Physicist Imre Mikoss presented his data on tests on the August 15th. recall vote and comparison to Benford’s law two weeks ago at the third Simon Bolivar University seminar. His presentation is now online. While Pericchi and Torres have done similar tests, Mykoss does a couple of very interesting things which are worth posting for their implications.
–Results from the 2000 election: Mikoss has looked at the data from the 2000 Presidential vote. This is interesting because even though electoral results would seem like a natural set to test Benford’s law, nothing guarantees that it works in Venezuela or everywhere. Below is a graph the first digit in the number of votes obtained by Chavez’ challenger Francisco Arias Cardenas in the 2000 elections. :
Frequency of occurence of the first digit for the votes in favor of Arias Cardenas at each machine in the 2000 as a function of the digit.
The graph shows the frequency of occurrence expected from Benford (green bar) and the frequency seen in the election (red bar) in the vertical axis versus each of the digits in the horizontal axis. The graph not only looks like Benford’s law, but the author performed statistical tests and obtained in the case of the number of votes for Arias across the nation to have a parameter S (which I believe is chi^2, but the presentation does not define)=0.003. Chavez’ votes in the same election, as well as the difference between the two numbers at each machine were all found to follow Benford’s law with S<0.014. Thus, Venezuelan electoral results did follow well Benford’s law in 2000, which needed to be established and seems to be established by this comparison.
–Results from the recall vote: The same test on the results from the recall vote do not agree well with Benford’s law as sown below for the Si and the No frequencies. As in the case of Pericchi’s analysis for the second digit presented here earlier, Mykoss finds that the Si votes agrees better (S=0.33) than the No vote (S=0.97) as seen below:
Frequency of occurence of the first digit for the Si votes at each machine as a function of the digit.
Frequency of occurence of the first digit for the Np votes at each machine as a function of the digit.
-“Reverting the data”: Mikoss then studies rather than the Si or No numbers, the set of differences (No-Si) for each voting machine. This apparently has the advantage that it provides a more uniform set of numbers that is not as bound as the pure set in which machine size bounds numbers. In fact, this difference for the recall vote shows the best comparison to Benford’s law with S=0.1.
But there is an additional reason for doing this. If you want to “simulate” tampering with the data and you calculate (NO-Si) at each machine, then it is very easy to “transfer back” No votes to the Si votes and measure chi^2 as a function of this “reversion” of the votes. Mikoss tested this, “reverting” votes by equal percentages in all machines and obtains the following graph:
Chi squared of the comparison between Benford’s law for the difference (No-Si) as a function of the percentage of No votes “reverted” to the Si
The suggestion is a) the fits is much better if you shift votes from No to Si, with a very well defined minimum in which chi^2 goes down sharply by two orders of magnitude, corresponding to about 18% of the votes being shifted from No to Si. b) The work of Mikoss shows that you can use such testing to test for this reversion. C) There are suggestions that this was done given that the work assumes all machines were altered, which would seem surprising.
For completeness, below are the results for the second digit of Arias and Chavez in 2000 as well as the Si and the No in the recall referendum, which have also been studied by Pericchi and posted here before:
Frequency second digit Arias 2000 Second Digit Chavez 2000
Second digit Si vote RR Second digit No vote RR
The Devil's Excrement 
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