How is the g-factor of intelligence calculated? I am currently reading Dr. Richard Haier's book The Neuroscience of Intelligence and I am a little confused about the g-factor. Note that I have no education in psychology outside of general ed courses at university.

I have two questions regarding the g-factor.  What confuses me is for one, he does not go into technical detail about how the g-factor is calculated.

He does include this illustration:



Here we see that 15 mental tests are taken in groups of 3, where each group is correlated to a certain factor. So tests 1-3 are for reasoning, 4-6 for spatial ability, and so on.  These factors then correlate to the g-factor.

I am wondering what is the methodology used to find the correlations between the tests and factors/factors and the g-factor? For example, I don't quite understand how they measure the correlation between the test and the factor given that the factor is abstract (ie. reasoning). I hope what I am asking makes sense. I am not asking how to perform linear regression or calculate Pearson's r. 

My second question is about the statement in the book: 

"Different mental abilities are not independent. They are all related to each other and the correlations among mental tests are always positive.... It strongly implies that all the factors derived from individual tests have something in common, and this common factor is called the general factor of intelligence."

Wouldn't this imply that all of the factors are multicollinear and make it difficult to distinguish which factors are most significant?
PCA or EFA: A standard approach to calculating a g factor would be to use principal components analysis (PCA) or exploratory factor analysis (EFA). Basically, you administer a broad battery of cognitive ability measures. Then you use these tests as variables in a PCA or EFA. This will partition the variance in the tests into a set of uncorrelated factors. The first factor will be the factor that explains the most variance in the tests. It is this first factor that represents "g". Typically, each test will "load" (or correlate) with this first factor, some tests more than others. The first factor will capture something that is common across most or all of the tests. I.e., it is g, for general factor, that measures cognitive ability.

An empirical finding is that almost all tests of specific cognitive abilities correlate with the general factor. And typically, the loadings are fairly substantial. I.e., the first factor is quite substantial relative to second and subsequent factors.

Confirmatory factor analysis: It seems that what you present is a confirmatory factor analysis. This is a variant on exploratory factor analysis. Basically, you specify which observed variables load on latent factors and all other possible loadings are constrained to zero. More specifically, this is a higher-order confirmatory factor analysis (CFA). It is higher-order because, there are two-levels of latent factors. I.e., tests load on domain-specific latent factors (e.g., reasoning, spatial ability), and these domain specific latent factors load on g.

The CFA estimation procedures will find the parameter estimates that best model the data (i.e., the covariance/correlation structure of the tests).

So in this case, we see that all the tests load fairly well on their domains (i.e., > .70) and all the domains load on g (> .65), although reasoning and spatial ability load a little stronger.

In general, multivariate statistics training in psychology (typically presented at postgraduate level and sometimes latter undergraduate-level) will cover these statistical techniques. Here's Bill Revelle's notes that cover some psyhometric theory: http://www.personality-project.org/r/book/

Wouldn't this imply that all of the factors are multicollinear and make it difficult to distinguish which factors are most significant?
That's why researchers often focus more on g (i.e., the general factor). But yes, you are in a sense correct. Of course, you can still do multiple regression and such with correlated predictors; you can then look at things like semi-partial correlations to look at the unique prediction of any one domain.