Your argument was atheism is weakly correlated with vocab. Vocab is weakly correlated with intelligence. Therefore, atheism is weakly correlated with intelligence.
Ah, I see your point. However, a) vocab is highly correlated with intelligence, not weakly so, b) vocab is not just highly correlated with a single intelligence metric, but is correlated with such in a variety of different metrics of intelligence. While it is possible to construct variables such that A and B are correlated, with B and C correlated, and A and C anti-correlated, it is quite difficult to do so with a large set of distinct variables that all have such correlations with each other and have a single pair be anti-correlated, especially when one has the same set of correlations even when one controls for a variety of other variables. Moreover, as a probabilistic matter if one as three variables with two pairs correlated, it is much more likely that the remaining pair will be correlated than anti-correlated, assuming that variables don’t have too pathological a distribution.
Moreover, as a probabilistic matter if one as three variables with two pairs correlated, it is much more likely that the remaining pair will be correlated than anti-correlated, assuming that variables don’t have too pathological a distribution.
Where you you get that? The intended probability space isn’t clear, but if I take three random directions in N-dimensional space for large N, I find that the chance of two pairs having an angle less than pi/2 and the third an angle greater than pi/2 is about 1.4 times the chance of all three being less than pi/2. The ratio rises to about 3 if I add the requirement that the corresponding correlations are in the range +/- 0.8 (the upper liit of correlations generally found in psychology).
Hmm, that’s a good point. I’m aware vaguely of theorems that say what I want but I don’t have any references or descriptions off hand. It may just be that one is assuming somewhat low N, but that would be in this sort of context not helpful. I do seem to remember that some version of my statement is true if the variables match bell curves, but I’m not able at the moment to construct or find a precise statement. Consider the claim withdrawn until I’ve had more time to look into the matter.
Your argument was atheism is weakly correlated with vocab. Vocab is weakly correlated with intelligence. Therefore, atheism is weakly correlated with intelligence.
Ah, I see your point. However, a) vocab is highly correlated with intelligence, not weakly so, b) vocab is not just highly correlated with a single intelligence metric, but is correlated with such in a variety of different metrics of intelligence. While it is possible to construct variables such that A and B are correlated, with B and C correlated, and A and C anti-correlated, it is quite difficult to do so with a large set of distinct variables that all have such correlations with each other and have a single pair be anti-correlated, especially when one has the same set of correlations even when one controls for a variety of other variables. Moreover, as a probabilistic matter if one as three variables with two pairs correlated, it is much more likely that the remaining pair will be correlated than anti-correlated, assuming that variables don’t have too pathological a distribution.
Where you you get that? The intended probability space isn’t clear, but if I take three random directions in N-dimensional space for large N, I find that the chance of two pairs having an angle less than pi/2 and the third an angle greater than pi/2 is about 1.4 times the chance of all three being less than pi/2. The ratio rises to about 3 if I add the requirement that the corresponding correlations are in the range +/- 0.8 (the upper liit of correlations generally found in psychology).
Hmm, that’s a good point. I’m aware vaguely of theorems that say what I want but I don’t have any references or descriptions off hand. It may just be that one is assuming somewhat low N, but that would be in this sort of context not helpful. I do seem to remember that some version of my statement is true if the variables match bell curves, but I’m not able at the moment to construct or find a precise statement. Consider the claim withdrawn until I’ve had more time to look into the matter.