Time changes things and ideas. Lately I was thinking that maybe the chart I’ve described three years ago for a two-way table could be not so impressive as I was expecting. Nevertheless I thought continually if in some cases it could be useful to feature its two simple properties: different bar widths and equivalence of their areas with the underlying cluster area to show the contribution of each row (or column) to the overall total and the contribution of each value to the row (or column) total.
Eventually I figured out that the visualization of Simpson’s paradox is one of these instances. Wikipedia page shows three known different graphic representations of Simpson’s paradox: a correlation scatterplot for the continuous case, and two diagrams corresponding to its vector and physical interpretation for the discrete case.
Let’s see the chart I want to speak of for the same (revised, since at present there is a discrepancy between data and diagram) Wikipedia example corresponding to physical illustration.
Lisa and Bart edit document articles for two weeks. In the first week, Lisa improves 60 of the 100 articles she edited, and Bart improves 9 of the 10 articles he edited. In the second week, Lisa improves 1 of 10 articles she edited, while Bart improves 30 of the 100 articles he edited.
Like in any other bar chart, heights represent values, that is, in this example, percentages of improved documents. In addition, bar widths represent the total number of articles so that bar areas correspond to the number of improved articles. For example, about the bars on the left referring to Lisa, the light blue one has width proportional to 100 and area proportional to 60, while the dark blue one has width proportional to 10 and area proportional to 1. The same for the bars on the right referring to Bart. The bars in the background correspond to their average rates and their areas equal the sum of the bar areas for the first and second week. For example, for Lisa it is 100×0.6+10×0.1=110×0.556 because 100×0.6+10×0.1/110=0.556.
It should be easy to understand visually why Lisa overall improvement rate is higher than Bart one, although her rates are lower for both the first and second week: it’s for her overwhelming number (and high rate) relative to the first week compared to his overwhelming number (but low rate) relative to the second week. So Simpson’s paradox arises for the very different weight of first week and second week in Lisa and Bart data.
More generally the paradox depends on the different sample size and value for confounding factor among categories, which in the chart combine in the different mass for bars of the same color. See the following other examples drawn from various sources.
A common example of Simpson’s Paradox involves the batting averages of players in professional baseball. It is possible for one player to hit for a higher batting average than another player during a given year, and to do so again during the next year, but to have a lower batting average when the two years are combined. This phenomenon can occur when there are large differences in the number of at-bats between the years. […] A real-life example […] involves the batting average of two baseball players, Derek Jeter and David Justice, during the baseball years 1995 and 1996 [and 1997].
This is a real-life example from a medical study comparing the success rates of two treatments for kidney stones. The table shows the success rates and numbers of treatments for treatments involving both small and large kidney stones, where Treatment A includes all open procedures and Treatment B is percutaneous nephrolithotomy. The paradoxical conclusion is that treatment A is more effective when used on small stones, and also when used on large stones, yet treatment B is more effective when considering both sizes at the same time.
|Treatment A||Treatment B|
|Small Stones||93% (81/87)||87% (234/270)|
|Large Stones||73% (192/263)||69% (55/80)|
|Both||78% (273/350)||83% (289/350)|
One of the best known real life examples of Simpson’s paradox occurred when the University of California, Berkeley was sued for bias against women who had applied for admission to graduate schools there. The admission figures for the fall of 1973 showed that men applying were more likely than women to be admitted, and the difference was so large that it was unlikely to be due to chance. But when examining the individual departments, it appeared that no department was significantly biased against women. In fact, most departments had a “small but statistically significant bias in favor of women. The data from the six largest departments are listed below. The research paper by Bickle et al. concluded that women tended to apply to competitive departments with low rates of admission even among qualified applicants (such as in the English Department), whereas men tended to apply to less-competitive departments with high rates of admission among the qualified applicants (such as in engineering and chemistry).
In a 1991 study by Radelet and Pierce of the effect of race on death-penalty sentences, […] we see Caucasian defendants received the death penalty more often than African-American defendants.
Now, we consider the very same data, except that we stratify according to the race of the victim of the murder. Below is the table.
Here we see that when considering the cases involving Caucasian victims separately from the cases involving African-American victims, that the African-American defendants are more likely than Caucasian ones to receive the death penalty in both instances (22.9% vs 11.3% in the first case and 2.8% vs. 0.0% in the second case).
|Caucasian defendant||African-American defendant|
|Caucasian victim||11.3% (53/467)||22.9% (11/48)|
|African-American victim||0% (0/16)||2.8% (4/143)|
|Both||11% (53/483)||7.9% (15/191)|
Almost 50 years ago, Congress passed the landmark Civil Rights Act of 1964. An analysis of party voting patterns shows a good example of Simpson’s Paradox. Eighty percent of Republican senators voted in favor of the Act compared to just 61% for Democrats. This despite the conventional notion that Democrats are more supportive of civil rights legislation. When a third lurking variable is added we see Simpson’s Paradox at work. If the voting patterns are further broken down by region (North vs. South) a different pattern emerges. Northern Democrats were more supportive of the Act than their Republican counterparts (94% vs. 85%). Similarly, Southern Democrats were more supportive (7% vs. 0%).
As shown in the chart below, region was a more predictive variable for voting preference than party affiliation. Because a much greater number of Democratic senators were from the south (94 vs. 10) combining regions distorted the total outcome.
|Northern||94% (145/154)||85% (138/162)|
|Southern||7% (7/94)||0% (0/10)|
|Total||61% (152/248)||80% (138/172)|