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.

The blue mist

The New York Times printed several charts about Twitter "blue checks," and they aren't one of their best efforts (link).

Blue checks used to be credentials given to legitimate accounts, typically associated with media outlets, celebrities, brands, professors, etc. They are free but must be approved by Twitter. Since Elon Musk acquired Twitter, he turned blue checks into a revenue generator. Yet another subscription service (but you're buying "freedom"!). Anyone can get a blue check for US$8 per month.

[The charts shown here are scanned from the printed edition.]

The first chart is a scatter plot showing the day of joining Twitter and the total number of followers the account has as of early November, 2022. Those are very strange things to pair up on a scatter plot but I get it: the designer could only work with the data that can be pulled down from Twitter's API.

What's wrong with the data? It would seem the interesting question is whether blue checks are associated with number of followers. The chart shows only Twitter Blue users so there is nothing to compare to. The day of joining Twitter is not the day of becoming "Twitter Blue", almost surely not for any user (Nevetheless, the former is not a standard data element released by Twitter). The chart has a built-in time bias since the longer an account exists, one would assume the higher the number of followers (assuming all else equal). Some kind of follower rate (e.g. number of followers per year of existence) might be more informative.

Still, it's hard to know what the chart is saying. That most Blue accounts have fewer than 5,000 followers? I also suspect that they chopped off the top of the chart (outliers) and forgot to mention it. Surely, some of the celebrity accounts have way over 150,000 followers. Another sign that the top of the chart was removed is that an expected funnel effect is not seen. Given the follower count is cumulative from the day of registration, we'd expect the accounts that started in the last few months should have markedly lower counts than those created years ago. (This is even more true if there is a survivorship bias - less successful accounts are more likely to be deleted over time.)

The designer arbitrarily labelled six specific accounts ("Crypto influencer", "HBO fan", etc.) but this feature risks sending readers the wrong message. There might be one HBO fan account that quickly grew to 150,000 followers in just a few months but does the data label suggest to readers that HBO fan accounts as a group tend to quickly attain high number of followers?

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The second chart, which is an inset of the first, attempts to quantify the effect of the Musk acquisition on the number of "registrations and subscriptions". In the first chart, the story was described as "Elon Musk buys Twitter sparking waves of new users who later sign up for Twitter Blue".

The second chart confuses me. I was trying to figure out what is counted in the vertical axis. This was before I noticed the inset in the first chart, easy to miss as it is tucked into the lower right corner. I had presumed that the axis would be the same as in the first chart since there weren't any specific labels. In that case, I am looking at accounts with 0 to 500 followers, pretty inconsequential accounts. Then, the chart title uses the words "registrations and subscriptions." If the blue dots on this chart also refer to blue-check accounts as in the first chart, then I fail to see how this chart conveys any information about registrations (wbich presumably would include free accounts). As before, new accounts that aren't blue checks won't appear.

Further, to the extent that this chart shows a surge in subscriptions, we are restricted to accounts with fewer than 500 followers, and it's really unclear what proportion of total subscribers is depicted. Nor is it possible to estimate the magnitude of this surge.

Besides, I'm seeing similar densities of the dots across the entire time window between October 2021 and 2022. Perhaps the entire surge is hidden behind the black lines indicating the specific days when Musk announced and completed the acquisition, respectively. If the surge is hiding behind the black vertical lines, then this design manages to block the precise spots readers are supposed to notice.

Here is where we can use the self-sufficiency test. Imagine the same chart without the text. What story would you have learned from the graphical elements themselves? Not much, in my view.

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The third chart isn't more insightful. This chart purportedly shows suspended accounts, only among blue-check accounts.

From what I could gather (and what I know about Twitter's API), the chart shows any Twitter Blue account that got suspended at any time. For example, all the black open circles occurring prior to October 27, 2022 represent suspensions by the previous management, and presumably have nothing to do with Elon Musk, or his decision to turn blue checks into a subscription product.

There appears to be a cluster of suspensions since Musk took over. I am not sure what that means. Certainly, it says he's not about "total freedom". Most of these suspended accounts have fewer than 50 followers, and only been around for a few weeks. And as before, I'm not sure why the analyst decided to focus on accounts with fewer than 500 followers.

What could have been? Given the number of suspended accounts are relatively small, an interesting analysis would be to form clusters of suspended accounts, and report on the change in what types of accounts got suspended before and after the change of management.

***

The online article (link) is longer, filling in some details missing from the printed edition.

There is one view that shows the larger accounts:

While more complete, this view isn't very helpful as the biggest accounts are located in the sparsest area of the chart. The data labels again pick out strange accounts like those of adult film stars and an Arabic news site. It's not clear if the designer is trying to tell us that most of Twitter Blue accounts belong to those categories.

***
See here for commentary on other New York Times graphics.

 

 

 

 

A graphical compass

A Twitter user pointed me to this article from Washington Post, ruminating about the correlation between gas prices and measures of political sentiment (such as Biden's approval rating or right-track-wrong-track). As common in this genre, the analyst proclaims that he has found something "counter intuitive".

The declarative statement strikes me as odd. In the first two paragraphs, he said the data showed "as gas prices fell, American optimism rose. As prices rose, optimism fell... This seems counterintuitive."

I'm struggling to see what's counterintuitive. Aren't the data suggesting people like lower prices? Is that not what we think people like?

The centerpiece of the article concerns the correlation between metrics. "If two numbers move in concert, they can be depicted literally moving in concert. One goes up, the other moves either up or down consistently." That's a confused statement and he qualifies it by typing "That sort of thing."

He's reacting to the following scatter plot with lines. The Twitter user presumably found it hard to understand. Count me in.

Why is this chart difficult to grasp?

The biggest puzzle is: what differentiates those two lines? The red and the gray lines are not labelled. One would have to consult the article to learn that the gray line represents the "raw" data at weekly intervals. The red line is aggregated data at monthly intervals. In other words, each red dot is an average of 4 or 5 weekly data points. The red line is just a smoothed version of the gray line. Smoothed lines show the time trend better.

The next missing piece is the direction of time, which can only be inferred by reading the month labels on the red line. But the chart without the direction of time is like a map without a compass. Take this segment for example:

If time is running up to down, then approval ratings are increasing over time while gas prices are decreasing. If time is running down to up, then approval ratings are decreasing over time while gas prices are increasing. Exactly the opposite!

The labels on the red line are not sufficient. It's possible that time runs in the opposite direction on the gray line! We only exclude that possibility if we know that the red line is a smoothed version of the gray line.

This type of chart benefits from having a compass. Here's one:

It's useful for readers to know that the southeast direction is "good" (higher approval ratings, lower gas prices) while the northwest direction is "bad". Going back to the original chart, one can see that the metrics went in the "bad" direction at the start of the year and has reverted to a "good" direction since.

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What does this chart really say? The author remarked that "correlation is not causation". "Just because Biden’s approval rose as prices dropped doesn’t mean prices caused the drop."

Here's an alternative: People have general sentiments. When they feel good, they respond more positively to polls, as in they rate everything more positively. The approval ratings are at least partially driven by this general sentiment. The same author apparently has another article saying that the right-track-wrong-track sentiment also moved in tandem with gas prices.

One issue with this type of scatter plot is that it always cues readers to make an incorrect assumption: that the outcome variables (approval rating) is solely - or predominantly - driven by the one factor being visualized (gas prices). This visual choice completely biases the reader's perception.

P.S. [11-11-22] The source of the submission was incorrectly attributed.

Painting the corner

Found an old one sitting in my folder. This came from the Wall Street Journal in 2018.

At first glance, the chart looks like a pretty decent effort.

The scatter plot shows Ebitda against market value, both measured in billions of dollars. The placement of the vertical axis title on the far side is a little unusual.

Ebitda is a measure of business profit (something for a different post on the sister blog: the "b" in Ebitda means "before", and allows management to paint a picture of profits without accounting for the entire cost of running the business). In the financial markets, the market value is claimed to represent a "fair" assessment of the value of the business. The ratio of the market value to Ebitda is known as the "Ebitda multiple", which describes the number of dollars the "market" places on each dollar of Ebitda profit earned by the company.

Almost all scatter plots suffer from xyopia: the chart form encourages readers to take an overly simplistic view in which the market cares about one and only one business metric (Ebitda). The reality is that the market value contains information about Ebitda plus lots of other factors, such as competitors, growth potential, etc.

Consider Alphabet vs AT&T. On this chart, both companies have about $50 billion in Ebitda profits. However, the market value of Alphabet (Google's mother company) is about four times higher than that of AT&T. This excess valuation has nothing to do with profitability but partly explained by the market's view that Google has greater growth potential.

***

Unusually, the desginer chose not to utilize the log scale. The right side of the following display is the same chart with a log horizontal axis.

The big market values are artificially pulled into the middle while the small values are plied apart. As one reads from left to right, the same amount of distance represents more and more dollars. While all data visualization books love log scales, I am not a big fan of it. That's because the human brain doesn't process spatial information this way. We don't tend to think in terms of continuously evolving scales. Thus, presenting the log view causes readers to underestimate large values and overestimate small differences.

Now let's get to the main interest of this chart. Notice the bar chart shown on the top right, which by itself is very strange. The colors of the bar chart is coordinated with those on the scatter plot, as the colors divide the companies into two groups; "media" companies (old, red), and tech companies (new, orange).

Scratch that. Netflix is found in the scatter plot but with a red color while AT&T and Verizon appear on the scatter plot as orange dots. So it appears that the colors mean different things on different plots. As far as I could tell, on the scatter plot, the orange dots are companies with over $30 billion in Ebitda profits.

At this point, you may have noticed the stray orange dot. Look carefully at the top right corner, above the bar chart, and you'll find the orange dot representing Apple. It is by far the most important datum, the company that has the greatest market value and the largest Ebitda.

I'm not sure burying Apple in the corner was a feature or a bug. It really makes little sense to insert the bar chart where it is, creating a gulf between Apple and the rest of the companies. This placement draws the most attention away from the datum that demands the most attention.

 

 

 

Another reminder that aggregate trends hide information

The last time I looked at the U.S. employment situation, it was during the pandemic. The data revealed the deep flaws of the so-called "not in labor force" classification. This classification is used to dehumanize unemployed people who are declared "not in labor force," in which case they are neither employed nor unemployed -- just not counted at all in the official unemployment (or employment) statistics.

The reason given for such a designation was that some people just have no interest in working, or even looking for a job. Now they are not merely discouraged - as there is a category of those people. In theory, these people haven't been looking for a job for so long that they are no longer visible to the bean counters at the Bureau of Labor Statistics.

What happened when the pandemic precipitated a shutdown in many major cities across America? The number of "not in labor force" shot up instantly, literally within a few weeks. That makes a mockery of the reason for such a designation. See this post for more.

***

The data we saw last time was up to April, 2020. That's more than two years old.

So I have updated the charts to show what has happened in the last couple of years.

Here is the overall picture.

In this new version, I centered the chart at the 1990 data. The chart features two key drivers of the headline unemployment rate - the proportion of people designated "invisible", and the proportion of those who are considered "employed" who are "part-time" workers.

The last two recessions have caused structural changes to the labor market. From 1990 to late 2000s, which included the dot-com bust, these two metrics circulated within a small area of the chart. The Great Recession of late 2000s led to a huge jump in the proportion called "invisible". It also pushed the proportion of part-timers to all0time highs. The proportion of part-timers has fallen although it is hard to interpret from this chart alone - because if the newly invisible were previously part-time employed, then the same cause can be responsible for either trend.

Readers of Numbersense (link) might be reminded of a trick used by school deans to pump up their US News rankings. Some schools accept lots of transfer students. This subpopulation is invisible to the US News statisticians since they do not factor into the rankings. The recent scandal at Columbia University also involves reclassifying students (see this post).

Zooming in on the last two years. It appears that the pandemic-related unemployment situation has reversed.

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Let's split the data by gender.

American men have been stuck in a negative spiral since the 1990s. With each recession, a higher proportion of men are designated BLS invisibles.

In the grid system set up in this scatter plot, the top right corner is the worse of all worlds - the work force has shrunken and there are more part-timers among those counted as employed. The U.S. men are not exiting this quadrant any time soon.

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What about the women?

If we compare 1990 with 2022, the story is not bad. The female work force is gradually reaching the same scale as in 1990 while the proportion of part-time workers have declined.

However, celebrating the above is to ignore the tremendous gains American women made in the 1990s and 2000s. In 1990, only 58% of women are considered part of the work force - the other 42% are not working but they are not counted as unemployed. By 2000, the female work force has expanded to include about 60% with similar proportions counted as part-time employed as in 1990. That's great news.

The Great Recession of the late 2000s changed that picture. Just like men, many women became invisible to BLS. The invisible proportion reached 44% in 2015 and have not returned to anywhere near the 2000 level. Fewer women are counted as part-time employed; as I said above, it's hard to tell whether this is because the women exiting the work force previously worked part-time.

***

The color of the dots in all charts are determined by the headline unemployment number. Blue represents low unemployment. During the 1990-2022 period, there are three moments in which unemployment is reported as 4 percent or lower. These charts are intended to show that an aggregate statistic hides a lot of information. The three times at which unemployment rate reached historic lows represent three very different situations, if one were to consider the sizes of the work force and the number of part-time workers.

 

P.S. [8-15-2022] Some more background about the visualization can be found in prior posts on the blog: here is the introduction, and here's one that breaks it down by race. Chapter 6 of Numbersense (link) gets into the details of how unemployment rate is computed, and the implications of the choices BLS made.

P.S. [8-16-2022] Corrected the axis title on the charts (see comment below). Also, added source of data label.

Funnels and scatters

I took a peek at some of the work submitted by Ray Vella's students in his NYU dataviz class recently.

The following chart by Hosanah Bryan caught my eye:

The data concern the GDP gap between rich and poor regions in various countries. In some countries, especially in the U.K., the gap is gigantic. In other countries, like Spain and Sweden, the gap is much smaller.

The above chart uses a funnel metaphor to organize the data, although the funnel does not add more meaning (not that it has to). Between that, the color scheme and the placement of text, it's visually clean and pleasant to look at.

The data being plotted are messy. They are not actual currency values of GDP. Each number is an index, and represents the relative level of the GDP gap in a given year and country. The gap being shown by the colored bars are differences in these indices 15 years apart. (The students were given this dataset to work with.)

So the chart is very hard to understand if one focuses on the underlying data. Nevertheless, the same visual form can hold other datasets which are less complicated.

One can nitpick about the slight misrepresentation of the values due to the slanted edges on both sides of the bars. This is yet another instance of the tradeoff between beauty and precision.

***

The next chart by Liz Delessert engages my mind for a different reason.

The scatter plot sets up four quadrants. The top right is "everyone gets richer". The top left, where most of the dots lie, is where "the rich get richer, the poor get poorer".  This chart shows a thoughtfulness about organizing the data, and the story-telling.

The grid setup cues readers toward a particular way of looking at the data.

But power comes with responsibility. Such scatter plots are particularly susceptible to the choice of data, in this case, countries. It is tempting to conclude that there are no countries in which everyone gets poorer. But that statement more likely tells us more about which countries were chosen than the real story.

I like to see the chart applied to other data transformations that are easier. For example, we can start with the % change in GDP computed separately for rich and for poor. Then we can form a ratio of these two percent changes.

 

 

What does Elon Musk do every day?

The Wall Street Journal published a fun little piece about tweets by Elon Musk (link).

Here is an overview of every tweet he sent since he started using Twitter more than a decade ago.

Apparently, he sent at least one tweet almost every day for the last four years. In addition, his tweets appear at all hours of the day. (Presumably, he is not the only one tweeting from his account.)

He doesn't just spend time writing tweets; he also reads other people's tweets. WSJ finds that up to 80% of his tweets include mentions of other users.

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One problem with "big data" analytics is that they often don't answer interesting questions. Twitter is already one of the companies that put more of their data out there, but still, analysts are missing some of the most important variables.

We know that Musk has 93 million followers. We already know from recent news that a large proportion of such users may be spam/fake. It is frequently assumed in twitter analysis that any tweet he makes reaches 93 million accounts. That's actually far from correct. Twitter uses algorithms to decide what posts show up in each user's feed so we have no idea how many of the 93 million accounts are in fact exposed to any of Musk's tweets.

Further, not every user reads everything on their Twitter feed. I don't even check it every day. Because Twitter operates as a 'firehose" with ever-changing content as users send out short messages at all hours, what one sees depends on when one reads. If Musk tweets in the morning, the users who log on in the afternoon won't see it.

Let's say an analyst wants to learn how impactful Musk's tweets are. That's pretty difficult when one can't figure out which of the 93 million followers were shown these tweets, and who read them. The typical data used to measure response are retweets and likes. Those are convenient metrics because they are available. They are very limited in what they measure. There are lots of users who don't like or retweet at all.

***

The available data do make for some fun charts. This one gave me a big smile:

Between writing tweets, reading tweets, and ROTFL, every hour of almost every day, Musk finds time to run his several companies. That's impressive.

 

The gift of small edits and subtraction

While making the chart on fertility rates (link), I came across a problem that pops up quite often, and is  ignored by most software programs.

Here is an earlier version of the chart I later discarded:

Compare this to the version I published in the blog post:

Aside from adding the chart title, there is one major change. I removed the empty plots from the grid. This is a visualization trick that should be called adding by subtracting. The empty scaffolding on the first chart increases our cognitive load without yielding any benefit. The whitespace brings out the message that only countries in Asia and Africa have fertility rates above 5.0. 

This is a small edit. But small edits accumulate and deliver a big impact. Bear this in mind the next time you make a chart.

 

P.S.

(1) You'd have to use a lower-level coding language to execute this small edit. Most software programs are quite rigid when it comes to making small-multiples (facet) charts.

(2) If there is a next iteration, I'd reverse the Asia and Oceania rows.

 

Visualizing fertility rates around the globe

The following chart dropped on my Twitter feed.

It's an ambitious chart that tries to do a lot. The underlying data set contains fertility rate data from over 200 countries over 20 years.

The basic chart form is a column chart that is curled up into a ball. The column chart is given colors that map to continents. All countries are grouped into five continents. The column chart can only take a single data series, so the 2019 fertility rate is chosen.

Beyond this basic setup, the designer embellishes the chart with a trove of information. Here's a close up:

The first number is the 2019 fertility rate, which means all the data encoded into the columns are also printed on the chart itself. Then, the flag of each country forms the next ring. Then, the name of the country. Finally, in brackets, the percent change in fertility rate between 2000 and 2019.

That is not all. Some contextual information are injected in those arrows that connect the columns to the data labels. A green arrow indicates that the fertility rate is trending lower - which is the case in most countries around the world. Once in a while, a purple arrow pops up. In the above excerpt, Seychelles gets a purple arrow because this island nation has increased the fertility rate from 2000 to 2019.

Also hiding in the background are several dashed rings. I think only the one that partially overlaps with the column chart contains any information - the other rings are inserted for an artistic reason. To decipher this dashed ring, we must look at the inset in the top left corner. We learn that the value of 2.1 children per woman is known as the replacement fertility rate. So it's also possible to assess whether each country is above or below the replacement fertility rate threshold.

[I'm presuming that this replacement threshold is about the births necessary to avoid a population decline. If that's the case, then comparing each country's fertility rate to a global fertility rate threshold is too simplistic because fertility is only one of several key factors driving a country's population growth. A more sophisticated model should generate country-level thresholds.]

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Data graphics serve many functions. This chart works well as an embellished data table. It does take some time to find a specific country because the columns have been sorted by decreasing 2019 fertility rate but once we locate the column, all the other data fields are clearly laid out.

As a generator of data insights, this chart is less effective. The main insight I obtained from it is a rough ranking of continents, with African countries predominantly having higher fertility rates, followed by Asia and Oceania, then Americas, and finally, Europe which has the lowest fertility rates. If this is the key message, a standard choropleth map brings it out more directly.

***

Here is a small-multiples rendering of the fertility dataset. I chose 1999 values instead of 2000 to make a complete two-decade view.

The columns represent a grouping of countries based on their 1999 fertility rates. The left column contains countries with the lowest number of births per woman, and the fertility rate increases left to right - both within an individual plot and in the grid.

If you're wondering, the hidden vertical axis sorts the countries by their 1999 rank. The lighter colors are 1999 values while the darker colors are 2019 values. For most countries the dots are shifting left over the 20 years. There are some exceptions. I have labeled several of these exceptions (e.g. Kazakhstan and Mongolia), and rendered them in italic.

 

 

 

Convincing charts showing containment measures work

The disorganized nature of U.S.'s response to the coronavirus pandemic has created a sort of natural experiment that allows data journalists to explore important scientific questions, such as the impact of containment measures on cases and hospitalizations. This New York Times article represents the best of such work.

The key finding of the analysis is beautifully captured by this set of scatter plots:

Each dot is a state. The cases (left plot) and hospitalizations (right plot) are plotted against the severity of containment measures for November. The negative correlation is unmistakable: the more containment measures taken, the lower the counts.

There are a few features worth noting.

The severity index came from a group at Oxford, and is a number between 0 and 100. The journalists decided to leave out the numerical labels, instead simply showing More and Fewer. This significantly reduces processing time. Readers won't be able to understand the index values anyway without reading the manual.

The index values are doubly encoded. They are first encoded by the location on the horizontal axis and redundantly encoded on the blue-red scale. Ordinarily, I do not like redundant encoding because the reader might assume a third dimension exists. In this case, I had no trouble with it.

The easiest way to see the effect is to ignore the muddy middle and focus on the two ends of the severity index. Those states with the fewest measures - South Dakota, North Dakota, Iowa - are the worst in cases and hospitalizations while those states with the most measures - New York, Hawaii - are among the best. This comparison is similar to what is frequently done in scientific studies, e.g. when they say coffee is good for you, they typically compare heavy drinkers (4 or more cups a day) with non-drinkers, ignoring the moderate and light drinkers.

Notably, there is quite a bit of variability for any level of containment measures - roughly 50 cases per 100,000, and 25 hospitalizations per 100,000. This indicates that containment measures are not sufficient to explain the counts. For example, the hospitalization statistic is affected by the stock of hospital beds, which I assume differ by state.

Whenever we use a scatter plot, we run the risk of xyopia. This chart form invites readers to explain an outcome (y-axis values) using one explanatory variable (on x-axis). There is an assumption that all other variables are unimportant, which is usually false.

***

Because of the variability, the horizontal scale has meaningless precision. The next chart cures this by grouping the states into three categories: low, medium and high level of measures.

This set of charts extends the time window back to March 1. For the designer, this creates a tricky problem - because states adapt their policies over time. As indicated in the subtitle, the grouping is based on the average severity index since March, rather than just November, as in the scatter plots above.

***

The interplay between policy and health indicators is captured by connected scatter plots, of which the Times article included a few examples. Here is what happened in New York:

Up until April, the policies were catching up with the cases. The policies tightened even after the case-per-capita started falling. Then, policies eased a little, and cases started to spike again.

The Note tells us that the containment severity index is time shifted to reflect a two-week lag in effect. So, the case count on May 1 is not paired with the containment severity index of May 1 but of April 15.

***

You can find the full article here.