(Cross-posted with various interactive elements (videos, expandy/contracty boxes) that required slightly awkward substitutes here.)

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Is there a general factor of intelligence?

This question is a trap. If you try to answer it, you’ll find yourself beset by semantic questions. What’s intelligence? What’s a factor? And if you get past those, you’ll then find a bleak valley of statistical arcana. What do the eigenvalues look like? Do they imply causality?

This is all backward. If your goal is to understand the external world, you can skip the hand wringing and start by looking at the damn data. So let’s do that.

Physical tests are correlated

To start, let’s forget about intelligence, and ask an easier question: Does it make sense to refer to some people as “more physically fit” than others? Is that reasonable, or do we need to break things down by strength, speed, coordination, etc.?

To answer this, I looked for studies that gave people batteries of different physical tests. I found three that make enough information available to analyze.

Paper	Population
Baumgartner and Zuidema, 1972	283 male and 336 female college students in Michigan
Marsh and Redmaye, 1994	105 students at two private girls' schools in Sydney
Ibrahim et al., 2011	330 Malaysian students aged 12-15

Here are the correlations among the different tests in the first study. The columns are the same as the rows---so the 3rd square in the first row is the correlation between hand grip and pull-ups.

This is males. (Females are similar, except with lower correlations in upper-body strength.)

Here are links for Marsh and Redmaye and Ibrahim et al..

Most tests are positively correlated, and none are negative. Is this surprising? I don't know. For our purposes, we just want this as a point of comparison.

Mental tests are correlated

We can do the same analysis with batteries of mental tests. For whatever reason, many more studies have been done with mental tests, so I picked four of the best studies.

Paper	Population
Alderton et al. 1997	12,813 members of the US Navy, Army, and Air Forces
Deary, 2000	365 representative Scottish people
Chabris 2011	111 Boston adults 18 - 60 years old
MacCann et al., 2014	688 students from colleges around the US.

Here are the correlations in the first study.

This test battery is designed to measure aptitude for various military tasks. Some of these, like tracking and target identification, are partly physical.

Here are links for Deary and Chabris and MacCann et al..

What do we see?

The same basic pattern holds for both physical and mental tests.

First, almost everything is positively correlated. You might imagine that people with more upper-body strength would be worse runners---what with the extra muscle they need to carry around. You might imagine that people who are good at paragraph comprehension would be worse at math. But that's not what happens.

Second, more similar stuff is more correlated. It's natural that chin-ups are strongly correlated with pull-ups, or that arithmetic reasoning is strongly correlated with mathematics knowledge. It's more surprising that hand-grip strength is correlated with the 75-yd dash or that paragraph comprehension is correlated with target identification. These more surprising correlations are weaker (but still positive).

Third, the results are robust. The tests span several decades, different countries, and many different test batteries. The basic pattern doesn't change.

Things are correlated. No one seems to seriously dispute this. So why all the debates?

For one thing, the discussion sometimes ascends into meta-controversy. There are many arguments to be had about the definition of "general intelligence". Some people even debate if there is anything controversial! (I take no position here, but note that the "not surprising" camp doesn't seem to agree on why it's not surprising...)

On the lower planes of argument, the main issue is if the tests are just correlated or if there's something deeper going on underneath of them. Here, the burden of proof falls on whoever claims there is something deeper.

Aside: The mental correlations are somewhat stronger than the physical ones, but don't take that too seriously. The mental tests used more diverse populations than the physical tests. Imagine doing physical tests on a group of 20-year-olds. If you throw in a bunch of 80-year-olds, they'll be worse at everything and correlations will shoot up.

Factor analysis is like a cigar

The typical argument that there's something deeper happening relies on a statistical technique called factor analysis. This is usually described with fancy technical language, but the basic idea is just that you can summarize all the tests for each person using a single number.

Let's make this concrete. Say you go out and grab random people and test them, and get this data:

Person	Test 1	Test 2	Test 3
Antonio	1	-1	1.5
Bart	.67	-.67	1
Cathy	-0.5	0.5	-0.75
Dara	-2	2	-3

You can visualize the data as a magical rotating point-cloud:

Now, notice something important: This data is special, in that the points fall along a straight line. This means that even though each person took 3 tests, you can represent each person using a single number, namely their position on the line. If you tested lots of people, and the data looked like this, then each person's position along the line would be the "general factor".

Of course, real data would never exactly look like this. It would have noise! To reflect that, we need to build a "model". That is, we will try to build a "simulator" that can make fake data that (hopefully) looks like real data.

Simulations

The simplest simulator would be to just generate people along a line. First, pick some direction of variation. Then, to simulate a person (i.e. a set of test scores), draw a random number g from a standard "bell curve" Normal distribution to represent their position along the main direction.

Here's an example, where we choose a direction of variation similar to the dataset above. If you simulate a bunch of people, you'll get a dataset that looks like this:

Of course, real data will never look like that---there will always be "noise", either from measurement error, or from certain people randomly being good/bad at certain tasks. To account for this, let's update our simulator, by adding some random noise to each point. This produces data that looks like a cigar.

The critical thing here is that cigars are rotationally symmetric. If you "roll" the point cloud along the main axis of variation, it still looks basically the same.

Now we can finally say what factor analysis is. It's an algorithm that takes a real dataset and adjusts the shape of the cigar so that the simulated data will look as much like the real data as possible. It can modify the direction of variation, and how "thick" the cigar is, but that's it. (Note: all this describes the simplest possible variant of factor analysis, which is all we need here.)

If your dataset looks like a cigar, factor analysis will fit well. If not, it will fit poorly. Here's an example of the kind of data factor analysis can't represent:

The meaning of cigars

Factor analysis tries to approximate your data with a cigar. Why should you care about this?

Let's back up. As we saw earlier, physical and mental tests are correlated. If you learn that Bob scored well on paragraph comprehension that raises your estimate for how Bob will do on coding speed.

But say your data was a cigar. Take Bob's position along the length of the cigar, and call it g. Say Bob's value for g is low. If that's all you know, and you had to guess Bob's coding speed, you'd give a low number.

Now, suppose that in addition to g, you learn that Bob did well on paragraph comprehension. How does this change your estimate of Bob's coding speed? Amazingly, it doesn't. The single number g contains all the shared information between the tests.

In a cigar distribution, once you know g, everything else is just random noise---one test no longer tells you anything about any other. (Mathematically, once you control for g, the partial correlations of the tests are zero.)

In a non-cigar distribution, this doesn't happen. There's no single number that will make all the tests uncorrelated. Some interesting structure would remain unexplained.

Mental tests aren't not cigars

So, what does real data look like? Is it a cigar? Can we capture all the structure with a single number?

Here I took the earlier cigar data, and manually drew three lines to capture the "shape" of the data:

The blue line corresponds to the main direction of variation, while the shorter red and green lines correspond to the random noise added to each point. You can see that the shorter lines are the same length. This happens because factor analysis models are rotationally symmetric.

In contrast, here's the earlier "non-cigar" data:

Here, the shorter green and red lines are different lengths, reflecting that there is no rotational symmetry.

OK, I lied. I didn't draw the lines manually. There's a simple algorithm that can automatically compute these for any dataset. (By computing a singular value decomposition of the covariance matrix, if those words mean anything to you.) The details don't particularly matter, just that we can automatically find lines that span a point cloud. This will be important when we move beyond three dimensions.

So now we have a plan: We will take a real dataset, compute these lines, and see how long they are. If we have one long line and a bunch of equal-length short lines, then the data is cigar-like, meaning that a single variable explains all the "interesting" latent structure. If we have a bunch of lines of random lengths, then the data isn't cigar-like, meaning that we can't summarize things with one number.

I'd like to show you the real data from the datasets above, but none of them seem to be publicly available. Still, we can approximate it by generating data from a multivariate Normal with the known covariance.

Here are the first three tests of Alderton et al.'s (Paragraph comprehension, work knowledge, and general science).

It's not a perfect cigar, but it's not exactly not a cigar either. Here are the relative lengths of the three directions in decreasing order:

1st direction (blue line):   0.890
2nd direction (red line):    0.362
3rd direction (green line):  0.279

What if we use all seven tests? We can't make pretty pictures in seven dimensions, but we can still do the math. With N tests, a factor analysis model always produces 1 "long" direction and N-1 "short" directions. If we plot the length of the directions, it should look like this:

In contrast, here's how things would look if all the tests were completely uncorrelated.

What do the lengths on real data look like? Well, judge for yourself. Here's Alderton et al.:

And here are Deary and Chabris and MacCann et al..

Do these look exactly like what factor analysis can produce? No. But it's a reasonable approximation.

Directions of variation

Here's another way of visualizing things. For any dataset, we can take the principal direction of variation (the blue line) and look at its length along each of the tests. This says, essentially, how much each of the tests contributes to the main direction of variation. Here's what we get if we do that for Alderton et al.:

Calculating g is similar to taking a simple average of the test scores, though the weights are slightly higher on some tasks than others.

If we calculate g like this for each person, we can then compute the partial correlations. These are the correlations once you control for g. Here's what that gives for Alderton et al.:

Mostly it's a sea of gray, indicating that the partial correlations are all quite small. The other studies look broadly similar.

If factor analysis was a perfect fit, these would all be zero, which they aren't. But they are pretty small, meaning that in each case, the single number g captures most of the interesting correlations.

What would g look like?

Factor analysis is a decent but not perfect model of mental tests. What does this tell us about how intelligence works? Well, suppose that factor analysis was a perfect model. Would that mean that we're all born with some single number g that determines how good we are at thinking?

No. A perfect fit would only mean that, across a population, a single number would describe how people do on tests (except for the "noise"). It does not mean that number causes test performance to be correlated.

This is a point that often comes up in "refutations" of the existence of g. People argue, essentially, that even though tests are correlated, they might be produced by many independent causes. I'd go further---we know there are many causes. While intelligence is strongly heritable, it's highly polygenic. Dozens of genes are already known to be linked to it, and more are likely to be discovered. How "broad" the effects of individual genes are is an active research topic. It's harder to quantify environmental influences, but there are surely many that matter there, too.

So, no, the above data doesn't imply that there's no magical number g hidden in our brains, just like it doesn't imply that there's single number in our bodies that says how good we are at running, balancing, or throwing stuff. But that doesn't change the fact that a single number provides a good description of how good we are at various mental tasks.

Suppose you're hiring someone for a job that requires a few different mental tasks. (Arithmetic, sequential memory, whatever.) If you knew someone's g, you could guess how well they'd do at each task. But it would only be a guess! To really know, you still need to test the skills individually. That's the key word: Individually. It's not that g tells you everything---it doesn't---it's just that once you know g, how good someone is at one task doesn't tell you anything about how good they'll be at another.

Again, that's assuming factor analysis were a perfect fit. Which it isn't. Though it's close.

Takeaways

1. Skill at mental and physical tasks are positively correlated. More similar stuff is more correlated.
2. A factor analysis model tries to model data with a "cigar" shape. These models fit mental and physical tests reasonably well, but not perfectly.
3. Call the position along the "long axis" of the cigar g. A perfect fit wouldn't mean that g contains all the information about how good someone is at different tasks---only that it contains all shared information.

(Thanks to Aaron Bergman for comments on a draft of this essay.)