Personality & Individual Differences · Simplified

A General Factor With No General Ability

Cognitive tests all correlate positively with one another. That fact is a hundred years old, and it is much weaker evidence than it looks.

Step 1 of 2

First, a judgement

Give a large group of people any collection of cognitive tests and two things reliably happen. Every test correlates positively with every other, and a single factor accounts for a large share of the variance between them. What does that establish?

Step 2 of 2

A mind with no general ability in it

Here is a model of a mind. It has fifteen hundred small independent processes and nothing else: no general ability, no central resource, nothing that all the tests draw on. Each of the six tests draws on a share of the pool, and the shares overlap by chance rather than by design.

Key terms

Positive manifold: The observation that scores on cognitive tests correlate positively with one another, whichever tests you pick.
General factor: The first factor extracted from such a set of correlations. It is a summary of the pattern. Whether it corresponds to anything is a further question that the pattern does not answer.
Sampling account: The proposal that tests correlate because they draw on overlapping sets of many small processes, rather than because they share one big thing. It is what this model implements.
Residual: How far a fitted model's implied correlations sit from the real ones. A small residual means the model reproduces the data, not that it is true.
Fit, and strength: two different things: The fit of a one-factor model is how closely it reproduces the correlations. The strength of the factor is how much of the variance it carries. A weak pattern can still be a regular one, so a factor can fit closely and account for very little. Watch the two lower figures move apart as you lower the slider.

Which processes each test happens to use was fixed once and never changes. This slider changes only how many of them each test uses, and so how often two tests happen to share one.

Loadings on the single general factor fitted to the correlations above, and how far that factor's implied correlations sit from the real ones.
Test	Loading on the general factor

What this shows

The pattern is real. What it is evidence for is the open question.

Key idea: The positive manifold is one of the most robust findings in psychology, and nothing here disputes it. What this model shows is what it can and cannot be used to argue. There is no general ability anywhere in the model you have been moving. There are many small processes and some accidental overlap in which of them each test happens to use, and that alone produces every correlation positive and a single factor that reproduces the whole matrix to within a couple of hundredths. So a well-fitting general factor is not evidence that there is a general ability, because a mind built with no such thing produces one anyway.

And fit is not strength. Lower the slider to about five per cent and read the two lower figures together. Every correlation is still positive, the one-factor model still reproduces them closely, and that factor carries about six per cent of the variance. A close fit says the pattern is regular enough for one factor to describe it. It says nothing about how much the factor accounts for, and nothing at all about what produces it.

A positive manifold, and a one-factor model that fits it closely, are evidence about the pattern of covariance among test scores. They are not proof of a single causal general ability. The factor is a compact description of a pattern in a set of scores. Whether anything in a person corresponds to it is a further question, and answering it takes a different kind of evidence from the correlations themselves: developmental, neural, genetic and experimental work on what actually produces the overlap.

This runs the argument in one direction only, and the other direction matters just as much. Showing that a sampling model can produce a positive manifold does not show that a general ability does not exist, or that the sampling account is the right one. Both accounts fit these correlations, which is exactly the point being made, and neither is established by them. Three further cautions. The model is a toy: real cognitive processes are not fifteen hundred independent things, real tests do not draw on them at random, and real correlation matrices are not this tidy. General-factor scores do predict a range of outcomes, and that predictive record is untouched by anything here, since a description can predict perfectly well without naming a cause. And nothing on this page is about any person or group: it is a statement about what a correlation matrix can be used to infer.

The longer version compares the sampling account directly against a common-factor model and a mutualism model, lets you generate data from each, and works through hierarchical and bifactor solutions. It is at Positive Manifold Visualiser in the main collection.