Subtext is not invariant under linear transformations

by PhilGoetz1 min read23rd Mar 201013 comments

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Probability & Statistics
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You can download the audio and PDFs from the 2007 Cognitive Aging Summit in Washington DC here; they're good listening.  But I want to draw your attention to the graphs on page 6 of Archana Singh-Manoux's presentation.  It shows the "social gradient" of intelligence.  The X-axis is decreasing socioeconomic status (SES); the Y-axis is increasing performance on tests of reasoning, memory, phonemic fluency, and vocabulary.  Each graph shows a line sloping from the upper left (high SES, high performance) downwards and to the right.

Does anything leap out at you as strange about these graphs?

What leapt out at me was, "Why the hell would anybody make a graph with their independent variable decreasing along the X-axis?"

Socio-economic status (SES) basically means income.  It has a natural zero.  The obvious thing to do would be to put it on the X-axis with zero towards the left, increasing towards the right.  These graphs have zero off somewhere on the right, with income increasing towards the left.  That's so weird that it couldn't happen by accident.  It would be like "accidentally" drawing the graph with the Y-axis flipped.

What could be the intent behind flipping the X-axis when presenting the data?

If you drew the data the normal way, it would suggest that there's a natural zero-level to both SES and cognition; and that increasing SES increases cognition, possibly without limit.

But when you flip the X-axis, you're limited.  You can't go too far to the right, or you'd hit zero.  And you can't go off to the left, because we don't think that way in the West.  We start at the left and move right.

By flipping the X-axis, the presenter has communicated that SES and intelligence have natural bounds.  Instead of communicating the idea that higher SES is a good thing that leads to higher intelligence, this presentation of the data suggests that the leftmost point on each graph (the anchoring point) is "normal", and a lack of wealth has caused a deficiency in people of lower SES.

(And, of course, rotating around the line Y=X would suggest that intelligence makes you rich.)

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Brilliant title.

It seems to me that it's affine transformation, not linear, insofar as we consider the drawn intersection point of the axes to be the zero point of display-space. (It would be linear if the data were on the left of the axis.)

(Context: I am currently taking a course on linear algebra, so this came readily to mind, but I may lack further relevant information.)

Also, looking at the slides, the x axis is increasing rightward in its (unfortunately black-on-dark-blue) labels. So it's not so much the graph as the scale being used that is flipped. Perhaps they simply plotted the data using default sort-numerically-increasing software settings and didn't think about it too hard since they're used to working with that scale.

It seems to me that it's affine transformation, not linear

Hah! You're right.

These charts are just extremely poor. What's the point of using all those 3D effects and obnoxious colors? Charts should be boring and minimalistic by default, so that the patterns in the data can stand out.

A redditor in r/Anarchism just posted a semi-scholarly article on this topic.

Socio-economic status (SES) basically means income.

That might be true in a given society at a given time. But there is a major difference in that income is absolute and positive sum, while SES is relative and zero sum. So there is an upper bound to SES: When you dominate everyone else. And if you are looking for social benefits rather than competitive ones, increasing income is an option, but increasing SES is not.

David D. Friedman (and I) disagree with you: http://daviddfriedman.blogspot.com/2006/10/economics-of-status.html

(The details of the social and political system you live in can change your ability to take advantage of those things Prof Friedman describes. Over most of the developed world you have lots of opportunity to multiply the universes across which SES is zero sum.)

Great link, thanks!

I think the presenter took the data from an earlier study. The numbers on the x scale are categorical.

That's a good point about SES having a natural upper bound. It's not really a natural upper bound, because you have to know the population size and choose the number of categories you want in order to see where the mean of your top cluster falls. (Or else you have to plot Bill Gates on your graph.)

So you think that this was deliberately done in order to obscure the implication that high SES was a good thing? Very possible.

I wouldn't say it that way, because I think "deliberate" denotes one rarely-used corner point in a 2D space, where one dimension is the degree to which an action was affected by a desired outcome, and the other dimension is the degree to which the agent was aware of this.

What leapt out at me was, "Why the hell would anybody make a graph with their independent variable decreasing along the X-axis?"

Yup, that's pretty much what I thought too -- and it was just your description of the graph that prompted the thought, not even the graph itself.

Yep, that was my immediate question too, before reading on past your first paragraph - "Why decreasing?"