A couple of days ago, on Twitter, @alung mentioned an article I wrote (in French) about open data, explaining how difficult it was to get access to data in France. @alung wondered if it was still so difficult to access nice datasets. My first answer was that I now know more people willing to share their data. And on the internet, amazing datasets can be found now very easily. In France, for instance, you can find detailed information about qualitifications, houses and jobs, by small geographical areas on http://www.recensement.insee.fr. And that's great for researchers.

But one should be aware that those aggregate data might not be sufficient to build up econometric models, and to infer individual behaviors. Supposing that relationships observed for groups necessarily hold for individuals is a common fallacy -- the so-called ecological fallacy. 

In a popular paper, Robinson (1950) discussed "ecological inference," stressing the difference between ecological correlations (on groups) and individual correlations (see also Thorndike) He considered two aggregated quantities, per American state: the percent of the population that was foreign-born, and the percent that was literate. One dataset used in the paper was the following:

> library(eco)
> data(forgnlit30)
> tail(forgnlit30)
Y          X         W1          W2 ICPSR
43 0.076931986 0.03097168 0.06834300 0.077206504    66
44 0.006617641 0.11479052 0.03568792 0.002847920    67
45 0.006991899 0.11459207 0.04151310 0.002524065    68
46 0.012793782 0.18491515 0.05690731 0.002785916    71
47 0.007322475 0.13196654 0.03589512 0.002978594    72
48 0.007917342 0.18816461 0.02949187 0.002916866    73

The correlation between foreign-birth and literacy was

> cor(forgnlit30$X,1-forgnlit30$Y)
[1] 0.2069447

This suggests a positive correlation, so one quick interpretation could be that in the 1930's, Americans were illiterate, but literate immigrants got the idea to come to the US. But here, like in Simpson's paradox, the sign should be negative, as obtained on individual studies. In the state-based data study, correlation was positive primarily because foreign-born people tend to live in states where the native-born are relatively literate.

So the problem lies in the way that individuals were grouped. Consider the following set of individual observations:

> n=1000
> r=-.5
> Z=rmnorm(n,c(0,0),matrix(c(1,r,r,1),2,2))
> X=Z[,1]
> E=Z[,2]
> Y=3+2*X+E
> cor(X,Y)
[1] 0.8636764

Consider now some regrouping, e.g.

> I=cut(Z[,2],qnorm(seq(0,1,by=.05)))
> Yg=tapply(Y,I,mean)
> Xg=tapply(X,I,mean)

Then the correlation is rather different:

>  cor(Xg,Yg)
[1] 0.1476422

Here we have a strong positive individual correlation, and a small positive correlation on grouped data, but almost anything is possible.

Models with random coefficients have been used to make ecological inferences. But that is a long story, and I will probably come back with a more detailed post on that topic, since I am still working on this with @coulmont (following some comments by @frbonnet on his post on recent French elections on http://coulmont.com/blog/).