Yes, conditioning on g makes all the observations anti-correlated with each other (assuming for simplicity that the coefficients that define g are all positive). The case of two factors has come up on LW a few times, but I don’t have a reference: if you can get into an exclusive university by a combination of wealth and intelligence, then among its members wealth and intelligence will be anticorrelated, .
When all the eigenvalues after the first are equal, what has zero correlation conditional on g is the measurements along the corresponding principal axes, which are other linear combinations of the observations.
Sort of. The problem with the post is that it doesn’t distinguish between g (the underlying variable that generates the cigar shape) and IQ (your position along the cigar, which will be influenced by not just g but also noise/test specificities). If you condition on IQ, everything becomes anticorrelated, but if you condition on g, they become independent.
g is the thing that factor analysis gives you, and is your position along the cigar. When all other eigenvalues are equal, conditioning on g makes all the observed variables anticorrelated, and all the other factors uncorrelated.
The cigar is also constructed by factor analysis. Factor analysis is the method of finding the best fitting ellipsoid to the data. There is nothing in factor analysis about underlying “real” variables that the observations are noisy measurements of, nor about values along the principal axes being noisy observations of “real” factors.
g is the thing that factor analysis gives you, and is your position along the cigar.
g is generally defined as being the underlying cause that leads to the different cognitive tests correlating. This is literally what is meant by the equation:
x = wg + e
Factor analysis can estimate g, but it can’t get the exact value of g, due to the noise factors. People sometimes say “g” when they really mean the estimate of g, but strictly speaking that is incorrect.
Factor analysis is the method of finding the best fitting ellipsoid to the data.
This is incorrect. Principal component analysis is the method for finding the best fitting ellipsoid to the data. Factor analysis and principal component analysis tend to yield very similar results, and so are usually not distinguished, but when it comes to the issue we are discussing here, it is necessary to distinguish.
There is nothing in factor analysis about underlying “real” variables that the observations are noisy measurements of, nor about values along the principal axes being noisy observations of “real” factors.
Factor analysis (not principal component analysis) makes the assumption that the data is generated based on an underlying factor + noise/specificities. This assumption may be right or wrong; factor analysis does not tell you that. But under the assumption that it is right, there is an underlying g factor that makes the variables independent (not negatively correlated).
Yes, conditioning on g makes all the observations anti-correlated with each other (assuming for simplicity that the coefficients that define g are all positive). The case of two factors has come up on LW a few times, but I don’t have a reference: if you can get into an exclusive university by a combination of wealth and intelligence, then among its members wealth and intelligence will be anticorrelated, .
When all the eigenvalues after the first are equal, what has zero correlation conditional on g is the measurements along the corresponding principal axes, which are other linear combinations of the observations.
Sort of. The problem with the post is that it doesn’t distinguish between g (the underlying variable that generates the cigar shape) and IQ (your position along the cigar, which will be influenced by not just g but also noise/test specificities). If you condition on IQ, everything becomes anticorrelated, but if you condition on g, they become independent.
g is the thing that factor analysis gives you, and is your position along the cigar. When all other eigenvalues are equal, conditioning on g makes all the observed variables anticorrelated, and all the other factors uncorrelated.
The cigar is also constructed by factor analysis. Factor analysis is the method of finding the best fitting ellipsoid to the data. There is nothing in factor analysis about underlying “real” variables that the observations are noisy measurements of, nor about values along the principal axes being noisy observations of “real” factors.
g is generally defined as being the underlying cause that leads to the different cognitive tests correlating. This is literally what is meant by the equation:
x = wg + e
Factor analysis can estimate g, but it can’t get the exact value of g, due to the noise factors. People sometimes say “g” when they really mean the estimate of g, but strictly speaking that is incorrect.
This is incorrect. Principal component analysis is the method for finding the best fitting ellipsoid to the data. Factor analysis and principal component analysis tend to yield very similar results, and so are usually not distinguished, but when it comes to the issue we are discussing here, it is necessary to distinguish.
Factor analysis (not principal component analysis) makes the assumption that the data is generated based on an underlying factor + noise/specificities. This assumption may be right or wrong; factor analysis does not tell you that. But under the assumption that it is right, there is an underlying g factor that makes the variables independent (not negatively correlated).