gwern comments on Rationality Quotes September 2011 - Less Wrong

7 Post author: dvasya 02 September 2011 07:38AM

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Comment author: lukeprog 02 September 2011 05:01:50PM 3 points [-]

Sounds plausible. If anybody finds the citation for this, please post it.

Comment author: gwern 28 February 2012 11:09:29PM 13 points [-]

Here's another one: "National IQ and National Productivity: The Hive Mind Across Asia", Jones 2011

...cognitive skills—intelligence quotient scores, math skills, and the like—have only a modest influence on individual wages, but are strongly correlated with national outcomes. Is this largely due to human capital spillovers? This paper argues that the answer is yes. It presents four different channels through which intelligence may matter more for nations than for individuals: (i) intelligence is associated with patience and hence higher savings rates; (ii) intelligence causes cooperation; (iii) higher group intelligence opens the door to using fragile, high-value production technologies; and (iv) intelligence is associated with supporting market-oriented policies.

Comment author: gwern 03 February 2013 01:41:12AM *  5 points [-]

"Exponential correlation of IQ and the wealth of nations", Dickerson 2006:

Plots of mean IQ and per capita real Gross Domestic Product for groups of 81 and 185 nations, as collected by Lynn and Vanhanen, are best fitted by an exponential function of the form: GDP = a * 10^b*(IQ), where a and b are empirical constants. Exponential fitting yields markedly higher correlation coefficients than either linear or quadratic. The implication of exponential fitting is that a given increment in IQ, anywhere along the IQ scale, results in a given percentage in GDP, rather than a given dollar increase as linear fitting would predict. As a rough rule of thumb, an increase of 10 points in mean IQ results in a doubling of the per capita GDP.

....In their book, IQ and the Wealth of Nations, Lynn and Vanhanen (2002) present a table listing for 81 nations the measured mean IQ and the per capita real Gross Domestic Product as of 1998 (their Table 7.7). They subsequently extend this to all 185 nations, using estimated IQs for the 104 new entries based chiefly on IQ values for immediate neighbors (their Table 8.9). In both cases they observe a significant correlation between IQ and GDP, with linear correlation factors R^2 = 0.537 for the 81-nation group and 0.389 for 185 nations. McDaniel and Whetzel have extended the examination of correlations to quadratic fitting in a paper that demonstrates the robustness of these correlations to minor variations in individual IQ values (McDaniel & Whetzel, in press). But an even stronger correlation is found if the fitting is exponential rather than linear or quadratic.

Comment author: [deleted] 03 February 2013 08:42:30AM *  3 points [-]

It peeves me when scatterplots of GDP per capita versus something else use a linear scale -- do they actually think the difference between $30k and $20k is anywhere near as important as that between $11k and $1k? And yet hardly anybody uses logarithmic scales.

Likewise, the fit looks a lot less scary if you write it as ln(GDP) = A + B*IQ.

Comment author: gwern 09 February 2013 08:43:05PM *  0 points [-]

Yes, Dickerson does point out that his exponential fit is a linear relationship on a log scale. For example, he does show a log-scale in figure 3 (pg3), fitting the most reliable 83 nation-points on a plot of log(GDP) against mean IQ in which the exponential fit looks exactly like you would expect. (Is it per capita? As far as I can tell, he always means per capita GDP even if he writes just 'GDP'.) Figure 4 does the same thing but expands the dataset to 185 nations. The latter plot should probably be ignored given that the expansion comes from basically guessing:

In their book, IQ and the Wealth of Nations, Lynn and Vanhanen (2002) present a table listing for 81 nations the measured mean IQ and the per capita real Gross Domestic Product as of 1998 (their Table 7.7). They subsequently extend this to all 185 nations, using estimated IQs for the 104 new entries based chiefly on IQ values for immediate neighbors (their Table 8.9).

Comment author: Vaniver 09 February 2013 10:19:33PM 1 point [-]

Is it easy to compare the fit of their theory to the smart fraction theory?

Comment author: gwern 09 February 2013 11:32:07PM *  4 points [-]

I dunno. I've given it a try and while it's easy enough to reproduce the exponential fit (and the generated regression line does fit the 81 nations very nicely), I think I screwed up somehow reproducing the smart fraction equation because the regression looks weird and trying out the smart-fraction function (using his specified constants) on specific IQs I don't get the same results as in La Griffe's table. And I can't figure out what I'm doing wrong, my function looks like it's doing the same thing as his. So I give up. Here is my code if you want to try to fix it:

lynn <- read.table(stdin(),header=TRUE,sep="")
Country IQ rGDPpc
Argentina 96 12013
Australia 98 22452
Austria 102 23166
Barbados 78 12001
Belgium 100 23223
Brazil 87 6625
Bulgaria 93 4809
Canada 97 23582
China 100 3105
Colombia 89 6006
Congo 65 822
Congo 73 995
Croatia 90 6749
Cuba 85 3967
CzechRepublic 97 12362
Denmark 98 24218
Ecuador 80 3003
Egypt 83 3041
EquatorialGuinea 59 1817
Ethiopia 63 574
Fiji 84 4231
Finland 97 20847
France 98 21175
Germany 102 22169
Ghana 71 1735
Greece 92 13943
Guatemala 79 3505
Guinea 66 1782
HongKong 107 20763
Hungary 99 10232
India 81 2077
Indonesia 89 2651
Iran 84 5121
Iraq 87 3197
Ireland 93 21482
Israel 94 17301
Italy 102 20585
Jamaica 72 3389
Japan 105 23257
Kenya 72 980
Lebanon 86 4326
Malaysia 92 8137
MarshallIslands 84 3000
Mexico 87 7704
Morocco 85 3305
Nepal 78 1157
Netherlands 102 22176
NewZealand 100 17288
Nigeria 67 795
Norway 98 26342
Peru 90 4282
Philippines 86 3555
Poland 99 7619
Portugal 95 14701
PuertoRico 84 8000
Qatar 78 20987
Romania 94 5648
Russia 96 6460
SierraLeone 64 458
Singapore 103 24210
Slovakia 96 9699
Slovenia 95 14293
SouthAfrica 72 8488
SouthKorea 106 13478
Spain 97 16212
Sudan 72 1394
Suriname 89 5161
Sweden 101 20659
Switzerland 101 25512
Taiwan 104 13000
Tanzania 72 480
Thailand 91 5456
Tonga 87 3000
Turkey 90 6422
UKingdom 100 20336
Uganda 73 1074
UnitedStates 98 29605
Uruguay 96 8623
WesternSamoa 87 3832
Zambia 77 719
Zimbabwe 66 2669
em <- lm(log(lynn$rGDPpc) ~ lynn$IQ); summary(em)
Call:
lm(formula = log(lynn$rGDPpc) ~ lynn$IQ)
Residuals:
Min 1Q Median 3Q Max
-1.6124 -0.3866 -0.0429 0.3363 2.0311
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.77760 0.51848 3.43 0.00097
lynn$IQ 0.07876 0.00583 13.52 < 2e-16
Residual standard error: 0.624 on 79 degrees of freedom
Multiple R-squared: 0.698, Adjusted R-squared: 0.694
F-statistic: 183 on 1 and 79 DF, p-value: <2e-16
# plot
plot (log(lynn$rGDPpc) ~ lynn$IQ)
abline(em)

successful reproduction of Dickerson 2006

# an attempt at La Griffe
erf <- function(x) 2 * pnorm(x * sqrt(2)) - 1
sf <- function(iq) ((69321/2) * (1 + erf(((iq - 108)/15) / sqrt(2))))
# check for sigmoid
# plot(c(85:130), sf(c(85:130)))
lg <- lm(log(lynn$rGDPpc) ~ sf(lynn$IQ)); summary(lg)
Call:
lm(formula = log(lynn$rGDPpc) ~ sf(lynn$IQ))
Residuals:
Min 1Q Median 3Q Max
-2.5788 -0.6857 0.0678 1.0521 1.5901
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.705620 0.126152 69.01 <2e-16
sf(lynn$IQ) 0.000121 0.000102 1.19 0.24
Residual standard error: 1.13 on 79 degrees of freedom
Multiple R-squared: 0.0175, Adjusted R-squared: 0.0051
F-statistic: 1.41 on 1 and 79 DF, p-value: 0.239
# same plotting code

A failed fit of smart fraction theory?

(In retrospect, I'm not sure it's even meaningful to try to fit the sf function with the constants already baked in, but since I apparently didn't write it right, it doesn't matter.

Comment author: Vaniver 10 February 2013 01:27:29AM *  0 points [-]

Hm, one thing I notice is that you look like you're fitting sf against log(gdp). I managed to replicate his results in octave, and got a meaningful result plotting smart fraction against gdp.

My guess at how to change your code (noting that I don't know R):

sf <- function(iq,c) ((69321/2) * (1 + erf(((iq - c)/15) / sqrt(2))))
lg <- lm(lynn$rGDPpc ~ sf(lynn$IQ,108)); summary(lg)

That should give you some measure of how good it fits, and you might be able to loop it to see how well the smart fraction does with various thresholds.

(I also probably should have linked to the refinement.)

Comment author: gwern 10 February 2013 03:33:50AM 0 points [-]

I can't tell whether that works since you're just using the same broken smart-fraction sf predictor; eg. sf(107,108) ~> 32818, while the first smart fraction page's table gives a Hong Kong regression line of 19817 which is very different from 33k.

The refinement doesn't help with my problem, no.

Comment author: Vaniver 10 February 2013 08:08:25PM *  0 points [-]

Hmmm. I agree that it doesn't match. What if by 'regression line' he means the regression line put through the sf-gdp data?

That is, you should be able to calculate sf as a fraction with

sf <- function(iq,c) ((1/2) * (1 + erf((iq-c)/(15*sqrt(2)))))

And then regress that against gdp, which will give you the various coefficients, and a much more sensible graph. (You can compare those to the SFs he calculates in the refinement, but those are with verbal IQ, which might require finding that dataset / trusting his, and have a separate IQ0.)

Comparing the two graphs, I find it interesting that the eight outliers Griffe mentions (Qatar, South Africa, Barbados, China, and then the NE Asian countries) are much more noticeable on the SF graph than the log(GDP) graph, and that the log(GDP) graph compresses the variation of the high-income countries, and gets most of its variation from the low-income countries; the situation is reversed in the SF graph. Since both our IQ and GDP estimates are better in high-income countries, that seems like a desirable property to have.

With outliers included, I'm getting R=.79 for SF and R=.74 for log(gdp). (I think, I'm not sure I'm calculating those correctly.)

Comment author: gwern 10 February 2013 10:03:09PM *  0 points [-]

Trying to rederive the constants doesn't help me, which is starting to make me wonder if he's really using the table he provided or misstated an equation or something:

R> sf <- function(iq,f,c) ((c/2) * (1 + erf((iq-f)/(15*sqrt(2)))))
R> summary(nls(rGDPpc ~ sf(IQ,f,c), lynn, start=list(f=110,c=40000)))
Formula: rGDPpc ~ sf(IQ, f, c)
Parameters:
Estimate Std. Error t value Pr(>|t|)
f 99.64 3.07 32.44 < 2e-16
c 34779.17 6263.90 5.55 3.7e-07
Residual standard error: 5310 on 79 degrees of freedom
Number of iterations to convergence: 4
Achieved convergence tolerance: 8.22e-06

If you double 34779 you get very close to his $69,321 so there might be something going wrong due to the 1/2 that appears in uses of the erf to make a cumulative distribution function, but I don't how a threshold of 99.64 IQ is even close to his 108!

(The weird start values were found via trial-and-error in trying to avoid R's 'singular gradient error'; it doesn't appear to make a difference if you start with, say, f=90.)

Comment author: Vaniver 11 February 2013 02:12:14AM *  1 point [-]

Most importantly, we appear to have figured out the answer to my original question: no, it is not easy. :P

So, I started off by deleting the eight outliers to make lynn2. I got an adjusted R^2 of 0.8127 for the exponential fit, and 0.7777 for the fit with iq0=108.2.

My nls came back with an optimal iq0 of 110, which is closer to the 108 I was expecting; the adjusted R^2 only increases to 0.7783, which is a minimal improvement, and still slightly worse than the exponential fit.

The value of the smart fraction cutoff appears to have a huge impact on the mapping from smart fraction to gdp, but doesn't appear to have a significant effect on the goodness of fit, which troubles me somewhat. I'm also surprised that deleting the outliers seems to have improved the performance of the exponential fit more than the smart fraction fit, which is not what I would have expected from the graphs. (Though, I haven't calculated this with the outliers included in R, and I also excluded the Asian data, and there's more fiddling I can do, but I'm happy with this for now.)

> sf <- function(iq,iq0) ((1+erf((iq-iq0)/(15*sqrt(2))))/2)
> egdp <- function(iq,iq0,m,b) (m*sf(iq,iq0)+b)
> summary(nls(rGDPpc ~ egdp(IQ,iq0,m,b), lynn2, start=list(iq0=110,m=40000,b=0)))
Formula: rGDPpc ~ egdp(IQ, iq0, m, b)
Parameters:
Estimate Std. Error t value Pr(>|t|)
iq0 110.019 4.305 25.556 < 2e-16 ***
m 77694.174 26708.502 2.909 0.00486 **
b 679.688 1039.144 0.654 0.51520
Residual standard error: 4054 on 70 degrees of freedom
> gwe <- lm(lynn2$rGDPpc ~ sf(lynn2$IQ,99.64)); summary(gwe)
Call:
lm(formula = lynn2$rGDPpc ~ sf(lynn2$IQ, 99.64))
Residuals:
Min 1Q Median 3Q Max
-10621.6 -2463.1 442.6 1743.4 12439.7
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1345.7 886.6 -1.518 0.133
sf(lynn2$IQ, 99.64) 40552.7 2724.1 14.887 <2e-16 ***
Residual standard error: 4241 on 71 degrees of freedom
Multiple R-squared: 0.7574, Adjusted R-squared: 0.7539
F-statistic: 221.6 on 1 and 71 DF, p-value: < 2.2e-16
> opt <- lm(lynn2$rGDPpc ~ sf(lynn2$IQ,110)); summary(opt)
Call:
lm(formula = lynn2$rGDPpc ~ sf(lynn2$IQ, 110))
Residuals:
Min 1Q Median 3Q Max
-11030.3 -1540.1 -416.8 1308.6 12493.5
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 676.4 731.5 0.925 0.358
sf(lynn2$IQ, 110) 77577.0 4869.6 15.931 <2e-16 ***
Residual standard error: 4025 on 71 degrees of freedom
Multiple R-squared: 0.7814, Adjusted R-squared: 0.7783
F-statistic: 253.8 on 1 and 71 DF, p-value: < 2.2e-16
> his <- lm(lynn2$rGDPpc ~ sf(lynn2$IQ,108.2)); summary(his)
Call:
lm(formula = lynn2$rGDPpc ~ sf(lynn2$IQ, 108.2))
Residuals:
Min 1Q Median 3Q Max
-11014.0 -1710.0 -196.9 1396.3 12432.9
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 362.6 748.0 0.485 0.629
sf(lynn2$IQ, 108.2) 67711.5 4258.5 15.900 <2e-16 ***
Residual standard error: 4031 on 71 degrees of freedom
Multiple R-squared: 0.7807, Adjusted R-squared: 0.7777
F-statistic: 252.8 on 1 and 71 DF, p-value: < 2.2e-16
> em <- lm(log(lynn2$rGDPpc) ~ lynn2$IQ); summary(em)
Call:
lm(formula = log(lynn2$rGDPpc) ~ lynn2$IQ)
Residuals:
Min 1Q Median 3Q Max
-1.12157 -0.34268 -0.00503 0.29596 1.41540
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.812650 0.446308 1.821 0.0728 .
lynn2$IQ 0.089439 0.005051 17.706 <2e-16 ***
Residual standard error: 0.4961 on 71 degrees of freedom
Multiple R-squared: 0.8153, Adjusted R-squared: 0.8127
F-statistic: 313.5 on 1 and 71 DF, p-value: < 2.2e-16
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