Statistical graphics: When does it make sense to introduce deliberate distortion to counteract an expected perceptual illusion?

It’s been awhile since we’ve had a post entirely devoted to graphics!

Kaiser writes:

The link here contains an example of how the line-angle illusion can lead to misreading of trends on line charts:

Is there a bigger difference in revenue at Time 1 than Time 2? Many of us will think so but on careful judgment, I think all of us can agree that the difference at Time 2 is in fact larger. . . .

Studies have shown that humans tend to read not the vertical gaps but the angular gaps. Again, this issue is illustrated in the first mentioned paper:

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Matthias explained that their implementation of the hammock plot uses a strategy to counteract this line-angle illusion.

I take this to mean they distort the data in such a way that after readers apply the line-angle illusion, the resulting view would convey correctly the correct trend. A kind of double negative strategy. The paper linked above offers one such counter-illusion strategy.

I imagine this is a bit controversial as we are introducing deliberate distortion to counteract an expected perceptual illusion.

I’m not aware of any software that offers built-in functions that perform this type of illusion-busting adjustments. Do you know any?

I’ve actually thought about this question a lot! In some sense, just about all statistical graphs introduce deliberate distortion to counteract an expected perceptual illusion, in the sense that, with the exception of maps and astronomical charts, a graph is an abstract representation of data.

But to get closer to what Kaiser is asking: the analogy I’ve given is, suppose you’re building a wooden chair but using boards that are warped. In this case, the right thing to do is to incorporate the warp into the design, i.e. cut some pieces shorter than others and at different angles, etc., so that they fit together as is, rather than trying to go all rectilinear and then glue/nail everything together. The trouble with the latter strategy is that the wood will exert pressure on the joints and eventually the chair will break or distort itself in some way.

So, similarly, if you can anticipate that a graph will be misread, it’s a good idea to account for this possibility in the design.

Most simply we do this by just not making graphs such as tilted pie charts, 3-D bar charts, and other gimmicks that jump off the page and engage all sorts of visual illusions.

The other common option is to make an additional graph. For example, in you could keep the graph above with the two time series but just add another graph showing their difference. Why not?

But what about Kaiser’s original question: are there any graphs with deliberate distortions designed to counteract perceptual illusions of visual artifacts? (I guess that a log transformation doesn’t count here.)

I don’t have any perfect examples here, but I have one example from my applied research that comes close.

The example comes from my paper with Yotam on social penumbras. First there’s the explanatory diagram:

Then the data graph:

There are two things going on here.

First, the diagram is a circle but the data graphs are quarter circles, which we did for two reasons: (a) the quarter circle takes up only a quarter as much space, which is important when we’re displaying 14 of these at once (yes, you could just make smaller full circles but then you have only half the resolution when comparing sizes), and (b) for the goal of comparing one group to the next, quarter circles are better because you can compare the slices, as compared to full circles which all just look like bullseyes and are hard to tell apart.

Second, areas of shapes are notoriously difficult to compare. Why did we do these damn circle plots at all? Why not dot plots or repeated bar charts or some other visualization that would facilitate linear comparisons? The answer is that it was important to us to preserve the “feel” of the penumbra, the idea of concentric social groups. We were willing to pay a bit in statistical clarity in order to have this conceptual unity of the graph and the content of the paper.

But then the issue arises that, when comparing areas, people don’t really compare areas. Nor do they compare linear dimensions. At least according to Cleveland’s classic book, the implicit comparison is something in between. So by displaying the data as areas, we’re knowingly handing people a distortion. For example, if a certain group represents 1% of the population, then the core group (the yellow circle in the graph) will take up 1% of the area of the full circle and thus will be 10% in linear dimension.

That’s bad, right? Maybe not! Several of the groups in our study did have core populations, and if these were displayed as 1% in the linear dimension, they’d be really hard to tell apart. By using these intuitive-looking area graphs, we’re implicitly doing a square-root transformation without having to explain it. So I think it’s fair to say that we’re taking advantage of a perceptual illusion.

P.S. Pro tip: Do you see how in our graph, we order the groups by increasing size. Not alphabetically. You should almost never display your data alphabetically. OK, here’s a rare counterexample, with a slightly prettier visualization and some more graphs in Section 2.3 (“All graphs are comparisons”) of Regression and Other Stories:

For these we displayed the data straight up, no distortion.