But, it also seems reasonable to treat log(gdp) as a more meaningful object than gdp.
I’m not entirely sure… For individuals, log-transforms make sense on their own merits as giving a better estimate of the utility of that money, but does that logic really apply to a whole country? More money means more can be spent on charity, shooting down asteroids, etc.
It’s also bothersome that the primary empirical prediction of the smart fraction model (that there is some stable gdp level that you hit when everyone is higher than the smart fraction) is entirely from the extrapolated part of the dataset, and this doesn’t seem noticeably better than the exponential model, whose extrapolations are radically different.
The next logical step would be to bring in the second 2006 edition of the Lynn dataset, which increased the set from 81 to 113, and use the latest available per-capita GDP (probably 2011). If the exponential fit gets better compared to the smart-fraction sigmoid, then that’s definitely evidence towards the conclusion that the smart-fraction is just a bad fit.
Yeah; I’m curious what they’d have to say about the relative merits of the two models. I’ll see if I can get this question to them.
I’d guess that he’d consider SF a fairly arbitrary model and not be surprised if an exponential fits better.
It’s an offset, so that it’s an affine fit rather than a linear fit: the gdp level for a population with no people above 108 IQ doesn’t have to be 0. Turns out, it’s not significantly different from zero, but I’d rather discover that than enforce it (and enforcing it can degrade the value for m).
Why can’t the GDP be 0 or negative? Afghanistan and North Korea are right now exhibiting what such a country looks like: they can barely feed themselves and export so much violence or fundamentalism or other dysfunctionality that rich nations are sinking substantial sums of money into supporting them and fixing problems.
For individuals, log-transforms make sense on their own merits as giving a better estimate of the utility of that money, but does that logic really apply to a whole country?
The argument would be that additional intelligence multiplies the per-capita wealth-producing apparatus that exists, rather than adding to it (or, in the smart fraction model, not doing anything once you clear a threshold).
Why can’t the GDP be 0 or negative?
There’s no restriction that b be positive, and so those are both options. I wouldn’t expect it to be negative because pre-industrial societies managed to survive, but that presumes that aid spending by the developed world is not subtracted from the GDP measurement of those countries. Once you take aid into account, then it does seem reasonable that places could become money pits.
The argument would be that additional intelligence multiplies the per-capita wealth-producing apparatus that exists, rather than adding to it (or, in the smart fraction model, not doing anything once you clear a threshold).
That’s the intuitive justification for an exponential model (each additional increment of intelligence adds a percentage of the previous GDP), but I don’t see how this justifies looking at log transforms.
There’s no restriction that b be positive, and so those are both options. I wouldn’t expect it to be negative because pre-industrial societies managed to survive
The difference would be a combination of negative externalities and changing Malthusian equilibriums: it has never been easier for an impoverished country like North Korea or Afghanistan to export violence and cause massive costs they don’t bear (9/11 directly cost the US something like a decade of Afghanistan GDP once you remove all the aid given to Afghanistan), and public health programs like vaccinations enable much larger populations than ‘should’ be there.
That’s the intuitive justification for an exponential model (each additional increment of intelligence adds a percentage of the previous GDP), but I don’t see how this justifies looking at log transforms.
GDP ~ exp(IQ) is isomorphic to ln(GDP) ~ IQ, and I think log(dollars per year) is an easier unit to think about than something to the power of IQ.
[edit] The graph might look different, though. It might be instructive to compare the two, but I think the relationships should be mostly the same.
It’s worth pointing out that IQ numbers are inherently non-parametric: we simply have a ranking of performance on IQ tests, which are then scaled to fit a normal distribution.
If GDP ~ exp(IQ), that means that the correlation is better if we scale the rankings to fit a log-normal distribution instead (this is not entirely true because exp(mean(IQ)) is not the same as mean(exp(IQ)), but the geometric mean and arithmetic mean should be highly correlated with each other as well). I suspect that this simply means that GDP approximately follows a log-normal distribution.
I suspect that this simply means that GDP approximately follows a log-normal distribution.
This doesn’t quite follow, since both per capita GDP and mean national IQ aren’t drawn from the same sort of distribution as individual production and individual IQ are, but I agree with the broader comment that it is natural to think of the economic component of intelligence measured in dollars per year as lognormally distributed.
I’m not entirely sure… For individuals, log-transforms make sense on their own merits as giving a better estimate of the utility of that money, but does that logic really apply to a whole country? More money means more can be spent on charity, shooting down asteroids, etc.
The next logical step would be to bring in the second 2006 edition of the Lynn dataset, which increased the set from 81 to 113, and use the latest available per-capita GDP (probably 2011). If the exponential fit gets better compared to the smart-fraction sigmoid, then that’s definitely evidence towards the conclusion that the smart-fraction is just a bad fit.
I’d guess that he’d consider SF a fairly arbitrary model and not be surprised if an exponential fits better.
Why can’t the GDP be 0 or negative? Afghanistan and North Korea are right now exhibiting what such a country looks like: they can barely feed themselves and export so much violence or fundamentalism or other dysfunctionality that rich nations are sinking substantial sums of money into supporting them and fixing problems.
The argument would be that additional intelligence multiplies the per-capita wealth-producing apparatus that exists, rather than adding to it (or, in the smart fraction model, not doing anything once you clear a threshold).
There’s no restriction that b be positive, and so those are both options. I wouldn’t expect it to be negative because pre-industrial societies managed to survive, but that presumes that aid spending by the developed world is not subtracted from the GDP measurement of those countries. Once you take aid into account, then it does seem reasonable that places could become money pits.
That’s the intuitive justification for an exponential model (each additional increment of intelligence adds a percentage of the previous GDP), but I don’t see how this justifies looking at log transforms.
The difference would be a combination of negative externalities and changing Malthusian equilibriums: it has never been easier for an impoverished country like North Korea or Afghanistan to export violence and cause massive costs they don’t bear (9/11 directly cost the US something like a decade of Afghanistan GDP once you remove all the aid given to Afghanistan), and public health programs like vaccinations enable much larger populations than ‘should’ be there.
GDP ~ exp(IQ) is isomorphic to ln(GDP) ~ IQ, and I think log(dollars per year) is an easier unit to think about than something to the power of IQ.
[edit] The graph might look different, though. It might be instructive to compare the two, but I think the relationships should be mostly the same.
It’s worth pointing out that IQ numbers are inherently non-parametric: we simply have a ranking of performance on IQ tests, which are then scaled to fit a normal distribution.
If GDP ~ exp(IQ), that means that the correlation is better if we scale the rankings to fit a log-normal distribution instead (this is not entirely true because exp(mean(IQ)) is not the same as mean(exp(IQ)), but the geometric mean and arithmetic mean should be highly correlated with each other as well). I suspect that this simply means that GDP approximately follows a log-normal distribution.
This doesn’t quite follow, since both per capita GDP and mean national IQ aren’t drawn from the same sort of distribution as individual production and individual IQ are, but I agree with the broader comment that it is natural to think of the economic component of intelligence measured in dollars per year as lognormally distributed.