Junk Charts

Worthy of the Times

Andrew Gelman has put up a great post, discussing how he collaborated with the New York Times editors to transform his chart to a publishable form.  Some time ago, there was a discussion here about the publish-worthiness of my charts (actually, the lack thereof), and I explained that to take my charts to that level would require quite a bit of extra work.  Gelman speaks here from first-hand experience.

Here are the two versions side-by-side.  Readers on the Gelman blog are debating which version is "better" with Gelman himself voting for the revised NYT version:

Let's compile a list of the changes:

1. Removed 2000 data and relied on only 2004 data
2. Reduced number of groups of senators from 7 to 5 (with a special calling out of Senator Lieberman)
3. Re-ordered senators to facilitate classifying into Democrats, Independent, Republicans
4. Added annotation explaining the grouping of senators; in particular, qualified why the 49 Democrats and 32 Republicans were set aside
   
5. Removed scale labels on the top of the chart, retaining only the bottom labels
6. Vastly increased the area devoted to text labels, which now covered half of the chart
7. Added white vertical gridlines
8. Instead of coloring the cross marks, colored the background of the chart; introduced the yellow color for Lieberman (which Gelman originally colored blue)
9. Removed reference to the national average but explained clearly in the legend that the percentages are relative to the national average.
   

Of these, 2,3,4 are definite improvements. 1 is acceptable since 2000 and 2004 data are quite highly correlated (note, however, Iowa, Virginia and Ohio.) 7 looks brilliantly here although we typically don't recommend adding gridlines and such. 5 is also acceptable because of 7 -- with the gridlines, one doesn't need to have two sets of labels. 8 can be debated: I'd like to see a version with large colored squares instead of cross marks in lieu of the background; the current look is not bad. 9 can also be mulled over: would it be so bad as to insert in parentheses 73% after the words "national average" in the legend?  I suspect some feel that this may cause confusion because readers now need to understand relative versus absolute values. 6 is the most debatable because the data-ink ratio has been vastly reduced. 

There are several good features of the original chart that they left unaltered, which should also be duly noted:

• Allowed the positive side of the scale to extend to 10% even though the largest data only reached about 6%. While this creates empty space, it helps readers judge the magnitude of negative numbers relative to positive ones.
• Subdued horizontal gridlines
• Retained the title of the chart (I actually prefer that they use the title to summarize the finding as opposed to the current one which takes a neutral stance.)
• Retained the horizontal scale labels at 5% apart

One final comment on something omitted from both charts: I'd prefer that they plot all the dots for the 49 + 32 senators, on the top and bottom lines respectively, instead of just plotting the group averages.  It would be really interesting to see what the spread is around the 0% point.

Following one's nose 2

This is the second post on the immigration paradox study, first discussed on the Gelman blog.  My prior post on the graphing aspect is here; this post focuses on the statistical aspects. I am working backwards on Andrew's discussion points.

Which difference is most interesting?

5. Agree with Andrew; they should publish similar analyses on other minority groups as soon as possible.  One thing that strikes me when looking at the interaction plot is that the U.S. born non-Latino whites have a much higher incidence of mental illness.  The difference between different subgroups of Latinos paled in comparison to the difference between non-Latinos and the Latinos.  This latter difference is particularly acute among the U.S. born than the immigrants. The importance of the Latino analysis hinges upon whether the "paradox" is also found among other minority groups.

(Chris P also pointed this out in his comment on the previous post.)

Disaggregation, Practical Significance, and the Meaning of Not Significant

2. Andrew is also right in expressing moderate skepticism about this sort of disaggregation exercise.  He connects this to the subtle statistical point that "the difference between significant and not significant is not significant."  A related but less obtruse issue is that as one disaggregates any data, the chance of seeing variations that stray from the average gets higher and higher.  This is because the sample size is decreasing, and so the statistical estimates are less reliable.

(To give a flavor of the scale, there were a total of 2500 Latinos in the sample, with 500 Puerto Rican Latinos. The analysis drilled down to the level of different types of mental disorders, subgroups of Latinos, and also adjusted for demographics.  The details of the demographic adjustment are not available but in any case, one should be concerned about whether there were sufficient numbers of say, male immigrant Puerto Rican Latinos age 18-25 with income < $10,000 living in a rental apartment, for such an elaborate exercise.)

Expanding on this point further, one observes that the measured gap between U.S. born and immigrant Puerto Rican Latinos was about 5%.  But this 5% is probably of considerable practical significance since the base rate of incidence is about 30% (I say probably since I am not an expert in mental illness).  The current statistical analysis judged this to be insignificant -- if the sample size were larger, this difference could conceivably be statistically significant, and also practically significant.

But, doesn't the significance test deal with the small sample size problem?  Yes, if the authors merely described the Puerto Rico result as inconclusive.  Here, as is done very commonly, insignificance is equated to "no difference": they said

No differences were found in lifetime prevalence rates between migrant and U.S.-born Puerto Rican subjects.

In reality, a difference of 5% was found in the sample that was analyzed.  The statistical procedure found that this difference could have been a result of chance -- notice "could", not "must".  If the measured difference was 0.5% on 30%, then I might be willing to accept a finding of "no difference"; when it was 5% on 30%, I would like to see a larger sample analyzed.

The Meaning of Paradox

1. Andrew was perplexed by why the phenomenon is known as a "paradox". I had the same issue until I read the paper. The authors were a bit sloppy in the abstract. In the paper itself, they explained that the conventional wisdom has it that immigrants should be more likely to have mental illness because of the stress from the immigration process, and yet the statistics showed the exact opposite. That is the paradox.

Publication Bias

I was a little shocked to see the data tables that gave all the estimates of the various effects at the various subgroup levels: shocked because the authors were allowed (or asked) to include only the p-values that were below some unspecified level (which I surmised is 10% although a 5% significance level is used to judge significance as per convention). This is publication bias within publication bias. P-values that are not significant still provide valuable information and should not be omitted. They did provide confidence intervals but for each subgroup separately, rather than for the difference -- and as they noted, such intervals by themselves are inconclusive when they overlap moderately.

Life-enabling charts

In response to my call for positive examples, reader Merle H. sent in an example of how good charts can make our lives simpler and easier.

All of us have seen the following presentation of air travel data.

Not trying to pick on Travelocity - it's the same format whether you use Expedia or any of the airline sites.  For those customers who are looking to decide what dates to travel so as to minimize their air fare, this format is very cumbersome to use.

What about this fare chart at FuncTravel.com?

As you mouse along the line chart, the average fare for each day is visible.  Clicking on a particular day will fix the departure or return dates.

So much easier, isn't it?

A few caveats, though:

• Instead of just providing the historical averages, they should consider including information on variability, such as bars that indicate the middle 50% or 75% of prices.  Also, what about a sliding control for customers to decide which period of past history the averages should use?  More recent data may be more representative.
• This particular feature appeals to the price-sensitive, date-flexible customer segment.  Not everyone will pick itineraries based on those criteria.  There is an easy fix. If some controls are available for customers to indicate other preferences, e.g. exclude all British Airways flights, include only evening flights, etc., and the chart can update itself based on such selections, then the chart becomes a lot more flexible, and useful to many more customers.
• As with many automatically generated charts, the chosen labels on the vertical axis are laughable.  That should be relatively easy to fix, you'd think.
  

A great start.  I happen to notice that Travelocity has a beta feature that shows a similar chart.  A revolution in how travel sites present data to us is long overdue. 

Following one's nose 1

Andrew Gelman has a great post about a so-called Immigrant paradox here, which should be interesting to our readers too.

He posed a set of sharp questions.  My read, in reverse order:

6. The graph is pretty effective, I agree.  This is known as an "interaction plot".  The message the authors were trying to send was that the gap between immigrants and U.S. born in terms of prevalence of mental illness is not constant across sub-groups of Latinos.  For example, the gap for Mexicans (light blue) is larger than the gap for Puerto Ricans (pink).  Thus, the authors concluded that one should be careful about speaking of an aggregate (average) gap.

The graph lays this out clearly.  The steeper the line, the bigger the gap between the  immigrants and non-immigrants.

When Andrew showed this, I knew for sure someone will cry foul that a line is drawn between unrelated, discrete things.  Indeed, the very first commenter weighed in with this complaint.  In fact, whenever I show such charts to non-statisticians, a lot of people have this reaction.

So I'll take this as another chance to convince you to release interaction plots from jail.

Typically, a dissenter will offer up a dot plot as an alternative.  So let's look at the same chart without the lines.  Since the reader is supposed to figure out how the gap between U.S. born and immigrant groups across different subgroups of Latinos, the proverbial nose is tracing a line from a left dot to a right dot.  Thus, to follow one's nose is to mentally draw the lines I just removed.  The chart designer has done us a favor by making the lines explicit.

In addition, as Andrew pointed out, it is always better to try to get rid of the legend and put the line labels directly onto the chart.

One shortcoming of the interaction plot is that it does not disclose the relative importance of the different lines, which correspond to the relative proportions of people in these subgroups.  Without this information, the reader will likely assume the lines have equal weight.  This assumption, as I will explain in a future post, may be a problem.

This post dealt with the graphical aspect.  I will have more to say about Andrew's other points on the statistics in a future post.

Serving donuts

David Leonhardt's article on the graduation rates of public universities caught my attention for both graphical and statistical reasons.

David gave a partial review of a new book "Crossing The Finish Line", focusing on their conclusion that public universities must improve their 4-year graduation rates in order for education in the U.S. to achieve progress.  This conclusion was arrived at through statistical analysis of detailed longitudinal data (collected since 1999).

This chart is used to illustrate this conclusion.  We will come to the graphical offering later but first I want to fill in some details omitted from David's article by walking through how a statistician would look at this matter, what it means by "controlling for" something.

The question at hand is whether public universities, especially less selective ones, have "caused" students to lag behind in graduation rate.  A first-order analysis would immediately find that the overall graduation rate at less selective public universities to be lower, about 20% lower, than at more selective public universities.  

A doubter appears, and suggests that less selective schools are saddled with lower-ability students, and that would be the "cause" of lower graduation rates, as opposed to anything the schools actually do to students.  Not so fast, the statistician now disaggregates the data and look at the graduation rates within subgroups of students with comparable ability (in this instance, the researchers used GPA and SAT scores as indicators of ability).  This is known as "controlling for the ability level".  The data now shows that at every ability level, the same gap of about 20% exists: about 20% fewer students graduate at the less selective colleges than at the more selective ones.  This eliminates the mix of abilities as a viable "cause" of lower graduation rates.

The researchers now conclude that conditions of the schools (I think they blame the administrators) "caused" the lower graduation rates.  Note, however, that this does not preclude factors other than mix of abilities and school conditions from being the real "cause" of lower graduation rates.  But as far as this analysis goes, it sounds pretty convincing to me.

That is, if I ignore the fact that graduation rates are really artifacts of how much the administrators want to graduate students.  As the book review article pointed out, at the less selective colleges, they may want to reduce graduation rates in order to save money since juniors and seniors are more expensive to support due to smaller class sizes and so on.  On the other hand, the most selective colleges have an incentive to maintain a near-perfect graduation rates since the US News and other organizations typically use this metric in their rankings -- if you were the administrator, what would you do?  (You didn't hear it from here.)

Back to the chart, or shall we say the delivery of 16 donuts?

First, it fails the self-sufficiency principle.  If we remove the graphical bits, nothing much is lost from the chart.  Both are equally impenetrable.

A far better alternative is shown below, using a type of profile chart.

Finally, I must mention that in this particular case, there is no need to draw all four lines.  Since the finding of a 20% gap essentially holds for all subgroups, no information is lost by collapsing the subgroups and reporting the average line instead (with a note explaining that the same effect affected every subgroup).  

By the way, that is the difference between the statistical grapher - who is always looking to simplify the data - and the information grapher - who is aiming for fidelity. 

Reference: "Colleges are lagging in graduation rates", New York Times, Sept 9, 2009; "Book review: (Not) Crossing the Finish Line", Inside Higher Education, Sept 9 2009.

Pie Cubed

Omegatron also did away with a set of cascading pie charts on Wikipedia, a particularly ineffective use of this chart type.  Whenever there are more than two or three categories, the necessary use of many colors can really make one's head spin.

Here, the cascade is being used like a log scale, to artificially elevate the small pieces, which unfortunately are also the least significant pieces of the energy pie.  There is no reason for nuclear, bio-mass, hydro and "others" to add to 100% except that the author decides to group them together.  The 41% nuclear or the 41% solar heating in the second and third charts, respectively, have no meaning in the larger context.

In deference to the original author, Omegatron's new version preserves the arbitrary three-level cascade.  He converts to stacked bar charts, which brings out the differences better.


He also sensibly exposes the original proportions rather than the arbitrary relative proportions.  For example, nuclear energy accounts for 6% of the total, not 41% of the arbitrary "others" bucket which in turn contains 14% of the total.

I'd prefer an even cleaner presentation with unstacked bar charts.  This can be done in either one chart with all eleven categories, or in two charts, as shown below.  The two-chart version assumes that the reader have two key questions: alternative energy sources as a proportion of the total, and the mix of different sources within the alternative category.

With the ordinary bar chart, many fewer colors are needed, and there is no need to print out each data point, nor a need to use guides to point to labels and data.  The trouble of the latter is its tendency to draw attention to the least important aspects of the data.

With this further example, I continue to find the Wikipedia rule to discourage text annotations on graphs bewildering.   Such a rule apparently does not apply to data labels, as can be seen here.  Of course, a graph without any labeling of categories is robbed of meaning but if labels can be saved, so should annotations!

Degrees of likeness 2

We left off the other day with an interactive graphic with the ability to peer into subgroups.  This feature assumes implicitly that the overall average obscures differences within subgroups.  What statisticians do with this type of data is to compare the subgroups, and identify the factors that make someone different from the average.

For example, there is a clear distinction between the employed and the unemployed in how they spend the day (not surprising).

This happens to be what NYT printed in the paper edition that day.  (Note, though, that the graphic loses quite a bit without the interactivity.)

On the other hand, there appears to be little differentiation between men and women.

Nor is there much difference between blacks and whites.

One factor that matters is age.  Older people are not exactly like the young.  A lot of these factors (for example, age and employment status) are correlated, by the way.

I showed all these in order to talk about the statistical concept of "aggregation".  We noted that the distribution of time use of the employed is different from that of the unemployed.  Thus, we cannot use the "average" distribution to describe both groups, and so we show the data in disaggregated form.  Similarly for time use and age.  

But there is not much gain in disaggregating race and gender: the "average" is representative of the subgroups for these two factors.  This is one distinction I see between information graphics and statistical graphics: the former typically shows all possible subgroups while in the latter, the designer zooms in on the factors that matter.

Degrees of likeness 1

The NYT team just put up a fantastic visualization of the American Time Use Survey data, which purports to measure how the average American spends the time of a day.  (Apparently, thousands of people recalled what they did every minute of an average day.)  The amount of data collected is massive, and this graphic allows readers to explore the data in intuitive ways.

The chart shows for each minute of the day (horizontal axis) the proportion of people doing specific activities.  Not surprisingly, we spend more time sleeping than any other type of activity.  The axis and data labeling as well as gridlines are very restrained.

Normally, I am not a big fan of these proportional area charts because the only relevant dimension to look at is the vertical distance from one curve to the next but the focus on areas put equal weight on the horizontal and vertical distances.   The horizontal distance is meaningless, and thus the area is meaningless.

These designers found a solution to the problem, and good for them!  Because of the mouse-over effect, I could not save the actual appearance -- here, I show what it looks like.

By mousing over different parts of the graph (say, moving vertically), we can compare the actual proportions.  Terrific!

The key interest of this graphic is the following legend.

While the above graphic shows the use of time by all Americans in aggregate, this panel allows us to zoom in on specific groups of Americans.  How alike are Americans?

Reference: "How Different Groups Spend Their Day", New York Times, July 31 2009.

 

Space and time

When it comes to space or time in graphics, old habits die hard.   When we have spatial data, the default is to put it on a map.  When we have a time series, the default is to plot time along the horizontal axis.  Sometimes, these defaults work; other times, breaking up the map or straight-time-line works better.

Thanks to a reader, I noticed that Google put up a "Flu Trends" website to help us track the flu season.  They use two main charts to plot the data, as shown below.




On the right side is the time series, showing the severity of flu cases from month to month.  There are many great things about this chart and one serious flaw.  I love the fact that they did not plot time on the horizontal axis; they realize the seasonality and they create overlapping lines.  They make good use of foreground and background; it's easy for us to compare year to year differences.

The serious flaw: no vertical scale.  This was a problem with Google Trends from day one (see my post here).  They still haven't fixed it.  Because of this, we don't know if the peak shown was 5 cases or 5000 cases.  While for Google key word searches, one can excuse them for trying to protect commercial secrets.  I would imagine that this public health data is, well, public.  Since the apparent purpose of this chart is to allow citizens to declare a flu epidemic (say, when they see the current trend depart from the historical norm), not having the scale is a huge problem.

I also disagree with shifting the months around for the Northern Hemisphere so that the peaks of the graphs are aligned towards the middle.  It is better for the peaks to appear on the left and let the order of the months conform to our expectation.  (The "peak" would be split on the sides and the chart would look like a valley, which presumably is why they did it this way.)

The charts on the left side plot the spatial data, not surprisingly on maps.  Sadly, the standard exhibited on the time-series charts is nowhere found on these maps.

First, the legend is seriously deficient.

Second, the gradation of the colors is not fine enough, or put differently, the aggregation to the state/province level is much too coarse for any interesting pattern to be seen.

This poorly aggregated map becomes a farce when applied to the U.S.  There is not much left to be said, is there? 

Round up

Here are some interesting reading from other places:

Tag clouds have caught on since we approved them a while ago.  One interesting use was at the Life Vicarious blog.  They use it to compare the inclinations of three New York-based restaurant reviewers.  What they should have done is to remove irrelevant words like  "one", "also", "many", "make"/"made", etc.  In statistics, this is called removing "noise" which helps bring out the "signal".

Andrew Gelman discussed the NYT article that reported the finding of unexpected male bias in the children of Asian American families.  He can be counted on to make useful comments on any accompanying graphics.  He rightly pointed out that this is one example of not starting at zero: the relevant baseline is 100 since the metric is essentially the over-age of males relative to females.  I also agree that a line chart with a longer time series plotting percentages rather than over-age would work better.

The racetrack chart made an appearance at Flowing Data.  This one is even more busy and just as impossible to decipher.