 2021 number minus the average of 2015-19, skipping the 2020 year.