Junk Charts

Don't mess with the scale

My friend Patrick pointed out the single biggest issue with the chart below -- that the designer chose a scale that precisely undermines the message of the chart.  Undermine may be too mild a word to use here; annihilate may be more apt.

The lines in this chart are anchored at the zero point on the time line (horizontal axis) used to indicate the bottoms of various bear markets in the Dow from 1929 to 2007.  From that anchor, time runs to the left showing the amount of time for the Dow to go from peak to bottom (the decline); time runs to the right showing the amount of time for the Dow to climb back to the prior peak (the recovery).  As the caption said, the point of the chart is "if the decline was fast, the recovery took a considerable time".

Funny thing then that the distances from the zero point are roughly comparable on the left as on the right.

This illusion resulted from some very convoluted and perplexing messing around with the horizontal scale.  First, the left-of-center scale is in months while the right-of-center is in years.  Second, the left-of-center scale has normal spacing while the right-of-center seemingly was suffering from spasms.  Take a closer look:

The first five years (0-5) took up about half the scale while the next five (5-10) took maybe one-eighth.  The first year (0-1) took about as much space as the next two years (1-3).

I am not quite sure what is the logic behind this but since the message of the chart has everything to do with the time duration, it is most unfortunate to introduce such distortions.

There is yet another "innovation" in this chart.  Notice that on the right side, the axis labels are irregular (more spasms)... 0,1,3,4,5,10,15,20, 25...  This is as if the designer is posing one of those IQ questions requiring readers to figure out the next number in the sequence.  The specific time intervals selected may have meaning: note that all the lines are straightened out in between these tick marks.  Given that each line represents a different historical sequence, it is difficult to comprehend the regularity of these intervals across history.  Perhaps this will prove to be the key to unlocking the secret of this chart.  Please comment below if you are able to unravel this mystery.

Besides, the same type of "innovation" was not applied to the left side of the chart.  Here, the designer opted to throw out all the data between the peak and the bottom and straightened out all the intermediate fluctuations.

Below are two different versions of this chart, basically restoring the time scale to the normal, equally spaced, symmetric appearance.  The top one used monthly Dow returns where the volatility obstructed our understanding of the trends, requiring the use of color to differentiate the lines.  In the next version, I used R to generate the loess estimates (a type of smoothing) and the trends became clearer.  (There was a prior discussion of loess on Junk Charts here.)

 

Now, these pictures are very different from the original graph!

I'd be very cautious about reading into these charts anyway.  This question is not one suitable for statistical analysis.  The sample size of six is far too small.  Each recession is different in terms of causes, remedies and context.  The fact that we call them recessions do not make them comparable.  Further, it is also impossible to know at this stage if the 2007 decline has reached bottom.  The chart designer essentially assumed this to be the case but who knows?

PS. Nick Rapp, one of the designers of the chart, responds in the comments.  He has started a blog to feature the work of his graphics team at AP.  His colleague has created an interactive version.  More than anything, this post highlights an aspect of the chart that Nick and his team clearly spent a lot of time doodling over.  The concept of the chart itself is wonderful actually, if I didn't say so already; it is essentially the same chart as the oft-printed chart where the anchor point is the start of each recession, only here the anchor is the bottom of each recession.

Disseminating junk

Tuesday was an up day in the stock markets.  What did the headline writers think?

Wall Street resumes rally following housing report  (Associated Press; link)

What did this great housing report say?  Here was a version from Marketwatch, with my comments.

Housing starts surge 22% on apartment building

It was the largest percentage gain in 19 years and was the first increase in eight months in the sector that was at ground zero in the global economic recession.

At the minimum, they claimed to see a local maximum, perhaps even a global maximum

The housing data in winter months are especially volatile because of the weather.

Was someone hedging?  It was unclear at this point how important this piece of information was.  Some professionals expressed skepticism that the news was really positive:

"We're inclined to write this off as a weather-related fluke for now, " wrote economists for Wrightson ICAP.

Notice they hedged too, using the words "inclined to".  Perhaps they had a reason to doubt the news.  The article now cited less rosy numbers:

But despite February's gain, housing starts are down 47% from a year ago, and are down 74% from the peak in early 2006. Permits are down 44% in the past year.

And even more bad news:

The National Association of Home Builders reported Monday that its sentiment index was stuck at 9 on a scale of 1 to 100 in March.

Did that really say 9 out of 100?  Sounded really bad for a nation of optimists and grade inflators.  One had to start wondering about the headline.  Was that 22% growth a fluke?  Did it mean anything?

Now, two-thirds of the way down, the article revealed its essence:

The government cautions that its monthly housing data are volatile and subject to large sampling and other statistical errors. In most months, the government can't be sure whether starts increased or decreased. In February for instance, the standard error for starts was plus or minus 13.8%. Large revisions are common.

The key news about the sampling variability was a throw-away line ("for instance").

Here, the author was confused about standard error and margin of error.  Standard error, being a standard deviation, cannot be a negative number.  The range of 22% plus or minus 13.8% is a margin of error.  This is a gigantic range, between minus 8.2% to 35.8%.  While we conclude that the growth was statistically larger than zero, we have to wonder whether this number has historical significance (best in 19 years?)

Worse than that, in any month when the growth is less than 13.8%, this level of sampling error means the growth rate is not statistically different from zero.  Thus, the most newsworthy sentence of the entire piece was:

In most months, the government can't be sure whether starts increased or decreased.

In other words, this statistic of growth in housing starts is junk.  If we really want to examine this statistic, the survey needs to use a much larger sample.

MarketWatch is to be commended for noticing the volatility issue.  Of the dozens of articles that came up in a Google search about the housing starts data, none others mentioned this problem.

The original government press release is here (pdf), with all the fine print.

Eternal optimist

Chris P pointed us to the "Financial Comeback" calculator, surely a well-meaning joke from the folks at the Times.  Here is how one gets to make a 40% loss back in just six years!


Surely, someone has to tell them about simulation.  They have to assume a probability distribution on the annual returns, and show us some sample paths.  Using the average annualized historical return in essence wipes out all variability and no wonder it's smooth sailing upwards.  Eternal optimist.  

Here is Chris' comment:

The bad news is that the range of values it offers does not include the return on the market from last year (-31% to -36%).  I guess they are optimistic.

The interactive features of this chart, however, impressed me.  The smooth adjustment to the chart as one slides the control, including the automatic choice of appropriate axis labels, is very nice indeed.

Okay, they want to trademark the name of the calculator so perhaps this is serious.

Reference: "Calculate your financial comeback", Jan 6 2009.

A Harvard mess 2

Jon's comment on the previous post pretty much anticipated this post.   The prior post concentrated on graphical matters.  However, the biggest issue with that chart is the choice of metrics.  If the idea is to explore the potential adverse effect of a sharp decline in endowment investment performance, then it is not clear why one should be comparing the proportion of endowment funds and the proportion of operating revenues paid for by endowment funds.  A missing element from these two series is the relative size of the budgets of these different departments.

The next chart shows the proportion of each school's operating costs accounted for by endowment funds together with the total size of its operating costs.


We can turn the ratio around and directly compute how much of the total amount of endowment funds distributed for operating costs is accounted for by each school.  This is really the simplest metric that gets to the question.

 

There are really two possible worries: for the School of Arts and Sciences, who pays for just about $1 billion costs with endowment funds, any significant reduction in distribution will leave a gaping hole; for a department like Radcliffe that pays for over 80% of its operating budget out of endowment funds, obviously a reduction in distribution can cause problems but we are talking about a base of $18 million rather than $1 billion.

Reference: Harvard University Financial Report, 2008; Harvard Fact Book 2007

PS. The first source did not contain any data on operating budgets so the first set of graphs (now replaced) did not show what I intended to show.  The new ones used the right data and had the right order of magnitude in terms of budgets ranging from millions to 1 billion.  The 2008 data are not available as of yet.

R power, math stats power

Amusingly, the New York Times finally got wind of the R software.   See the article here.

We in the statistics community owe these folks a lot of gratitude for developing such a flexible, powerful software.  It is unfortunate that they didn't mention graphing as one of the great strengths of the software.

Equally amusingly, the Wall Street Journal told us what we already know, that we have the best jobs in the world.  Their discovery here.


 

Opinion graph

Sharon P. pointed us to this graph, which can make one speechless.

Reference: "Bulls, Bears, Donkeys, Elephants", New York Times, Oct 14 2008.

A joke

For light entertainment, see this chart sent in by Chris P.  Original here.

Chris said: "his graph breaks a few rules, but it has a clear message".

The shocking, out of the box column certainly grabs attention, and it is probably true that football coaches earn too much money.  But the chart really falls down on this one issue:

What's the median salary of these football coaches?

Reference: "Academic salaries", PHD Comics.

Dealing with skew

Bernard L pointed us to this income distribution chart printed in the Economist.

The accompanying paragraph points to the range of the bars, that is, the gap between the top decile average and the bottom decile average, as evidence of income disparity, concluding that the US and Britain are among the worst.

Bernard likes the use of vertical sections to represent the average incomes by decile and dislikes the USA-Today style background image.  Agreed.  But why plot the middle deciles at all when the only worthy data involve the endpoints of the bars?

A close examination of the spacing of the middle deciles leads to more befuddlement.  There does not appear to be much difference between the countries.

The answer to this is that decile statistics are not appropriate for data as skewed as incomes.  At the high end, the 10% intervals are too coarse.

One clue to this is that the top 10% in the US only earns $90,000 on average but we have all heard of the billion-dollar hedge fund managers and Wall Street bankers and $30 million a movie celebrities.  The problem is that within the top decile, the income distribution is also tremendously skewed. 

The neat idea of plotting the vertical sections indicates an awareness that the red dots (average income) are insufficient because of the skew.  Alas, there remains a lot of skew above the top decile and the designer inadvertently falls back into the same trap by considering the average income within the top 10%.  Thus, the amount of disparity on the right side of the chart is grossly underestimated.  Roughly speaking, we are looking at 10 samples of the distribution, nine of which at the low end of the range and only one at the top end (long tail).  Here is the idea:

Reference: "Spreading the wealth", Economist, Oct 21 2008.

Mind the gap

When comparing two time series, one typically wants to discuss the size of the gap as it changes over time.  This Business Week chart, for example, depicted for readers the expanding gap between intra-day high and low prices of the S&P 500 for 2008.


This chart construct is effective at pointing out large changes but lacks precision in conveying smaller differences, or trends.  It is always a good idea to plot the gap directly, as we will show below.

More importantly, a better choice of scale can help a lot.  By focusing exclusively on variability (extreme values), this chart hides the relevant information of the closing prices of the S&P.  A point spread of a 100 points means more when the index is at 800 than at 1200.  In order to capture this, we can divide the point spread by the opening price of that day so we say the gap is one-eighth or one-twelfth of the opening price. 

The junkart version makes both changes.  The top chart fixes the scale, plotting the point spread as a percentage of daily opening prices.  Relative to the original chart, the variability in the front part of 2008 was muted because the index was at higher levels back then. 

The bottom chart plots the gap sizes (lengths of the high-low lines).  It is without doubt that directly plotting the gaps showcases the key message.  The current level of volatility is more than double what occurred at the beginning of the year.

If one wants to illuminate the trend as opposed to daily fluctuations, a further improvement will be using moving averages.

For those interested, shown below is a scatter plot that compares the original point spread and the derived point spread, which shows that the change is not trivial.

 

Reference: "The Market: A Daily Roller Coaster", Business Week, Oct 27 2008.

How to read a graph

Via Gelman, here is a nifty book-buying map from Amazon, displaying the split between "red books" and "blue books" bought by Amazon users in each state in the months leading up to the 2004 and 2008 presidential elections.



Gelman noted the similarity between the Amazon map and the red-blue split of rich voters.

This post is about how to read a graph.  Here are some things that come to mind looking at the map:

• Sampling bias: how does Amazon's customer base compare with the U.S. population, or rich voters?  It would be prudent to check this before making generalizations.  Gelman's point may be that Amazon customers behave like rich voters.

• Sampling period: is the period long enough to capture the average inclination of the book buyers?  As is well known, book sales follow a long-tail distribution (Chris Anderson wrote an entire book based on this observation.)  Best-sellers have a disproportionate influence on average values.  If the time period is too short, the data may only represent the best-sellers.  Consider the following two maps in successive periods in 2004:

Much of the red in the first map was due to John O'Neill's "Unfit for Command", published in August 2004, and much of the blue in the second map was due to John Dean's "Worse Than Watergate", published in April 2004.  If one of these two-month periods was used to draw conclusions, we would make big mistakes!

• Classification: The long-tailed nature of book sales has wide-reaching implications on interpreting the data.  The most essential feature is that single books (bestsellers) have a disproportionate impact on average sales.  Since the key metric here is proportion of red (or blue) books, it follows that whether a best-seller is classified as red or blue makes a huge difference. 
  

Thus, one of the first things to look at is Amazon's helpful explanation of how they classified books as "red" or "blue".  We learn that they also have "purple" books which are those they could not decide if it's red or blue.  Each red or blue book is given equal weight but it appears that purple books are not tallied.  Glancing at the list of purple books, I see some hugely important books, e.g. Ron Paul's "The Revolution: A Manifesto" (Amazon rank #56  among all books), Tom Friedman's "Hot, Flat and Crowded" (#15).

If the purple books include best-sellers, then the decision to call it purple rather than red or blue causes an influential book to be excluded from the calculation.  We often forget that the decision to exclude is not a neutral decision; it is an active decision that says the excluded data contains no useful information.
 
This is not to say that excluding those books is the wrong decision.  We must make these decisions with considerable care, and realize that excluding best-sellers when book sales have a long-tailed distribution must not be taken lightly.

• Causality: Lets say we are sufficiently satisfied that we can make a statement about book buying habits and voting behavior.  Then we need to think about the direction of causality.  Is the map saying that red book buyers are likely to vote red?  Or that red voters are likely to buy red books?  No prolonged staring at this data set will resolve this issue as other data would be needed to address it.

The more data is used to create a graph, the harder our task is to interpret it.  But the pay-off for spending the time is all the sweeter.  Happy graph-reading!

One final note: there is no doubt that this interactive map feature is a brilliant marketing move by Amazon.  This is a great and fun way for readers to find interesting books.

Reference: "Amazon, U.S.A.", Gelman blog, Oct 5 2008.