This appears to be Hausmann and Rigobon's reply to Dan's Really Obvious Objection. Personally, I find it hard to follow.
How does the Carter Center answer our claim? They make three propositions:
1. They check whether the mean of the votes in the two samples are similar
[...]
With respect to the first point, the question that the Carte Center asks is whether the unconditional means of the two samples are similar. By unconditional we mean that they do not control for the fact that precincts are different in the four dimensions we include in our equation or in any other dimension.
To see the importance of conditioning, let us imagine that there is fraud and let us suppose that the fraud is carried out in a large number of precincts but not in all of them. The question is: is it possible to choose an audit sample of non-tampered centers that has the same mean as the universe of tampered and un-tampered precincts? The answer is obviously yes. Let us give an example using a population with a varying level of income, say from US$ 4,000 per year to several million. Assume that half of them have been taxed 20 percent of their income while the other half has not. Is it possible to construct an audit sample of non-taxed individuals whose average income is similar to that of those that have been taxed? Obviously the answer is yes. However, if one controls for the level of education, the years of work experience and the positions they hold in the companies they work in, it should be possible to find that the audited individuals actually a higher net income than the non-audited group. That is the essence of what we do.
Now, lets go back to the case in point. Precincts vary from those where the Yes got more than 90 percent of the vote and those where it got less than 10 percent. This is a very large variation relative to the potential size of the fraud, say 10 or 20 percent. It is perfectly feasible to choose a sample that has the same mean as the rest of the universe.
However, the non-random nature of the sample would be revealed if we compare the means but controlling for the fact that each precinct is different. That is what we do and this is the randomness test that the audited sample failed.
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