Junk Charts

Turning in his grave 1

(Thanks to reader Josh R. for the tip.)  The "plucky statisticians" at Urbanspoon decided to tackle the political hot potato: is Barack Obama an elitist?  Scratch that -- what they actually did was to determine if Obama supporters were elitists (of course, Obama would then be, due to guilt by association.)  Scratch that -- what they actually analyzed was if there tended to be more Starbucks per capita in those states in which Obama won Democratic primaries.

Suffice it to say, even if it can be proven that most states with high densities of Starbucks are more likely to have more Democratic primary voters who prefer Obama to Clinton, it is a far cry from proving Obama an elitist.  However, we take the leap of faith and look at the evidence presented to us.

The star witness was this chart plotting the "vote spread" of Obama minus Clinton and the per-capita Starbucks density.  The black line was a linear fit to the Starbucks data as shown in green dots.  Since the black and blue lines both pointed northeast roughly speaking, we were told: "States with more latte-purveying Starbucks stores are more likely to have gone for Obama."  (So Obama is indeed an elitist.)

To cover all bases, the creator of this chart suggested that "my statistics professor might be rolling over in his grave to hear me say it, but there's a mild but real correlation here!".

Mr. Urbanspoon, the statistics professor is here and he disapproves.  As discussed before (and here), plotting two series of data on the same chart and applying two different scales is a recipe for disaster.  Not reaching immediately for the scatter plot when one has two data series is another serious misstep.  (Indeed, Josh sent the link in with a note wondering why "people dislike scatter plots so much".)  So here is the appropriate graphic:

A quick first glance at the left chart indicates that any correlation, if it exists, is very weak indeed.  A simple linear regression analysis shows that Starbucks density explains only 14% of the variability in vote spread.  Note especially the wide dispersion of dots around the line.  Further, for the vast majority of the states (say those with vote spread between -20% and 40%), there appears to be no correlation.  This is seen on the right chart.

To the extent that there is a linear correlation, the points (orange dots) would be most influential.  The top cluster included Alaska, Kansas, DC, Hawaii and Idaho in which Obama had a large winning margin while the Starbucks density was above average.  The bottom cluster included Arkansas and Olkahoma where Obama was wiped out and where Starbucks had the lowest density.  These two clusters alone explained the mild relationship; removing them wiped it out.

Following Nyhan, we should remove some obvious outliers, such as Arkansas, Illionois and New York (home states), Michigan and Florida (disputed) and New Hampshire and Iowa (Edwards territory).  The result is also mild correlation (R-sq = 0.075).

Till next post, when the professor rolls over again ...

 

Notice that I prefer the number of people per Starbucks metric, as opposed to the number of Starbucks per thousand people (See prior discussion on Gelman's blog.)  The reason is that every number on the former metric is reality-based while the latter metric produces imaginary numbers for small states, i.e. the imputed number of Starbucks is smaller than what actually exists!

Also note that I used a renormalized vote spread so that the Obama proportion and the Clinton proportion added up to 100%.  This made the assumption that Edwards and other voters would split among Obama and Clinton in the same proportions as those who explicitly voted for the two frontrunners.

Running in the rain

Reader Eduardo is unhappy about the embellishments in this Nikeplus chart of miles ran by day; "pretty but misleading" he wrote us to say.  This is a clear case of more is less.

As a data graphic, it doesn't work.  The reflections don't work.  Perhaps Nike wants to remind all you super-dedicated Nano-wearing runners what it's like to run in mist or rain!  To quote Eduardo: "The bars start at -1! I guess it is motivation."  An extra mile for everyone.  The rounded corners make it harder to read the level.

Speaking of bar charts, I want to follow up on an exchange from March.  In that example, we claimed that not starting bars at zero misrepresented the relative lengths of those bars.  The chart showed counts of baseball players implicated in the Mitchell Report by position.

This distortion arises from taking the same length off each bar regardless of the data.  As a result, the ratios of the lengths between the bars have been changed drastically.

For example, the ratio of P/3B in the top chart is 31/9 = 3.4 but in the bottom chart, it is 23/1 = 23!

Knit-picking

In celebrating the recent trend by "elite" colleges to lowering the cost of education, the Times printed this chart, the top part of which is shown here.

The three colors represent different levels of aid.  Blue means "grants replace loans"; red means "free tuition"; yellow means "parents pay nothing".  The colleges are grouped by the minimum qualifying income for the blue category.

The whole effect is of a knit.  We shall call this the "knit chart".

I believe a simple data table will do the job nicely.  If any reader has other ideas, please show us your work!

A few points to note about the original:

• Ordering by the minimum income to qualify for "grants replace loans" is arbitrary, as is alphabetizing colleges within each group
• Qualifying "at any income level" should be shown on the left of "$40,000 or below" rather than to the right of $100,000.  The current order is such that qualifying level increases with income from left to right, except from $100,000 to "any income", where it falls off a cliff.
• Qualifying at any income level is better shown as a separate column on the right disconnected from the income scale.  The current configuration devalues the effort spent in making a proper income scale.
  
• Too many lines of equal length, and too few yellow and red lines to make the knit chart effective
  
• Should the graph cater to parents interested in seeing what aid they qualify for given their income level?  Or should the graph highlight the breadth of aid available at individual colleges?

Reference: "The (Yes) Low Cost of Higher Ed", New York Times, April 20 2008.

PS. The original point about the "any income level" was incorrect as pointed out by Chris below.  I have replaced that with a different issue.

PPS. Matias' version (see comments) is a superb demonstration of the power of data tables, well-applied.   It is clean and simple, and addresses both the questions pointed out in the last bullet point.  The only thing sacrificed was the visual representation of the relative size of the income requirements, which I agree is the least valuable part of the original.  As usual, many thanks to our readers for coming up with great ideas!

Amazing baseballs

Reader Jonathan S. submitted this entry.

USA Today chartjunk:

Recycled junkart (his chart):

Jonathan noticed that the scales were off (more likely, they began with an axis that did not start at zero!  This is precisely why most graphs should start at zero).

As an aside, pitchers used to point to their (frequently untoned) physique as proof that steroids could not help; now we know better.

 

Don't believe what you see

Mankiw's blog linked to a press release by the Congressman Jim Saxton, using CBO data to show "middle income tax burden at lowest level in decades".  The attached graph, as Junk Charts readers will immediately recognize, is classic chartjunk.  Every time the vertical axis does not start at zero,  one suspects something is amiss.  And what with the gridlines and data labels?

"Don't believe it? Check out the data source yourself."  I followed Mankiw's suggestion and was indeed surprised... but not by the great fortune of the "middle class".  The surprise was how the chart painted a dishonest picture of the CBO data.

The original chart plotted only the tax rate experienced by the middle 20% of the population.  The CBO provided data for all five quintiles; why not plot them all?  In this new chart (right), the "surprise" windfall to the middle 20% proved not to be anything special at all!  All five quintiles, especially the middle three, followed pretty much the same trend over time.  The effect of singling out the middle 20% is to deprive the context by which the data should be interpreted.

Further, what might be the result of the declining middle income tax burden?  The CBO data painted an unexpected picture.  Paradoxically, as the middle 20% see their tax rate decrease, they also earn a smaller share of the nation's after-tax income (black line at right).  At the same time, the top 1% saw their share of after-tax income double from about 8% to almost 16% (blue line).  The top 20% line is also upward-sloping although less pronounced.  So, the implication that the middle class have had it good is plainly wrong.

What is going on?  Two factors were at play and the Congressman presented only one side of the story (the tax rate).  What he omitted was that during this period, the nation's wealthy took home larger and larger shares of the pre-tax income.  This shift in pre-tax income more than offset any relative reduction in tax rate for the middle 20%.

This distortion can be traced back to the use of quintiles (or more generally, ranks).  We use them to cope with data having extreme distributions but a by-product is losing information about how extreme are the extreme values.  As demonstrated here, the quintiles from old are really different from the quintiles from today because the underlying distribution has become much more extreme.

Finally, another bit of mystery (to me) is how the middle 20% came to be considered "middle class".  Is there a widely accepted definition?

Reference: "CBO Data Show Middle Income Debt Burden At Lowest Level in Decades", Feb 21 2008.

Redundancy

Nick B., who occasionally writes about statistical graphics, found some classic chart junk from a Canadian report on the Afghan army.  Here's one example, together with the junkchart version.

Redundancy is an enemy of good graphics, and incongruous redundancy is worse.  Here, troop level is variously described as "total force size", "strength" and "army growth"; the chart on the right uses only the army concept.  The data labels ("47000 Strength"), the axis labels ("50000 Total Force Size"), and the gridlines all germinate from the five grand data points underlying the entire chart!

Another distorting feature is that use of different-sized time intervals, which we space out appropriately on the right chart.

Ultimately, the key message should be growth in the army size, not the absolute number of troops.  The slopes of the line segments encode this information.  Alternatively, a data table can be rather powerful for simple data like this:

By what is called the "end state", there would be 70% more troops than those as of December 2007.

 

Points of comparison

In light of the current housing crisis, arising from mortgage defaults, I pulled this graphic from a Jan 2007 opinion piece that plotted historical default rates of mortgages.  Notice the high degree of stretching on the vertical axis that exaggerates the volatility: essentially, the annual delinquency rate ranged from 1.75% to 2.65% during the last six years or so.  One might be forgiven to think that a 2% default rate is quite acceptable.

Compare the above chart to the pair that showed up in the NYT in Oct 2007 (see right).  The default rates here are in the 10-20% range, very alarming indeed.

The two graphics illustrate a key issue of "aggregation" in statistical analysis.  The first graphic is super-aggregated: all types of mortgages of all ages are put together to calculate each year's default rate.  The second graphic hones in on subprime mortgages only.

More importantly, the second graphic presents data in "vintages".  Each line represents loans originated during a particular year (a "vintage").  This establishes comparability.  On the first chart, each point in time represents the default rate of mortgages averaged over all ages (some loans may be only a few months old; others may be 15 years old).  Since the default rate is much higher for very young mortgages than for older mortgages, such averaging hides crucial information.

Overall, the NYT graphic very effectively conveys the alarming trend of new mortgages performing much worse, especially those originated in 2007.

It can benefit from two slight edits: adding a few more years, and using vertical lines (the most critical comparisons are default rates for loans of a given age!)  Something like this...

Sources: "As Defaults Rise, Washington Worries", New York Times, Oct 16 2007; "Mounting Mortgage Credit Problems", economy.com, Jan 23 2007.

Cheers

This is an exemplary chart from the NYT Sports page.  It provides a clear, informative and exciting way to visualize how the baseball season has gone for the Mets this and last year.  It's been mostly up and not much down. 

We can observe the more subtle differences: last season was a steady rise with only two prolonged down periods; this season's curve is driven by two up periods (including right now), outside of which the record has hovered around two levels (0, +3).

Especially commendable is the judicious use of axis labels.  However, I'm not clear on how some of the labels were chosen.  For example, 14 games ahead seem to me a rather arbitrary one.

All in all, a job well done.

Source: "Not Only Yankee Fans Cheering for Week 22", New York Times, Aug 27, 2007

Non-elites

From Mikhail Simkin comes some intriguing analysis of "experts"; in this line of research, experts are compared to the "general public" and often "proved" to be shenanigans. Stock pickers don't do better than apes; economists don't do better than Big Macs; you get the idea.  In a new twist, Simkin puts twelve images of modern art on his website, and asks visitors to distinguish between those by grand masters and those "ridiculous fakes" produced by him apparently on a computer.

Since conventional wisdom says elite universities provide better education, Simkin attempted to find out if there is a difference between "elites" and "the crowd" in their ability to recognize modern art. (Elites, to him, meant the Ivy League and Oxbridge.)  The following pair of histograms clinched his point:

we see that there is not much difference between the elite and the crowd.

Since the shapes of the histograms are similar, one might be inclined to agree with the statement.  This is until one notes the wildly different scales used because only 143 of the 56,020 quiz-takers could be identified as "elites".

The shapes are clarified if we use a relative scale (percentages) rather than absolute scale.  Further, the difference is more easily seen when cumulative percentages are plotted.  In other words, we are interested in comparing the proportion of respondents who score at least X points out of 12.

Two features are worth noting:

• A gap opens up between 4 to 7: specifically, 40% of "non-elites" scored 7 points or below while only 25% of "elites" scored 7 points or below.
• The curves criss-cross around 11 to 12: this shows that "non-elites" were more likely to have perfect scores (although this difference is small).  Perhaps museum directors don't have .edu addresses.

Notice that I plotted Elite vs Non-Elite rather than Elite vs All Respondents.  While it seems innocuous to use "All Respondents", and in this case, there is no noticeable difference since Elites were a tiny proportion, when the test group accounts for a significant proportion of the total, the value for "All Respondents" will be influenced by that for the test group.  As a general rule, compare A to not A.

Simkin's exercise raises many statistical issues of design, which we won't discuss here.

Source: "Properly Prescribed" (via, RSS Significance)

On the bubble

A couple of you noticed this table of bubbles in the Times, and asked what I think of it.  Dustin J suggested that this could be considered a decent application of bubble charts.  I agree, with some reservations.

The data set is the best thing about this chart.  The riches that lay beneath!  Many questions can be addressed, including:

• Which Presidential candidates are getting the most face time?
• Are candidates seen equally often across the stations?
• Are there differences between network and cable stations in terms of total face time?  In terms of individual face time?
• Are there Democratic/Republican leanings by station?  by type of station?

The intrepid can even build a regression out of it.

The bubble chart contains answers to all those questions but nothing jumps out. Okay, it's easy to see the station that gives each candidate the most face time.  Anything else requires moderate to a lot of effort.  Here's the junkart version.

The list of things done to the data is long:

• Candidates are grouped together by party
• Candidates within each party are arranged in order of decreasing maximum face time
• Stations are arranged by increasing total face time, this order happens to retain the network vs cable divide
• A heat map construct is used instead of bubbles: the legend is missing but there are four hues for each color: darkest = top 10%; medium = 10th - 50th percentile; light = bottom 50th percentile excepting zeroes; white = no face time.  In raw numbers, 90th percentile = 81 minutes, 50th percentile = 19 minutes.
• The only data shown are the totals by candidate and totals by station.
• On the right margin are little bar charts that show the distribution of network/cable for each candidate.
• On the bottom margin are little column charts showing the distribution of party affiliation by station.

A few observations follow:

• Cable stations gave much more face time to the candidates in general.  Fox, no surprise, gives Republicans 85% of its time while all the others were roughly equal.
• The more mainstream the candidate, the balanced was the time spent on networks versus cable.  John McCain (R), Hillary Clinton (D) and John Edwards (D) had the highest proportion of network time.
• More time is not necessarily good since McCain was the clear winner but his campaign is struggling

Source: "Tracking Face Time", New York Times, August 1, 2007.